THE BOARD OF GRADUATE STUDIES 
APPROVED THIS DISSERTATION 
fOR THE Fa. I. DEGREE ON 27 JUl 1967 
SOME ASPECTS 
OF 
MAGNETOHYDRODYNAMICS 
by 
J.C.R.HUNT1 
Trinity College o 
A dissertation submitted for the 
Ph.D. degree at the Universi ty of 
Cambridgeo 
January 1967 o 
lu"lv~I 
I ,I kARV I ~GE 
Surrnnary 
This thesis is an account of vari rn}s phenomena. caused by the 
interaction of the motio!J. of electrica l ly conchlt::U.nc:;; fJ11.ids with magnetic 
fields. Such phenomena, t,he study of virh::l.ch i s, usuaUy known a.s 
Magnetohydrodynamics (MHD), occur on .";, galactic, planetary or laboratory 
length scale; however in th:Ls t hesis we concentrate on those phenomena. 
which can be reprodut1)d in the laborator y o 
In chapter 2 we study the laminar flow of uniformly conducting, 
incompressible fluids in rectangular ducts under the action of transverse 
magnetic fields. We ~gin by proving that when the duct has a constant 
cross-section the solution is tmique and then : analyse theoretically 
some o.f the curious effects on the flow of the duct 'S walls being 
electric ally conducting. We find close a greement between the results. 
of these theories and the experiments of Alty (1966) and Baylis (1966). 
We then analyse the flow in ducts wi t.h varying cross-sections. 
In chapter 3 we analyse some of t he curious ±"lows and current stream-
line patterns produced by placing electr od es on the non-conducting walls of 
a. container.,f.i.lled ta th a conducting flu i d1 and passing el~ctr.i.c currents 
between the electrodes in the presence of a s t rong :rm.gnetic field. 
In chapter 4 we analyse some of the theoretical limitations O!J. the 
use oi' Pi tot tubes and electric potential (eopo) probes in MHD flows, and 
provide some estimates of the errors t o be eA-pected. 
In chapter 5 we analyse the stability of parallel flows in parallel 
magnetic fields and also some aspects of the stability of the f101t1S 
analysed in chapters 2 and 3o 
In chapters 6, 7 and 8 we describe our experimental apparatus, 
the experiments to i nve s tip:a te directly some of the flows. analysed , 
theoretically in cha,pters 2 and 3 by me ans of Pitot and e.p. pr,obes, and , 
experime nts to measure the MHD errors inher ent in the use of these probes. 
We concluded that the curious phenomena pre dj_cted 9ctually exist. We 1',lso 
l~arnt much ru:iout the use of Pitot and e .,p o probes, especially as some of 
the experimental re~ilts were as predicted in chapter 4. 
.. ·······~··--
CONTENTS 
Content.s 
Index of figures 
Index of t. '3.bJ.es 
Preface 
1 Introdu ct,i on. 
1.1. What Ma.g-.r1etohydrodynamic s is all a.bout. 
The aspects of :MHD considered here. 
2 Magnetohydrodynamic flow in . r ectangul ,3.l" dncts 
2.1. Introduction. to ch?,pter 2. 
2.1.1. Aims. 
2.1.2. Rea.sons. 
2.1.3. Contents . 
2.2. Eqnatj_ons and boundary conditions for 
incornpress ible magnetohydrodynamie flo~ts. 
2.3. 
2. 5. 
2~2.1. The governil').g equa tions. 
2. 2.2. Approx:tms.tions made . in the equations. 
2, 2 .3. Boimqa!"'J Gondi ti ons. 
T'ne equations, b ound a.r y conditions, and uniqueness 
t heorem for MHD duct flows. 
2.3.1. 
2. 3.2. 
2, 3.3. 
Formuhtion of the problem. 
Uniqueness theorem ~ 
The MHD duct f l ow eqnat.ions. 
Magnet ohydrod;ynamic flow in r ect r.tngula.r ducts I. 
2.. 1.i.. .1. Introduct i on t o Htmt ( 1965) 
2 . 1, .• 2. The stability of flows ·,flhen M>> 1. 
2 .1; .• 3. Exper ir:1.enfa l r e sults of Alt;r (196.6 ) ~ 
}fagnetohydrodynamic flows in rectangular ducts II. 
2. 5.1. 
2. 5.2. 
2. 5 .3. 
Introd:u0ti on. 
Compari son of all t.ypes of duct flow. 
Compari.son vrl.th other s I results. 
-i-
i. 
v . 
vii:1,. 
ix. 
1. 
2. 
5. 
5. 
6. 
7. 
10. 
10. 
11. 
11. 
13. 
13. 
14. 
17. 
20. 
20. 
21. 
23. 
24. 
24. 
24. 
27. 
~ . 
FloH in Curved Che..nnels. 
2. 6. 1. 
2.6.2. 
2.6.3. 
Equa tions for cyUndric .tl fl ,J'vS. 
Flow in a. rectc=:.ngulttr annulus o 
Experimenta.l results of Baylis ( 1966). 
28. 
28. 
29. 
32. 
2.7. Nagnetohydrodyna.mic fl.01,·1 in channel s of ,m,.r:'5. a1) le 
cross~section 111. th strong transverse magnetic fi€lds. 34. 
2.1.1. 
2.7.2. 
2.7.3. 
2.?.6. 
2.7.7. 
Int roduc ti on. 
State1ren.t \'.f the problan. 
Gor·e f' l ow. 
The Tudford layer e qu:Jt:i. ons. 
Ifa.rtIB,;1.rm 1,cnmdary layers., 
E...xaJnple : flow through stre.ight w~J.led 
converging and di verging ducts. 
Discussion. 
31, .• 
36. 
39. 
41. 
45. 
3. Soml3 e lectrica.J!~ driven f lows in ma.gnetohy"drodynrunics. 
3.1. Introduction 8nd ~m.mm.ary. 53. 
?r.2. Two..,.dimensional electrode confiJ?Urations. 56. 
3.2. ·1. 
3.2 .. 2. 
The equations. 
A.lined line electrodes. 
D:i.splaced line ele ctrcdes. 
Circulcir elec~rodes. 
3.3. L Introduction. 
3.3.2. Alined point olectrod~c')S. 
l\.sym:ptctic arn:>J .. ysi.s c:f. a.lined. circulci.r 
56. 
57. 
61. 
63. 
63. 
63. 
66. 
ll-, On the use of nit.0t -t,ub,Js and electric potential probes in 
MHD flows. 
4. 1 • . Introduction and suminary. 
4.2. Pitot tubes. 
4. 2, 1. 
4.2.3. 
4.2.4. 
Non-dimensional equations, bound~ry 
conditions. 
'I'h.e generc.l probe1m and some special 01.'l.sesr 
A ph7sical discussion of Pi.tot tube errors. 
Conclu.sion. 
75. 
76. 
76. 
7S. 
8·1 •• 
a5. 
}_; .• 3. 3. 
5. ·1. 1 • 
.5 .1~ 2 . 
,- 1 , .. 
:,. I I j . 
fi elds . 
6, Ex.;.k::: riinent.a.l a ..ppar atu:s. 
... ... <'J 
:.) • I • I • 
6. 1. 2. 
6. L. '1h?- mechan:~s t, i:'~1r moving the P!''.) bes~ 
6. 5, 'l'he :flow ci rcri.it and its componm t s " 
6. 5.10 
6. 5 .:z. 
6. 5 . 3. 
Fump . 
Th,_,, flow ci rcuit. , 
6. 6. Instrm .. ,ients. 
6. 6. 1 • 
6,6.2 . Potenti ometer. 
7. Exper:i .. ments en eJp ct:cica.117 d:dven flm·rs , 
? .2 . 1. The duct mul 1ts e.l ectrodss 
7. 2.2 . Proba r.eteefJE..n'l.cm.~ 
7.2.J. Bl octrj_c pot entia l rrccbes. 
Pi. t:c ·t. tr..1 be: s " 
B6. 
g6. 
8'7 . 
89. 
90, 
90, 
90. 
91. 
92. 
93, 
99 . 
99. 
99. 
101. 
101. 
104. 
106. 
1 ('I'' ,;0 , 
107, 
108. 
11 O, 
110. 
111. 
112. 
113 . 
113. 
1 '1!~ . 
115. 
.. "' 
. 
~ 
........___ 
Experimental resultso 
7 .3 ,_1. 
7<'3.2. 
7.3 .3. 
Electric potential measurements . 
P:itot tub e .measurements. 
D:is cus sion . 
8. Experiments on flows in r ectangular dt.H,ts,. 
9. 
80 1. Introduction. 
8.1.1. 
8.1.2. 
Aims . 
Prev.i. ou s work . 
Et 1. 3 0 Summary • 1 .~ 
g1' x 2i'·' duct; non,=conducti:ng wal ls . 
8.2 ~1. Appar atus. 
8.2,2 . 
8 2. 3. 
Sta.ti.c pre ssure mea:suremer1t s. 
Pitot tube me asure:merrts. 
8.3. ·6 11 x 3n du ct~ non~conducti ng -i,,rd.lls. 
8.J . 1. 
8q3.3o 
8~3.4. 
8~1.i..1. 
8.4,, 2. 
8.4.3. 
Conclusionc 
References. 
Tables. 
Apparatus 
Stat i c pr e ssu.r~ measurernents. 
Pit ot tube meas-:.:i.l'e:mente ~ 
Electric pot ential measu.rements c 
condu ct.ing and non=eondurc:t::l ,·i,.i,,.. walls" 
(J 
Apparatus . 
Static press,.1re measccrements. 
Pitot t ube mea&·ux'en:ent s ·~ 
Electric pot ential measurements o 
Discu ssion. 
=iv-
119. 
119 0 
126,, 
129. 
129,, 
1:29. 
1350 
138. 
140 .. 
i4,L 
i43o 
1450 
1J+5o 
150 0 
1520 
1.57 ~ 
I.ridex o.f figllres. 
2.1. Cross-section of cin arbitarily shaped duct ,..tl. th 
external electrical connections. 13. 
2.2. 'I'he potential at the mid-point of one insulating 
wall agidnst Reynold's number. 23. 
2.3. Pressure gradient against Reynold 1s number 23. 
2.L... Potential profile a.long the insulatj,ng wa.11 . 23. 
2.5. Cross-section of the duct when dA=O, dB=oO.(M>">1). 25. 
2.6. Cross-sectior1 of annul~r rectangular cha..'1Tlel. 28. 
2.7. Flow regior;is, core (C), boundary layers (B) and Ludford 
l~yers (1), di. scussed in §2.? .2. 37. 
2.8. Illustra.tion of the 1:asic problem for the Ludford 
layer. 37. 
2. 9. Vi.riai:.:ton of 1..r* and / ... Y-\J""* 1·rl.th x· in the Lua.ford 
layer, at various values of Y. 45. 
2.10. Notation for the boundary layer analysis in §2.7. 5. 1~6. 
2.11. Velocity distributions in t.he core of div1;,rging and 
converging flows in straight walled ducts. 46,49. 
2.12. Veloc;i. ty profiles qf U80 and U81 in the boundary 
. 0 
layer ~here x = ±a, Ql= ± 45 • 1+9. 
3.1. .!\lined line electrodes. 56. 
3.2. Asymptotic regions between the electrodes when M.>~· 1. 56. 
3.3. Alined :tine.eiectrodes: velocit:;r profiles in region 
(1) at't= .5, 1.0. 
3.1.i.. Current stream Jines between the line electrodes when 
M,.') 1. 
3. 5 •. 
3.6. 
3.7. 
3 .s. 
3.9. 
Displaced line electrodes showing asymptotic regions. 
Velocity profiles and current stream lines for flow 
between displaced Jine eJ.ectrodes when M>'> 1. 
Notation for a.lined point electrodes. 
Alined circular electrodes wi\h asymptotic regions. 
Graph of ~UJ 2t'2 against ·r 1142 at ) = 1 .o . l; !l 1-/2./ ,,, 
-v-
59. 
60. 
6o. 
62. 
63. 
63. 
71. 
6,, 1 .. 
6.2,, 
6 .. 3. 
6.4 .. 
6.5. 
6.6. 
7. 1. 
7.2. 
7.3. 
Notation for Pi.tot tube or electric potential probeo 
Error...;free Pi.tot Stati .e tube . 
Current stream lines found P:i.tot tubes. 
Notation for ar,alysis of §5.2. 
I-B
0
, curve of electro=ma.gn.et calibration 
Main, 66 n long du et ~ 
Probe mechanism. 
Flowmeter caJibration rigo 
Mark II .f1 owmet er. 
Flow circuit 
Apparatus for examining ~ c:tricall;r . drl ~ren flows. 
Pi.tot and electric potentia+ prob~s • 
Potential between the discs v ._::V; 9 agc:i.inst current~ r~ 
1010 
1020 
10ho 
107 0 
mg. 
1090 
at various M,, n 9o 
I 7. 4,, 1) against I for tit.JO Vc1lues of Mo 120. 
7 .. 5. Variation of resistance between the elect1°odes with Mo 121" 
7 e6. Variation 9f potential along th.e centre lines o:f the 
electrodes ~ vihen M ':;:~ 1 run M = 0 o i 22 o 
7. 7 Q Graphs of potential against radius vihen M ,::., 0 o 122 o 
7.f3. Variation of potential with r:~cti us in regions (1\ 
when M , ';,- 1. 123. 
7.9 • . Variation of Pi.tot pressure with eu.rrent -,.hen M.;:.> 10 124" 
7 . 10. 'I'he variation of velocity with ra.di us in ( i) o 125 o 
7 .. 11 . Comparison of results of Pi tot md electrie potentia,l 
probes for velocity . 
·1 'l 8'' x 2i" duct (I ). 
Pi.tot tubes used in duct I. 
8 .. 30 Pressure drop near the end o.f the duct against the 
flow rate. 
=v.i..= 
126. 
B.l+. St~, t i c . pressure along the !n x 2i 11 duct when M = 0, 
M = J+6,. 
8,5. Variation o.f' pre ssure drop .wlth flow rate near the 
end of fu e duct 1r,hen lb},. 1 • 
Bf 6. Graph of Pitot pressure against (flow ratei when 
M = O; 
Measur. ed velocit y profiles in i11 x 2i11 duct when 
M = o. 
S.S. Graph showing negative pressure registered by Pitot 
when M ?>· 1. 
8.9. Measured vel ocity profiles ·when M = 2~36, 5.03 -
circular pitot tube. 
B~ 10Q Measured v elocity pro.file i\ihen M = 2 ,36 - flattened 
pitot tube. 
137. 
137. 
138. 
138. 
139. 
139. 
139. 
g.11. ,,6 11 x 3. 01 11 .duct (II). 11+0. 
S.12. Pitot and e.p .. probes used in duct II" 11+1. 
B.13. Variati9n of . ntot tube error with flow rate 1'hen 
M = 137, 253. 142. 
B, 11~. Universal plot of electric potential near x=b men 
M >i> 1. 1 h3. 
8.15. Var iatiot1 in potential drop in boundary layer at x=b 
· r-1 
when M ~ 11+1+. 
s.16.. Potential prof:i.les in boundary la,yer at x=by 
M.=.75~ 136. 144. 
8.17 .. E.,p. probe for 2 .. 5° x 3" duct (III). 11t-5. 
S1; 13. 
8.19. 
8.20 .. 
s.21. 
Static pressure drop in 2! 11 x 311 duct; M e::: 91+3. H-5. 
Velocity profiles in the botmda.i.7 layer at x='o, M=91+3 .. 1Li.6. 
Variation of potentia l with flow rate in 2! 11 x 3''du.ot ... 11+7. 
Potential profile in the boundary layer at :x.=b; M=91+3. 11t-7. 
8.2. 
s.3. 
Basic data for 'it' x 2ir1 duct, (I). 
Static pressure drop in ~~e t ~. M = O. 
Pi.tot tube readings in duct I, ·when E :::= O~ circula,r tip. 
a.1+. Pitot tube read:-i.ngs for duct I 1\hen H = o~ flattened 
tip. 
a. 5, 
8.6. 
8.7. 
a.a. 
a. 9, 
8.10. 
8.11 • 
8.12. 
Basic data for .6n x 3" duct. (II). 
Static pressure drop i n duct. II 1,vhen M ~, 135. 
Pi.tot tube errors in duct II, (circular tips), 
Pi.tot tube errors in duct II, (flattened tips P af'-'.'.bp). 
Pitot tube errors in duct IIP (flattened tips 9 aff"">bp). 
Electric potential measurements in core fll'.JW' of duct IL 
Basic data. for 2!n x 311 duct,. (:j:II). 
Pi tot tu be readings in du. et III P M = 91+3 ~ 
=viii-
157 
159 
159 
160 
, 61 
163 
163 
163 
The work described in this thesis be gem a t the DepartJ11.e nt 9f 
Engineering of C:;:1.mbridge University during t he ac;,.demic •re ar 1963-4 9 
w1:1,s mainly done at t.ht1 Sch.ool of Enginetn-tj_ ng Science of the Univ·ersi ty 
of Warwick from September 1961+ to December 1966. During this time I 
have bee.n given a generous salary by the f:entr al Electricity Generating 
Board. The Board als o ena,1-:, led me t. ,:i :r:1!.tke two visits to Franee and one 
to Salzburg in order tq attend .conferences a.nd meet r..ther r es~arch 
worlcers, Fu.rth ermore, by means of ft contra.et. 1,rl. t h the Univ-ersity of 
Waml!:i.ck, the Board helped to1rm.rds the costs o:r. thr1 experimental work 
a.t Coventry, I should li~e to ~:h'Press my ·~hanks to tho Board for this 
support and, in pa.rtimlar, to liir4II .J.L:rwe, Mr .L.Yourrta.nd Mr.D.J.W. 
Richards of the Central Electricity Research Laboratories at Lea.therhead 
for the ,ht;llp they have given me and for the interest th0Jr have shown in 
my 'vi.iork. I should also like to record 1WJ thanks to Professor J .A. 
ShercJif'f for bdr-_;g su ch art excellent supertrisqr in providing Judicious 
qua.nt:i.ties of both encouragement and criticismi · rmd to Dr.H.LMoffatt 
and Professor K.Stewartson . for rra11y interesting di scussil"l11s on the 
theoretical aspects of ~ITTD. As r egards the experiments :1 I must first 
of ~11 th ank I,h•.A.E. Webb for his e.xcellemt work in t he con~truc-t;ion. of 
the apparatus, his pa-~ience with the endless modificatL,ns j) and 9 • perhaps 
most important of all, his advice i n the de sign of th~ :9.pparatusc ('I'he 
design of the 66 11 duct was as much his M ntine). I should also like to 
thank Dr.C.J.N.Alty for sharing ·with us the b~nef'it of his experience 
with MHD duct flotrrn. Mr.E.P .Sutton and the Ca.nibridge Universi ty 
Aerona.utica.l Department ad,ris ed us in the. design of sma.D. Pi tot m bes j 
for whi.ch we are ve1::sr grateful. . Fi.nctlly, I must tharJc my wife for 
doing most of th e drrndngs and filling in thei mathematics between th e 
typing. 
The work descr.:i. bed i:n this thesis . is original except 1,mere 
references are gl ven to the work of otre rs. In particu1a r, the intro...-
ductory chapters 1 and 2 are not original except f or §2, 3.1. and 2.3.2. 
all the subsequent chapters being substanti l!l.lly originaL 
None of the the sis has b ean Sl'l. b:mi t.; !;ed at any other university 9 
although somi:'! parts lw.ve been published or have i-ie·"n submitted for 
1 . t. . . ,,l' • .!!>. • 1 " h ... 2 .., 5 d ,.., pub ica ion in sc1.enDl.1J_c JOD.rna. _s. ,.,,_ apuer:s , :; 9 &'1, i were 
subrni tted as a Fellowship dissertation at Tr:i.n1 t y College, Crunbridge. 
JlR.U--
1 
S"" . t' , ( ,1 . 
a~tr ~ IJO-lWStP. 
<=>.X-
1 o Introductio11." 
10 1 e l,..Jhat rnagnetohydrodypam.ic:5 is all about. 
Thi:s thesis is an examination into various aspects of t he dynamic~ 
of electrically conducting fluids under the action of ma.gnetie fields 9 
~ subject usually known a.5 Magnetohydrodynamic ~ (MHD).. Before 
describing the new w::irk presented in this the~i3 we first bl'.'ief1y discus~ 
the general principles of the subject and then mention a few of the 
reasons for its study. 
MHD is a combination of the two subjects 9 classical electromagnetimls> 
Le. the study of the interaction of electric currents , electric fields 9 
and magnetic: fields ll and fluid mechanics II Le. the study of the interaction 
of f orces and motion of liquids and gasesll the principles of which ar,e 
basically Newtoniano Normally9 the motion of a fluid is examined without 
considering any electromgnetic effects; however 1 when the fluid is 
electrically conducting and in the presence of electric and magnetic 
fields j the mechanical and electromagnetic effects become interdependent 
because the electric currents are changed by 'th e fluid 0s moti.cn through 
t he magnetic field and the forces on the fluid are altered by the electri c 
currents and magnetic fieldso The aim of MHD is to S1tudy t he phenomena 
created by these combined effects .. The method of the study is similar 
to those of fluid and solid mechanics, in that the phenomena are examined 
with a view t,o explaining thEin in terms of tr.ie basic prlneiples of t he 
su.bject 9 namely class:lcal eleetromagr1etism and fluid mechanics~ a:nd the 
physical properti es of the fluid, e .. g .. its electrical conductivity and 
viscosity, the phenomena. are not studied, as in plasma . physics 9 with a 
view to e:xplaining th em in terms of atoms and eleetrons o 
There are three main classes of reason for studying MHD, the first 
being that M"HD i s a c omparatively unexplored and intrinsically 
fascinating branch of classical physics and applied mathematics,, the 
second being t hat }lHD effects are often believed t o exert a controlling 
influence on many geo= and astro- physical phenomena 9 a:rrl t.he third being 
that many MHD phenomena can be used for practical purposes eog o measuring 
the flow of liquids and accelerating9 controlling, or generating 
electrical power frorn 9 streams of high tern erature gases o 
I f-P -
-1- j LltlKAKY 
1.2 0 The aspects of MHD considered here. 
In this thesis we concentrate on those MHD phenomena which can be 
reproduced in the l aboratory as opposed to those phenomena which can 
only occur with very large magnetic fields or on planetary length s~ales. 
Also we con.fined ourselves to examining situations which have somE': 
similarity with those found in practical MHD devicesl' though the 
conclusions we draw from these studies ha~,e some considerably 'Wider 
significance • 'Toe third limitation of our study was that we only 
consider situations which are simple enough to be analysed 
theoretically as well as experimentally. 
Since the four .nain aspects of MHD examined in this thesis form 
a rather motley collectio1J a w:>rd of introduction is necessary to 
explain . the actual reaso!1$ for choosing to study these particular 
aspects.. (Fo:rmal introductions to each aspect of the thesis 'With a 
review of the previous work and a summary of the main results are 
given at the be gfrmning of each chapter) o 
At the time of my joini..YJ.g the Carnbr1.dge University EngirH::!6:t>ing 
Laboratory MHD group under Professor J .,A.Shereliff in October 1963)} 
Messrs. C.,J .N.Alty an.d J .A.Baylis were engaged in experiments to 
investigate the l\1HD flow of mercury through rectangular ducts of 
constant eross=section whose walls are electrically conductingo At 
the time the theory of' such flmrs was ve:ry incomplete 9 but most of 
all we lacked a physieal insight into the processes involvedo 
Consequentlyi> with the encouragemmt of Pro±'erssor Sher©lif'i~ I began 
work on extending the existing thoo :ry of such i"lows ·which had been 
developed by Chang & Lundgren (1961) and Shercliff (1953\ concentrating 
on the interesting physical effects whi«::h occur when the mttgnetirc 
field strongly affects the flOW'., As a result of this U1eoretical work 
it became clear that the presence of electrically conducting walls 
radically alters the flow ·when the magnetic field is strongo This 
conclusion was justified b;f the experiments results of Alty ( 1965) 
and Baylis (1966) 1t-.ihose results agreed well wi t..11 the theory developed 
by myself (Hunt 9 i 965) and that developed in collaboration w'ith 
Professor K.Stewartson (Hunt & Stevra.rtsonj 1965) o See also chapter 2o 
The other interesting result of this work was that it showed that 
in some circumstances a magnetic field can make flow i:n a duct less 
stable$ contrary to all previous evider,iceo This concluron 
stimulated my interest in the problem of how a :rnagnet{~~ffects the 
stabilit y of a flow and some results of this study have been published 
(Hunt, 1966a) and others are mentioned in chapter 5. 
Further work on MHD duct flows has been concentrated on the more 
important practical problem of flow in ducts wiose cross-sections vary 
along their length (§2. 7) . This work has recently been extended to 
the study of compressible flows (Hunt, 1966b). 
In parallel with the work on MHD duct flow, at the instigation of 
Professor Shercliff ~ I studied the theory of some of the interesting 
effects found in electrically driven flows in MHD Le. those caused by 
current sources and si nks being placed round the boundary of a fluid 
placed in a strong magnetic field.. (This problem may not be altogether 
academic s.ince the walls of an MHD generator are divided into conducting 
and non=conducting strips lrhieh are like some of the situations 
considered i n the analysis of chapter 3). 
Since at the University of Warwick it was possible to obtain an 
electromagnet with a 3n gap, and therefore to have a duct wi. th an internal 
dimensicm of 2! 11 9 I decided to concentrate on investigating some :MHD 
flows internall:v 9 by using pressure and electric potential probes rather 
than investigating such flows by external measurements as performed by 
Alty and Baylis. Although some pioneering wo:ru: on the use of such 
probes had been be gun by East (1964), Lecocq (1964), and Moreau (1965), 
we have been involved i n a consid,erable amount of trial and error in 
the design of a suitable duct and apparatus for moving the probes and 
there is still a lot more development "Which needs doing .. We also 
extended the t hooiry of the measurement of MHD flows by pressure and 
electric potential probes (chapter 4). 
The experiments which have been performed have confirmed directly 
many of the phenomena predicted in the theory of MHD duct flows and 
electrically driven flows, as well as indicating the kind of errors to 
be expected in the use of pressure and electric potential probes,, 
-3-
Althougtl these internal measurements have not been anything like as 
accurate as the external ones normally ma.de~ e.,.go static pressure amd 
electrie potential on the boundaries, they show: that such measurements 
can certainly indicate the nature of the flow quite satisfactorily 
when no theoretical model exists and therefore should be of use in 
studying turbulent flows. 
=1+= 
2 . fi!tj ~~£~ o kJ dro '*a..,( o. wtt-~ J low 4,, 1:?ib7*1'1, ""[ rk cft-
2.10 Introduction to chapter 2. 
2.2 .. 1 .. Aims. 
In this chapter we e.:,camine the theory of the flow in rectangular 
ducts of electric1;].ly conducting fluids up.der the action of a trams-
verse magnetic field, confining ourselves to the study of steady 9 
laminar flow of incompressible fluids whose conductivity, viscosity, 
and density are asstnned to be constant. There are two main aims of 
the study. 
The first aim is to examine how the flow through a rectangular 
duct is affected by the electrical oonductivity of the walls of the 
duct and the external electrical connections made to them. To do 
this we make oo:rre further simpµ.fying restrictions to our study; we 
only study flows mich are fully developed, that is· to say the velocity 
in the duct does not var.J in the strea.mwise direction; we only consider 
uniform magnetic field which are pe~pendicular to two of the walls of 
the duct, and we only consider ducts whose walls have uniform conduct-
ivity, though the conductivity of different walls ~Y, vary. Although 
flows in rectangular duets, subject to the same restrictions, have 
been studied before the only type of duct studied at all completely is 
that with non".'"conducting walls, the .wo:rn: on flows in ducts vd th con= 
ducting walls being very incomplete. For this type of duct there has 
been no attempt to understand tne p4ysical implications of the 
mathematics nor to compare the solutiorsfor various types of ductso 
Our aim is to use the mathematical solutions to the problem in order to 
obtain a sound physical µnderstanding of the flOW" in different types of 
duct with various electrical connections, arnd to obtain useful formulae 
for volume flow rate through f1.. duct, electric potential difference 
across the duct etc, which can reacifly be tested e.xperimentallyo 
The second aim is to examine the flow in ducts whose cross= 
section varies in the streamwise direction, men the transverse 
magnetic field is very strong. In th~s case the restrictions we make 
are~ only the dimension of the duct in the direction of the magnetic 
field varies, we can igno:r,-e the variation. of the flow in the 
direqtion perpendicular to the streamwise and magnetic field 
-5-
directions, the mgnetic Reynolds nurnber1 Rm~ is low enough for induced 
magnetic fields to be ignored, and the magnetic field is unifonn. These 
approx::in:ations aRd restrictions lead to a great simplification i.n the 
otherwise very complicated problem of calculating the flow over a body 
olaced in a transverse magnetic field. We find that these same 
~pprnximations also lead to great simplifcations in colculating .internal 
f'lows and our aim is to use arrl develop the existing mathematical 
solutions to provide a physical understanding of this problem and to 
provide formulae which can be tested in the laboratory. 
2. 1 o 2. ReasoIJS~ 
There are two main reasons for studying the incompressible flow 
of fluids wi. th uniform properties through rectangular duets under the 
action of a tmifarm transverse magnetic field (MHD duct flc.w for short) .. 
Firstly there are p:ractica l reasons. The first practical use of 
MHD was in flow measu.rernent and this affected most of the original -work 
on MHD duct f low9 as revief by Shercliff (-1962)0 To find the flow 
rate through a duet, men a transverse magnetic field is applied 9 the 
voltage between two electrodes in the walls of t.he duct is measured by 
draw5.ng a very small cu rrent through the electrodes which has a 
negligible effect on the .flow or the currmt in the duet. Most of' 
such measurement:.s a re made in ducts wiose walls have low or zero 
conductivity in order that the yoltage should not be short circuited. 
Therefore much of the early work was on ducts ,..fuose walls were of low 
or zero conduetivi't.,v and on flows 1110t affected by ext.ernal electrical 
However 9 with the gr01.,_1i ng interest in 
using magnetohydrodynarn.ic means to pump liquid metals 9 accelerate ion.i.zed 
gases and generate electricity from moving streams of ionized gas~ it is 
now important to study the interaction of duct flows wt th external 
circuits when appreciable electric currents circulate between thano Also 
in order to minimise the electrical losses 9 the ducts must have highly 
conducting walls and the eff'ects of such walls should also be studiedo 
The analysis of the flow in these applications is extremely eomplfoated 1 
sine e the flows are usually turbulent arrl the fluids highly no?'l=Unifonn 1 
as well as being compressible in most cases. In order to make ar1y 
progress in our 1.:mderstanding9 various simplications are necessary ~ 
for example the flow velocity is assumed to be uniform across the duct 
with laminar boundary layers fanned on the wal~ (Kemt:.ibrock 1961 9 Hale 
and Kemebrock 196i,) 9 or to be inviscid and two=dimensional (Sutton and. 
carlsor1 1961 \ or the approximations of one=dimensional gas dynamicEi 
are used (Resler and Sears 1958)0 'Ihe simplification considered :in 
this dissertation that the flow is laminar an::i incompressible and tha t 
the f1uids n properties are unifarm is merely one amongst man.yo Each 
of the simpB.f'ications enable certain as:pects of the flow to be studied 
and taken together an understanding of the overall process may errBrge. 
The aspect of the flow which our simplif'ication chiefly reveals is the 
interaction of the electro=magnetic and viscous forces in the boundary 
layers on the wq]ls 9 an effect which becomes increasingly important as 
the size of the de·vice and the strength of the magnetic fiekl are 
inic:reasedo 
The se{';ol'ld reason f'or our exanrurl:t:ig MHD duct flows is that they 
are one of the few instances in MHD whereby the theory ma.y be critically 
tested by experiment. Most theoretical and experimenta.l ·work 4:e-
eer1;:eem.:r-1R>ea in MRD is concentrated 1i~compressible flows because of the 
greater pra.ctic:al and astrophysical interest in S!.t.eh flovm 9 and in s'trnh 
flows it is ·irery rare indeed fm· the ex.1)6riments to be accurate or 
repeatable enough frr.r the theory to be tested at all critically. On 
the other hand 9 using liquid meta.ls experimentalists have been able to 
achieve accuracies cf 1 or 2%9 and S'J. Ch exi.:er0 iJntmts provide r.eal tests 
for the theory o Sin~ the wh<',le st:r"'ur;ture of MHD theory must be judged 
by t he accuracy with which its predic)'.'tions a g:r·ee with those of 
e:x:periment 9 the theoretical and experimental study of MHD duct flows is 
@rilt;ially necessary for the further understanding of :MHD. 
2.1.30 Con~ 
82.,2. We state the equations ar.d boundary conditions of MHD for 
the incompressi.ble flow of fJuids with uniform properties 9 making some 
observations on- the appro.xi:rre.tions in the equationso 
§~.3o We present a novel derivation of the equations and boundary 
conditions for f ully developed flow in du~ts of comt,ant cross=sectional 
=7= 
area. mid er the action of a uniform magnetic field, first fonnulated by 
snercliff ( 1953). The essential feature of this derivation is a 
uniqueness theorem (a generalisation of one deduced by Moffatt (1964) 
for electrically driven flows), which p,:-oves conclusively that the 
assumptions made in deducing Shercliff 's equations are justified. 
§2.,4. We analyse the fully developed new in rectangular ducts 
under the action of a transverse magnetic field, concentrating on the 
effects of th.e duct having electrically conducting walls. To do this 
we gmeralise the mathematical solutions of Chang ~d Lundgren (1961) 
and Uflyand (1961), and Shercliff (1953), t,o cover flow in tw:> main 
types of duct: 68 
(i) The walLs perpendicular to the .field ~) perfectly 
conducting and the walls pg.rallel to the field .,. ~ 
of arbitrary conductivity. 6.8 
(ii) Walls AA non ... conducting and walls a of arbitrary 
conductivity. 
We then concentrate on the flow when the Hartmann number, M, = B
0
a(cr /i.} 
is Jarge, where B p is the imposed magnetic field, a half the duct width l> 
~ the conductivity ap.d VZ the viscosity of the fluid., Various 
interesting pqysi cal effects are f oun~ the boumda.ry layers on the 
walls parallel to the magnetic field as the conductivities of the 
walls are altered. The most interesting and unexpE:cted effect occurs 
when the walls of the duet perpendicuJar to the magnetic field are highly 
conducting and the walls pg.rallel to the rragnetic field are non-
conducting; then when M»1, large positive and negative~locities of 
order 1-Wc are irrluced in the bouncfa.ry layers on the wall~, where V
0 
is 
the uniform veloc;ty in the centre of tl'j.e duct, usually known as the core 
velocity. It is therefore like),y that, :i,n contrast to all previous 
evidence, the magnetic field may in some situations have a destabilizing 
effect on now in ducts. (R.atiier than copy out the author 1s paper 
(Hunt, 1965), we :refer the reader to the paper which is attached to this 
thesis). 
Finally in thi~ sect;ion we show that the experimental results of 
Alty ( 1966) agree rel'J'\B,rkably closely with the theoretical predicted 
va1ues and also that these results conclusively bear out the hypothesis 
-8-
that the ne.gnetic field destablizes the flowo 
~2 .. 5. We continue the analysis of the effects of conducting walls, 
this time investigating the flow i1:1 a duct whose walls A.A are non.-
conducting and walls BB are perfectly co:q.ducting. We also examine the 
effects of an external electrical circui t . F:i,nally we compare the 
flows in rectangular ducts with all combinations of conductii:ig and non-
conducting walls. . (In titj_s section we refer to the paper Hu:nt & 
Stewartson (1965)). 
We apply the results of §2.,5 to flow in a rectangular annulus 
which is driven by an electric current with an applied magnetic field 
parallel to the axis of the a.rmulus. Then we compare the theoretically 
predicted values with those follll.d by Baylis (1966) to find reasonable 
agreement between then. 
82 0 7. We analyse the steady, incompressible, two-dimensional flow 
of conducting fluids through ducts of arbitrarily varying cross=sectio:n 
when a strong, uniform, magnetic field is imposed.. The direction of 
the ma,gnetic field is perpendicular to the flow amd parallel to the 
direction in mich the ducts diverge.. It is assumed that the interaction 
parameter, N(~e)>>1, where M is the Hartmann number and Re is the 
Reynolds number,. and also that M:;),;,1 and R <..<.1 where R (=u.cr{J,e) is the 
· m m r 
magnetic Reynolds number , )A, is the :rm.gnetic permeability, er the 
conductivity, £\ ~ typical velocity and { a characteristic length o:f the 
flow. 
We examine the now in three separate regions g 
(i) The '~ore' regipn in which .the pressure gradient is balanced 
by electro-magnetic forces. 
(ii) Hartmann boundary layers where electromagnetic forces are 
balanced by vis~ous forces. 
(iii) Thin layers parallel to the magr,.et=\-c .field in which electro-
. magnetic forces, inertial forces, and the pressure gradient 
balance each other. These layers which have thickness 
O(N-V~) occur where the slope of the duct wall changes 
abruptly. 
By expanding the solution as a series m descending powers of N we 
calculate the veloc:tty distribution in regions (i) and (ii) for finite 
values of N attainable in the laboratory. 
-9-
2.2. ~guations and bound.fil:Y. conditions for incompressible magnetohydro-
dvnamic flows. 
2. 2. 1. The governing equations 
The equations governing the flow of incompressible fluids 11-r.i.th 
uniform electrical conductivity, viscosity and density have been derived 
in many text books e.g. Shercliff (1965). These equations describe the 
I 
behaviour of liquid rretals very accurately and in some circumstances may 
describ e the flow of conducting gases if their velocity is low enough. 
'!hey are:-
the momentum equation, ( {if + ([.v'))(] = -Vp + J x _@-t i v'\r, 
the equation of continuity 9 
Ohm' s Law:1 
. 
J = 
-Maxwell 's Equations, 
''° ":[ = 0 > 
cr(§-t~x~), 
v"><€:: - "?.J~/Jt, 
v.~ -- o > 
. J :, V·x H ~ - ') 
2.2.5 • 
2.2.6 . 
1.fuere C is the density, ~ the velocity ll p the p:resst!t'e 9 ~ the 
vi.seosity, j the current density, e the :rragnetic flux density, -c:r 
- -the ele<.!trlcaJ. condu0tivi ty 1 e the . eleet:ric field strenght 9 and t-,t 
- ~ the magnetic field . Since we will only be c onsidering materials 9 
whose permeability, )A--o is that of a vacuum 
(3 =AAo H 2o2.7. - ,-· ,,,.....,, 
Hmce, using (2.2.5) and (2o2.6) we have: 
v. J, =- 0 2 .. 2080 
-
Hereafter we will use the suffices x, y, z to refer to the components 
o:f vector quantities e .go V'"'.x.. ~ \f j , V"'i!' refer to the compoenents 
of \J" 0 
-
2. 2
0 
1 0 AEProximt ions made in the eguat ions o 
In these equations certain effects are igmored which we now state 
along with the co:raditions ira which these effects are truly negligible. 
For further justification of the equations see, for exa.mple1 Shercli.ff's 
book. 
(1) Compressibility. The velocities must be sufficiently low 
comps.red to the speed of somi.d b. the fiuid. 
(2) Variation in fiuid properties due to heating by electric curremts 
(3) 
(4) 
and viscous dissipation. This effect is negligible in most. 
experimental situations with liquid metals~ but it is an importa:rit 
effect with gases. 
Hall effect and 'ion slip'. These effects lltlich alter the 
relation between the electric current arrl the electric field 9 
equation (2.2.3) are appreciable in gases 9 but nay be ignored 
in liquid metals 1 unless the magnetic field is exceptionally high 
by laboratory standards. (Le. greater than 105 gauss). 
nDisplacement CuITent'. This effect, which produces a modifica-
tion i:ro. the rel~tion between current azrl magnetic flux density 
(equatio~ (2.2.6)), is only significant for very high frequen cy 
electromgnetic . oscillations and is qµ ite negligible in laboratory 
MHD experiments .. 
(5) Cha~ge ooiac:entration. . It may be shown that although charge 
c:o!'}.ce ntrations exist (i.,e. V .. E' .{0) ~ the forces on the fluid due 
-to ~ and ~ , the charge density v a.re negligible in all practical 
or experiment&J_ situations. 
2.2o3o Boundary condit ions at a rigid surfaceo 
For i'u tur e reference we state here the boundary C01'.'1di ti ons at an 
interface between a solid and i'luid with finite viscosity 9 both of which 
have fini te conductivity~ 
'J"":. 0 
[,J.~~ ]so 
[[»'~] = 0 
, since 
' 
since 
, since 
there is no slip at the wall 2.2.90 
y'., j = 0 /le- C(•, ,~ I' 2 .. 2.100 
'V 
v 'a(~ = _a~ I at. 202. 11.. 
-11-
di I A..,_ 
, si:mce 
[["~ J $ 0 
The brackets [ J refers to the ehal!'lge across the boundary and 
n. is the vector normal to the bqumdary o For detailed derivation of ,....._ ' 
these conditions see Shercliff (1965)0 Note that in steady flow, since 
, \J)( E = 0 1 we can writef: =-V<J 1 where ~ is the scalar, electric 
potential O Then (2o2o 11) becomes~ 
0 
CDhY1LCkioV\S 
to uw:,moJ GU'<I'~. 
cy Pot10V\t\CAL I ij 
..._,_---..-t-____ ----.1,._ ,,,~ ,/ ,,_._ ~ curre;nt \e.o.vij , 1 
dud: w-u,.,I I 
-%1.L Crcr,,s, ~°"' CMb]:f(lf 1~~b ~ 
_d_ud ___ )µ\tiA ___ ,wle.r~CA{ __ <le:dxicqj C6hVl(?,,G-t16]!~, 
2 3 The equations, boundary conditions anq. uniqueness theorem for O O - - -
MHD duct flowso 
2.3. 1. Formulation of the pr:oblem. 
We now consider steady flows through duets ooder the action of a 
transverse magnetic field. In this and the next tm sections we will 
con.fine ourselves to examining fiow-s in ducts of constant c.ross~sectio:nal 
area which are fully developed, that is to say the velocity, the rmgnetic 
field, the cJ.ect;cic field and the electric cur rent do not vary along the 
length of the duct. Thus all tile variables except pressure are 
independemt of z, (see fig. 2.1). (We show subsequently that :'Lt follows 
dp/dz is also independent of z). 
The duct may be connected to external electric circuits. 
if the current leaving the duct per unit length and the electric 
potential at each of these connections are IJ and ij respectively 9 
we assume that the external circuits uniquely define ~j in terms of 'Ij 
or vie~ versa; a condition satisfied in all practical circuitso Now9 
given the flux density of the imposed :rragr:i.etic f:i. eld l' we want to fi~d 
the distribution of velocity; pressure, electri c potential and electric 
current in the duct given the following data: 
(1) eit~ the mean pressure drop per unit lel!'lgth in the Z= 
direction LJ.r 9 (we subsequently show that the pressure drop must 
be indep,mdent of x and y) or the volume flowrate 9 Q9 and 
(2) either -~i or ~J at each connectio:n of the duct with its 
external circuitsa 
The problem may now be expressed mathematically as follows~= 
Find ~ '=> [!:!)p)~> j ) ~ } which satisfiellftthe . equations (2.2. 1 .. = 
2.2 .. B) when d/dt = 0 gi,,';n Zir or Q and Ij or iJ 9 provided the 
following boundary conditions are satisfied~ 
(i ) at the fluid-wall interface of the duct: 
}{ = 0 } 
(j.~)f = (J~~lw 
_, - ) 
(v4><~} = ('1~ )( ~)w > 
-13-
where 'l'\ is in the direction into the fluid a:nd the subscripts f , VJ' 
,,...,,, 
indicates the value of the quantity inside the bracket at the fluid side 
and wall side of the interface respectively;; 
(ii) The boundary conditions at the exterior of the wall of the duc:t 
is~ 
~ D ) J . V) ~ """' 
except at the connections to the external electric circuitsa 
(The equations for the wall are the same as 2.2.3 - 2.2 .. g with 
u- = 0 ). 
(iii) As ) ';)c../ ) j ~ I ~ ""° J § ~ ~-Z> ( ::c, 1_:j) , 2.3.5. 
mere ~ is the imposed magnetic field. . Since ~ is produced 
by currents outside the duct, it satisfies the equations: 
v' ;,(' ;o == b 
0 
We assume 13,:i- to be given in our problem. 
By considering the energy di ssipg.ted in the duct we now prove that 
thereis a m.iique solution for 'y" • (This analysis is similar to 
that of Moffatt (1964), thou~ ~e general in th~t we consider inertial 
terms and make no restriction on \i_, the magnetic Reynolds :aumber). 
From (2.2.1) and (2 .. 2.3) we cam elimim.te B to obtain~ 
~ I /j ) ~ - ~ : . v ~ ~ ::: _.. ~ . V' I - d . v7 ~ = f '-!:. ( ( ':!: . zl) ~) 
Now" integrate this equation unit distance along the duct and across the 
duct but not including the doot walls.. Call this volume. Vf and its 
surface Sf'° Ther:i. 9 us~g Gauss I theorem an.d equations (2o2o2) and 
(2o2o8)J) we obtaing l 
Jvf "-J " d v - 1 f v / \!; 'v '"d J v = - fsf f ( ~· '.}) r + 1' Lt C'; J cJ s 
_ f i [ !!; . V ( v- '"/,_) - '.: ( ~ -s ~) J J \/ 2.3.6. 
\J;_ 
where v-.r = \J't.. If O Now 
-., 
-14-
rate, and thence (2o3o6) becomes: 
\ [ "-,. /,1,I + ~ 1~1'] d ~ • a L'lr-J:f (~ J, "c) d s 2,3. 7. 
Now consider the walls of the ducto If _the walls are non-conducting, 
{ • -:Q-,= 0 cm the walls and (2.3o7) becomes: 
Sv (o-'/l/2+~/~/'Jd'J =-~6r 
If t}1e ~lls are conducting, the potential c:f is comtinuous across the 
wall=fiuid interface andj . n is also continuouso By integrating in a ~ --I ' 
volume Vw of the wall also of unit length along the duct, with an 
external surface area of Sw1 its internal surface area being Sf' we 
obtain: 
cc fs/1> 11 · ~J) c:1 s- t L¥ cr~D Js) 
==- J ~ C.i . ~ \ d s - , ~J· 'J. SJ '"" -.J C.. ..... ) 
.J 
since J , \'\ = 0 on the external surface of the duct except at the 
..... ..._, 
connections with the external circuits~ 
Thence (2 o3o 7) becomes: 
r r l I I , I I I 
Now suppose that ! ~ l ~ f i tfa > I > J2, J is a second 
solution of the equation (2o2o1 - 20208) satisfying the same boundar y 
corrlitions as q. with 8 p1 = ,Qr or Q' = .e. and I j ' = 1 j or 
"' l ;i:; • -CJ xj = ~ j at each connection, depending on mich condition was 
specified for the solution 4 T.et 
~ , -.J t 
~ =-, -'.:t /\ 
be the di fference oetween the tm solutionso Then q, satisfies the 
equations (2.2.1 = 2.2.S) and the boundary conditions (2.3o 1o - 2.3.5.)p 
but / /'\ ,'\ J'\ 
6 p = 0 or ~ = 0 and JJ = 0 or i j = 0. 
Then (2.3.s) becomes~ 
·"\.. r·., A ""\ 
Therefore f. = ux_ = v-J = P 
and t hence, since j· = 0, p 
I • .
= 0 and since 
.I\ 
wj = o, u-z:. = O 
,,, 
= O and. ~, = o. 
Thus 9r = 0 and q/ --~ ~ , which shows that there is only one solution 
to the pr oblem. "' 
There are some irrteresting aspects of this uniqueness theorem • . 
Firstly i t is valid for all vaJnes of the ~etic Reynolds number, Rm., 
Therefor e even if the induced magnetic field is of the same order as the 
impos ed magnetic field, the result is not affected. Secondly the result 
is independent of the orientation and distribution of the imposed mgnetic 
fieldo Thirdly the result is not affected by havi:ng the walls of the 
duct conducting. Last ly we note that~ in general, specifying Q and dp/dz 
=16-
or ! j and ; does not uniquely determine the solution e.g. a current 
flowing betwee.n the wqlls perpendicular to the mgnetic field in the 
duct examined by Hunt (1965) does not affect the flow and therefore 9 of 
course, specifying lj and li j would gtve no inforna.tion about the flow. 
On the other hand in the flow examined by Hunt & Stewartson (1965) 
specifying the current and potentia~ between the electrically conducting 
wa.11s does uniquely determine Q and bf , given B~ • This case is the 
exception. 
2.3.3. 'Ihe MHD duct flow equations 
Since we have shown that there is only one solution to the problem 
of fully developed duct flow given suitab:J.e boundary conditions, if we 
assume a given flow and show that sueh a flow satisfies the equatio:rn.s and 
the boundary cond.it ions 9 then we have found the solution to the p:roblem9 
We assume that there is no secondary flow and that the imposed 
magnetic .field is constant i.,.e. 
~ : ( D, Ba) o) . 
Then 9 makµig the same assumptions as in ~2.3.1 9 that the cross=sectional 
area 9 u- ~ ~ 9 J. and f:> do not -vary in the z...clirection 9 and using ...., CJ _, 
the result of §2. 3. 2. that d p/ dz is a constant, and cip,/clx and 'ap/ay are 
functions of x and y only 9 the equations (2.2.1 - 2$2.S) reduce to~ 
2 .. 3.13., 
These equations nay be rewritten in terms of p 1 H and v : z z 
0 ~ ~h~ c pt )A o+-\..::: ) 
2. 
and 0 = f:, c)V-r a-
0 ;:i:_ 
Thus p has the fonn. deduced in §1,.3.2. Also note that, given the 
boundary conditions dp/dz or Q and Ij or J> j we can find vz and Hz 
by only considering t2.3. 17)and l2.3.1B)P which were first deduced by 
Shercliff (1953). When the walls of the duct are conducting 9 to find 
the value of Hz at the fluid-wall interface)) in addition to examining 
(20 3.17) and (2.3.18) we have to analyze the current distribution in 
the walls of the duct and use tli~ bounda...ry conditions (2 .. 3.1 = 2.3.,5) 
to natch the solution i:n the walls t.o that in the duct and to the 
external electrical circuit. We now write down the equation for Hz in 
the wall and these matching conditions in terms of H and v • Let z z 
s be the co=ordinate parallel to the walL Then, in the wall, H 
z 
satisfies: 
provided the c onducti vit,y of the wall is constant, and the matching 
conditions at the fiuid~wall interface are: 
and 
where tr w- is the conductivity of the walL 
At the outer boundary of the duct wall, 
'°aH /as = 0 z 
except where the duct wall connects with an external circuit. 
If the duct walls are non=conducting 1 the boundary conditions on 
H becomes simply 9 \ 
z ('aH:z / os J = 0 2.3., 23 " 
and the condition on dH~ /ori.. is then ignored because in the wall t h e 
electric fiel d (- arp/-;:;s) ~ __ kw- ( d H/o-n) . 
Sharcliff ( 1956) pointed o-ut that t hese boundary conditions al'ld 
the equation for Hz in t h e wall may be simplified when the thickness of 
the wall 9 t , is small compared to the duct sizeo 
Then )f-1-;a_ ) ) ~~ and (2o3o 19) becomes 
o.x os ?J-;;_ tt::z. /'cln'" -
~I' ~ .0 I 
(oil /;,").,.. =- (( H.,)., - ( Hz)J/t 
Consequ ently except where the duct wall connects with an external 
circuit 9 (H ) and (H ) being the value of H at t he outside and the 
-z W Z o Z 
inside of the td.u©t wall respectively. From (2.3o20\ 
(H ) = (H ) 
z w z f 
Therefore the condition (2.3.,21) becomes~ 
~ of':;. - t-4 ?.: iL 
- --- ·- ----
- -
·t 
20 4 0 Magneto~ydrodymamic flow in recta.n.gula:r ducts. I.. 
2.4. 10 Introduct ion to Hunt ( 1965) 
In §§2.4 and 2. 5 we merely introduce the work we have already 
published on steady fully developed now in rectangular ducts of fluids 
with uniform properties under the action of' a transverse magnet ic field. 
In this section we refer to the author 9s paper 1 Hunt (1965)~ in mich we 
generalize the mathematical solution of Chang & Lundgren (1961) and 
Sher~liff (195:3) to examine the !'low in two kinds of rectangular ducts~ 
(i) those vhose walls parallel to the nngr:ietic field (AA) are of 
arbitrary conductivity and whose walls perpendicular to the 
magnetic field (BB) are perfectly conducting, 
(:ii) those whose walls (AA) are non=eonducting and walls (BB). are of 
., 
arbitrary conductivity. In ihe paper we concentrated o:m the 
interesting physical effetts 1d1i~h occur men M>"')1 9 (ars have already 
been described in §2. 1 .3 ) and were led to make some speculation on the 
stability of the resulting flows in the ee~relusion of' the paper. Since then 
we ha.ve examined the stability of these flows in greater detail and O'U1" 
conclusions ai~e presented in §2&4.20 Finally in §2&4.3 we compare our 
theoretical results with the experimental results of Alty (1966)., 
Since the publicatiom of Hunt (1965)jl we have found tm papers by 
Chax1g~ Atabek and Lundgren (1961) and Uflyand (1962) in -which were 
analysed the ±"low in a duct whose walls (BB) are perfectly conducting 
am walls (AA) are non=condu~ting. However 9 owing to the form of 
solution used in the se papers:, the i:nteresti ng properties of the flow as 
M-+ao could not easily· be seen and no physi9al discussion of the 
p:t>oblem was attempted. A recent book by Hughe s and Young (1966) also 
analyses the same pt"oblem using the same eumbersomttechniqu es as Changg 
Atabek and Lundgren (1961\ but the book is of interest since velocity 
profiles and current stream lines have been computed in great detail for 
various vaJn.es of M., 
2°4.2. !fue stability of the flows when }.f,.:21. 
In t his se~tion we di seuss the stability of the high Hartmmm 
number flows analysed in our paper~ Hunt (1965)~ using the results of our 
general analysis of the stability of MHD duct flows in §5.3. 
In §-503 it is shown that the--analysis of the stability of' a fiO'W' 
in a rectangular duct is simpler when M;>>1 than when M = 0., The 
reason is that 1 1'hen M >-:'.> 1 a core flow develops in the centre of the 
duct and boundary layers form on the side walls and therefore the 
stability of the flow is determined by that of the bouti.dary layers 1 'Which 
are simpler to analyse than the flow found at M = 0 mich varies equally 
in two directions. Furthermore9 it is frumd that the most 1u1stable 
disturbancesj which determine the stability of the boundary layers., are 
unaffacted by the magnetic field. Therefore in examining the stability 
of these flows we can use our knowledge of the stability of boundary 
layers where there is no magnetic field. 
Let us examine the stability of the boundary layers on the walls 
M in a duct wi. th perfectly conducting walls perpendicular to the field 
and insulating walls parallel to the field (dA = O~ dB = r:P )o In tllis 
case as M~oo the velocity profile in the boundary layer becomes g 
v = t ~j ccs__b{ 'l)_ VL f (- 5' !¥- ) z;'.4c (- j' r;J) J 
:
01 
1 f' I j= 0 M ol ,J - ·i 
where S : M ~ j and thus (MV) becomes a function of J' only. 
The stability of this boundary layer is then determined solely by the 
value of the Ri=ryn.olds number for the layet" Rb,.Q, 9 since the velocity 
pr·ofile 1 sttitably expressed 9 is independent of Mo We now have to 
determine the value of rt.t.in terms of Mand R9 the overall Reynolds 
m.unb er (""' a ut-/ 1) ) 9 where Li T; is the me an ve loo it y jl for var1. ous 
shapes of' duc·co 
l 
If a/b<:<. M"'-'2, Le. a very thin duct with 1~lls AA much shorter 
than walls BB, the mean velocity in the duct 9 l~_ 9 closely approaches ~ 1 
the core velocity and most of the flow is in the coreo (For a/b '>- M=z 
most of the flow is in the boundary layers on AA 1 Section :3 of Hunt 
(1965)). Then the m3an velocity in the boundary lavers on the walls AA 
- " l is O(M) V-r-and since the thickness of these boundary layers i r: O(a.M"'-'2), 
1 -
the Reynolds 1-111mber of the boundary layer R - = O(a.M2 lf / 'l,) ; 9 where 
--b 9 • -l=-
v is the kinematic viscosity.. ~ence 9 
\,. Q. = O(M2 )R, 
=21= 
where R is the overall Reynolds number, (R = a tri! Iv ) . Thus for 
given R, 1b.{. increases with M: hence the critical overall Reynolds 
number at which the boundary layers become unstable is reduced by 
increasing M. Note, however, that away from the remote walls AA the 
flow would be very stable. 
Now consider an approximately square duct with alb = 0(1 ). We 
see from equation (24) of Hunt (1965) that in this case most of the 
flow is in the boundary layers on AA. 'Ihe ~an velocity in the 
boundary layers on AA is O(M) 1~ , where lfc is the core velocity an1, 
since the thickness of these boundary layers is given by S = O(a.M""z), 
lr2a :::- o [C tv\ u-c. o?- M -~2 + v ,_, o..b) /o-8 : O [M;-2 u-C- o/b] 
Hence, if alb = 0(1), R = O(M2 a tr:: liJ ), 
· 1 C 
and since 1b.t. = O(M2 a lf Iv), R ~ 1b.t. 
It is important to realise that the forms of the velocity profiles are a 
function of M and not R. Thus velocity over-shoot and reversed flew 
can occur in the boundary layers 011 AA at arbitrarily small Reynolds 
number. Using the usual sufficiency condition for boundary laye.T flows 
we can show that below a certain value of R the flow is stable. Since 
tne velocity profile is in the fonn of a jet with an infinite number of 
points of inflection and since the critical Reynolds numbers of free jets 
vary between about 4 and a few hundred ~t would seem that in this type of 
ductj when ~>.>1, Rcrrf-1000 whereas when M = O, Rc~~t~.3000.. If it cruld 
be shown that when M>>1, Rcr,~<1000 then, by definition, our hypothesis 
that the magnetic field bas a destabilizing effect would be verifiedo 
When the walls of the ducts are all perfectly conducting (dA = dB 
e:(71) ), the velocity p;rofile :-.of the boundary layer also contains points 
ot, ~ ... tQ"!6rJJ 
of inflection (fig.4)iand hence raisimg M reduces the Reynolds number at 
'Which the boundary ],ayers become unstable. But in this case, for M:~ 1, 
the velocity in the boundary layers on AA is of the same order as the 
1 
core velocity and since the boundary layer thickness is O(a.M""2), R l = 1 1 -Do o 
O(M""'z)R.. In this case, provided a/b <. M 2 , the shape of the duet is 
irrelevant. Raising M at a constant value of R may first te:t1d to 
destabilize the boundary flow and then stabilize it. 
-22-
Soo O Q =- 1>C\ u 
o e =- --11 ° 
2.oo 
0 
0 
() ---i----f---,.j----'----J..-- t R 
lbOO 2000 
~2,2.J~ ... fokvitia,L a± itii1.. 
~~1vil1lfor1e l~Jl~ ~ ~ II 
il'jO.tnst t<enVl'Dld S ~Y\UW\ ~£.r 
I I\ 
-2Mck; e~±qo~. JJl=-221 
(BH~ 
1 
1qCt) . 
5060 rs oo 1 0 00 
( 
" 
( 6 ~ Az) 16 
IN, G-/ir-.t RU"' 
0 I 
0 ( C>l)D 
0 
~.:..--- 0 
'\ 
-4=- B-~~\lr,, 
or I+ b·~ N'rt 
0 
D 
0 
0 
r , R 
Z.000 cSooo 4coo 5000 f,ood 1000 
· 1ij2,3 . ... l\e.$Sl)re wad~V\t ~Qins/:, . ~o(dg_ 
_n_umbec. _____ t~d».d~ ~~ ± 4o~.- ) M ~ 22:l 
(AH~ , i 1 tf t) 
-0• \ ~ 
- o·s _ \ 
0 
0 
3 
I 
0 
4 
0 
~ 2.4_'PcW"1bai ~le.e e. \6V13 :tk ·l~SM loJ~ l;l,-ll. 
31 II d vet : G '::: 96 a flt.=: lsq l {(. ~ i00 0 
-- --- ---
-- ( --- . --- ---- ----------from A I f.j ([J ( (,) 
Experimental results of Alty. 
Since the publication of Hunt (1965), Alty has performed a series 
of experiments on the effect of a transverse magpetic field on the flow 
of 1!1'l"CUI°Y through a square duct with two walls conducting a:m.d two non= 
conductingo He was able to vary the direction of the magi:ietic field 
relative to the duct and i.n one series of expel)'iments he investigated 
the flow whe:n the magnetic field was perpemdicular to the highly con= 
ducting walls • 
In his Ph.D. thesis Alty -has made a _detailed comparison between 
the results predicted by our theory and his experimental results. We 
will merely present the three relevant figures which show most of the 
results. Fig. (2.2.) is a graph of the electric potential differenGe 
between the mid-point of a noncondu cting wall and ~ conducting wall, 6 p II 
against the overall Reynolds number, R. Note the close agreement 
between the . the_oretical and exerimental values up to a value of R of 
6o(X). Fig.(2o3) is a graph of the variation of pressure gradient with 
R at a given value of M, (M))1), and shows that for R > 1000 the flow 
is not laminar, i.eo it ls unstable, and also that for R low enough the 
pressure gradient become~ very c l ose to that predicted theoretically. 
Fig.(2.4) sho-ws t he distribution of~ along the non=conducting wall at 
a given value of R and M., The theoretical values only agree with 
experimental values if the potential is calculated from the mean . velocity 
and~ the pressure gradient, though the explanation for this is not 
quite clear. 
'Ihese e::q)erimental results have proved the followi_ng~ 
(i) The ma.gn,etic field, if sufficiently large, can lower t he 
Reynolds number at which the flow in a duct becomes unstable. 
(ii) In the particular duct flow studied, for a given value of 
M(~'::>1)~ the f:Low can be stabilised if R is reduced low enough, as was 
shown. in §2. 1~. 2. 
(iii) The theory of MHD flow in a duet with conducting walls can 
accurately predict the values of ~ , (dp/dz) etc., found experimentally. 
The experiments have also shown that various interesting effects 
occur "When the boundary layer becomes unstable which we do not understand. 
-23-
M.agnetohydr odynam:i.c flows in rectangular duets II . 2. 5. -
2• 50 1. Introduc tion. 
In this section we conti nue our examinat i on of flow in r ectangular 
ducts whose walls are electricall y conducting. we oogin by r efer-ring 
to the paper ~itt en j oint l y wit h P:cof'essor K., St ewartson, Hunt & 
stewartson (1965) 1 herei nafter r eferred t o as H .'·. s. In this peper we 
considered the case where t he walls par allel to '::>~ magnetic field (AA) 
are perfectly conducting and tqose per pendicular to the field (BB ) are 
non-conducting, this being the kind of duct used in MHD pumps and 
generators, which are usually connected to an external electric circuit .. 
In our analysis, which is only valid when M»-1, we considered the e ffects 
of such circuits, though it is found that for this particular duet they 
do not make the problem more difficult.. (The demarcation of the worlc in 
the paper was precise in that Stewartson-performed the asymptotic 
analysis of §2 j wh:q.e I wrote the other sections). 
In the following sub-section we extend the order of magnit ude 
argument of the paper to the .duct flow examined in §204 and then compare 
the flow in re ctangular ducts with all eombinations of conduc t i ng and 
non-conducting walls. Finally: we compare our results with the exact 
numeric al solut ion of Tani ( 1962) o Since the publication of our pap er 9 
the trans lation of a paper by Berezin ( 1963) has become available , in 
which he c onsiders the same problem for all values of M. 
merely reduces the problem to a single infinite series of a lgebraic 
equations, which, he c laimed, can be solved by the method of successive 
approximation. 'l'his does not seem to us a great s t ep forwar d. 
2.5.2. Comparison of all types of duct flow. 
· In this section we use the .method of ~~4. 1 and 4.2 of H & S' t o 
deduce the main r esult of Hunt ( 1965\ namely that most of' the flow occurs 
in the boundary layers on t4e walls (AA) if the walls (AA) are non-.. .. .··. ·;' .. ·· ·, .. .. . · .. ' - · 
conducting and t he walls (BB) are perfectly conducting. We call this 
type of duct flow case (iii). 
Qg.se (iii) 
We use the same notation as for c ase (i) and note that f or case (:iii); 
( 
I 
- ,, Q. 
-~ --- --- ----'1!!11--- s' 
Bg 2 .S__ Ct~s9_::-_s~~:.:;J,.ovi __ ~~Jb~ _ _d1v..c.t_. li~k,e1.,\ 
_d_A::: o . cwd cl.B-=--oo _ _( M\ ~>!~----· 
as nay be seen from Hunt (1965) 1 the core vaJnes of curTent and velocity, 
jc... and LJc., 9 a~e also the same/, (Cl . 1 1 I 
. j ' : - f) _\T o -= FI d 2; I/ B o i . e . c.., 0 c. 
- 2 . 501 .. 
1eJe consider the integral§~. d l taken round the patn, PQRS in figo 
(2, ,. Then, since Ey = 0 in the core and E = 0 on the walls BB, and x. 
since § § ,c:Lt 
f \: .X,J L t s p E-_ _'.) J. j 
f' s 
=O 
I 
== 0 2. 5. 2 .. 
By considering current continuity in the s econdary boundary layerg 
- O [Jc ~ /J J J 
and therefore, f' J,\~~ ~ £ h;-~ --o[J,_ aic,-c)J. 
Since E = 0 in the core.? jc,/cr + Bo I.re..-= 0.9 and therefore x 
Q_ J ,' ·l l E.x. J ;,c. -= j J_s._d ~ --· + Bo j u; clx., 
p ' () (!_( D . [' ] Now since J ~ = 0 on the wall it follows thatJs = - 0 J e, 
The/ore, using (2.,5 .. 2) and (2. 5.3) 
Q. 
-6[J'ca./ er s] :::. 0 [- jc J /er J + BD i lrs dx. ) 
p 
whence E, u lf"5 S ,,, -'- O [j c. Q "/ o ~ , since 2> <.< a, . 
From the equation of motion 1 sincej = - 0 Qc.J 1 
and /~:;J I= o[ :;,L] 1 it follows ~at J!c
6
~s = () tJ'~ B0 J, 
where j1. is the viscosity in this case. 
Then dividing (2.5.li,) by (2.,5.5) 
J 4/a4=- () [p-/csBo2 Cil2J, 
or · [ _ 0 [ at M -V2-J . 
Thence (2 .. 5.,4) leads to~ 
=25= 
Therefore, from (2.5o 1) 
U-c_ = 
.:, 
,i..,1,' 
and S U--~ Jx. 
f.) 
whhm result is the same as that of Hunt (1965L 
As was m9ntioned in H. & Sy the form of the boundary layer on the 
walls AA is best explained in terms of the seconda,:"y currents induced in 
these layers o We now draw up a table showing the orders of magnitude of 
the secondary currents, relative to both the eore current and the core 
veloc:Lty~ including in this list the case (iv) wiere all the walls are 
non...conduc ting analyzed by Shercliff ( 19 53) o' dA and dB are as ' defined 
by Hunt (1965), being proportional to the conductivities of the walls AA 
and BB r espectively o 
Case Number dA dB 
(i) 0,0 ,:;,o ' 
= -Ou<- IVr~ = - o (f\,(:,..t) 0 ,re,) Js (ii) ( open e ircui t) (Ja C> . = -6(Jc) =- O(M- 1(}'~ 0 t.re, j Js (ii) (short circuit) C-0 0 Js = - t(IVl- '.j~) = -o(rV1-\1 B0 1rc) (iii) 0 qo J.s = -· o(jc) = ·- o ( tJ B ti is-e--) (iv) () 0 . 
-()(Jc) = - 0 (M-1 crlscl.ie-) Js = 
From case (i.i) w-e see that the value of the secondary currents relative 
~ to the c ore currents may vary 9 yet expressed as a fraction of v-L- 9 Js. 
is oi' the same order in both ca.seso This must be so for the viscous 
and electromagnetic forces to balance in the bounda.ry layer. 'l'he most 
significant result from this table is that J s = -0[ tvf 1"5' ..Bo Lie..) in 
every case except (iii) and, as we saw in §§1i. .. 1 and 4.,2 of H & 8 9 this 
means that i~ each of these cases tr5 = -0[ 1r~.. It is only i~ case 
(iU) where J 5 , relative to t.rc , is O (M) times the value ofJs. in the 
other cases and where, in consequencej the viscous stresses must be O(M) 
times as greatj that V--5 = O(M) lre-. O~her important differences are 
shown up by the order of magnitude arguments.. This crude table only 
indic ates the gross difference between the secondary boundary layer in 
case (iii) and the other cases. 
I I 
I I 
11 
Comparison with others' results. 
Although the analysis in H & S is only valid when M >:>1, Tani ( 1962) 
has provided a suitable variational method fat' calculating the velocity 
profiles and the volume flow rate, Q, at values of M below about 250 It 
is t ,·:: a.resting to compare his values for Q in a square duct when M = 25 
and E = 15, with those calculated from our result (1 .5.4S). 
M P:~ / ( (- vp2>~ )4,c{) 
Tani From (3 .. 5) of H & S 
We see that the difference at M = 25 is about 1%~ 
at least, the entire range of M is now covered. 
Thus for square ducts, 
It should be noted that in the analysis of H & . S we found that the 
velocity profile was unchanged by an external circuit. This is only 
true if the conductivity of .the walls AA is high enough. By analogy 
witi the result of Hunt (1965) that, for given dA, as M~~ , ruch that 
dAM2 7 oo , the solution becomes identical to that of the case mere 
dA = 00 , it is likely that a similar result will hold when dA is finite 
and dB= O. The physical reason is the same namely the relative 
resistance of the wall to the boundary layer on the wall AA decreases as 
M -;;i. o0 i because of the decreasing thiclmess and consequently conductance 
of the boundary layers. (This p'oint has been analysed more rigorously by 
Chiang_. ( 1965)). 
In the same series of experiments mentioned in §2 .. 1~.3 .. C.J .N.Alty 
examined the f1ow in his duct when the conducting walls were parallel to 
the field and the non-conducting walls were perpendicular. The con-
ducting walls, AA, were connected together, the resistance between them 
being very small so that they were virtually short circuited. Therefore 
in measuring Q as a function of (dp/dz), the flux deficit du~ to the 
boun:iary leyers on the walls AA ia O(M""'3/2) that of the core, as shown in (3o9) of H & s.. With M > 100, this tenn was too small to measure so 
these experiments gave no test to the theory of the secondary boundary 
layerso However, in some experiments on electrically driven flows in a 
curved duct with dp/dz = o, J.A.Baylis (1966) has provided a critical test 
of the theory. We describe the theory for the flow and Baylis' results 
in the next section. 
6 Flow in Curved Channels . 2. • 
2.6~1. ~guations for cylindrical flow. 
We now consider the extension of the theory of H & S to flow in 
curved rectangular ducts. (See fig.,2.6L We will only examine the flow 
whm the secondary or radial velocities due to the curvature of the due: t 
are very small compared to those in the -€,- direct.ion. Also, we assume 
that flow does not vro:y in the 0 or stream:wise diY·ection and therefar e } 
the pressure gradient in the strea.mwise direction must be zero, the . , f 
energy of the flow coming from electrical energy fed in at the walls~ 
The equations (2. 2.1 - 2.2.8) for steady flow written in cylindrical, 
(r, 6 , z) co-ordinates men °a/a., 'G = 0 are (Chandrasekhar, 1961 ) : 
( dV'. .l lr- d u f- u- ~) ·"" j · r B p \.T r ___:!.. ' . 2- - -::- - - · ~ :;::.-up ~ y +-j 0 0 L a< ~~ ~ I 
. 
to"L + __L d _ l +..i='] + -vi. b))· .,__ y O'{ , 2. 02.."'-- IJV) 2.6.1. 
(0 ( u-( du-;;:' + u-'i:. C2g-~, 
l QY Oi!:. 
e (., < d lJ' i: t l)7-. c)l(i!= ) 
cl.- d~; . 
~u-r T '?)0 -~ + .!:!:!-
or a?;. ' f 
a: (- c_;<p(o'I" + l)-e ba) jy \__ 
Je ::::. 6"" u--'( Bo 
J2 ;;'. 
-Ji' -
We now determjine the conditions under whieh lf.r , lf.l: << 1..rie j when 
M~)1. Differentiate (2.6.1) w.r.t. z,and (2.6.3) w.r.t. r
1
and subtract 
the equations. Assuming I~)>) \T...- , o-2 , we have in the primary or 
Hartmann boundary layers at z = :!: a, 
d ( \r • ~/t' ) ' P, 2. 
- r: d:. -· C! cr ~ c ~ 
-28-
4:·s i £~ 
/ 1) / t 
A , 2a. A 
ii B ~/ B 
c.,urrut I CM rrort I 'ir" Y, 
lwv'1~ 
1''21 
±ijzJ, Uossc~4--a·"41" -'.\Uk1lMr,k-
_chllnnei_(1,JalJ&___AA_(,\iL-f'~C&ilJud~ _ 
w:a..11~ B 5 a,e '.bo{/\ c~J~.:r-~-
Y)t 
6 Flow in Curved Channels. 2. • 
2.6~1. ~uations for cylindrical flow. 
We now consider the extension 9f the theory of H & S to .flow in 
curved rectangular ducts. (See fig .. 2.6)o We will only ex.amine the flow 
when the secondary or radial velocities due to the curvature of tbe c:loot 
are very small compgred to those in the B· direct ion. Also, we assume 
that flow does not va:cy- in the e or stream:wise direction and therefore 
the pr-essure gradient in the streamwise direction must be zero, the . 
energy of the flow coming from electrical energy fed in at the walls~ 
The equations (2. 2.1 - 2.2.B) for steady flow written in cylindrical, 
(r, e , z) co-ordinates when °a/a.. 9 = 0 are (Chandrasekhar, 1961 ) : 
( d\1'. _l_ u 2>u,= tr ~ ) ."' / r B . p lT 'r _.:.!.. 1 . ?.: -;:;::-- -·-· _Jj_ =-LJ t:, ';;;) y f-j €) {'.! ~ o< 0~ T f 
ro2- + -1 ~ _ l +~} · + 'vt Lor ">- Yo'{' ,-i. oz:·_ u-._, l 2.6.1. 
J. t.rr 
1-
..,,,.-
1(1 {- a <p(or 
\._ 
. 
jy 
Je ::::. •.r' 1;-v- Bo 
J?:: ;; 
j"f ::: 
We now deterrn:iine the conditions under which Lfy .. , lf.l; <~ \..~, when 
M~)1. Differentiate (2.6.1) w.r.t. z,and (2.6.3) w .. r.t. r,and subtr act 
the equations. Assuming I~ >) IJ"-r ., o-2 , we have in the primary or 
Hartmann boundary Jayers at z =±a, 
- '° d C ~5/-/r- ) (2!· tr- B c 2. 2i v- r-
~ d~. ~ 
-28-
!) l 
f. 
Since in these boundary layers ~/dz = O(M) and 1;-6 = O(\Tc..,) 1 where trc 
is the core velocity, we have: 
~-· =:_ 0 r- 0 I f__LJ:::....... J - ~-u- L '(' o-Bo2 CA. -- 2a6o7o (., 
Therefore provided K is small enough we can ignore V"<and lrccom:p9.red 
to \TQ , but it is important to realise that hcwever small ~is, uy 
is always present. 
2.6.2. Flow in a rectangular annulus. 
Consider the flow in a rectangular annulus of sides 2a and (r2 = r 1) with walls parallel and perpendicular to the field b~ ng perfect,ly 
conducting arrl. non-conducting respectively, as.shown in fig.(2.6)o The 
flow is driven by a total CUt'rent I and therefore from (2a6.6) the 
boundary conditions on He are: 
--+- °'-- 't ( 'f ~ G) = (_") at 2. -
at 1" - Y, ) Y -z.. 
Then let µB :::::- I, / z--rr,,...-
and ~\ -es :: T.-~. /!~ ·ri\,...· 
~ (-< \-\a);: 0 · 
at 2... -= + ~ l 
at "t. ·- - o..., 
when j : ~ I ) ? 
when \' =r, \ r '?.. . J 
-29-
2.6. 10 .. 
Dividing up the duct- into similar zones as in H & s, we can perform 
a similar as/,totie analysis when M>> 1 o 
S.· 6 ~ d of ,) (i) Core g .mce , -f' ~ :::__ l_ 
- (': j cip > 
~p\ '--- . r 
u-'(. _:: J \~'- I and h c., _;. J ( e) J 
where g and f are functions of f' to be determined by the primary 
boundary layers. 
'd , o · (ii) Primary bounda:r:y layers ; Sincea~ ,;-,)ie 9 \2o6o6J becomes g 
i'.:::~ + M "elf -o :!J• e... + M ~u-£-- - o 2.6.11. c\)2- ~ - -- ) 6! 2 6 S -
where \,= l F+ lTc ' ~I= 'Ac_+ k p and ht , \Tr are subject tog 
V-p:;-uc_ > hf = - & - he. at j ~ I I 
lrp =- - tr c... ', h P =- ~ - h c. t J? I I I \J a j == - , 
u- ~ b h _...... .,..__ away from walls. f I r / '-J 
Then in the boundary layer at f = 1 
I L - M {__1- ' ) ' I - M(t- j) 
,,- - I - e, ur= c ) V)r:C- e 
and at J -= =1 9 
. .Le -(lfl(l +j) Je.-M(l+J) 
e trr=e , h'f= 
It follows thatf(v= 0 and3((:l) = l.. so that, 
. (-? 
t.rc... :::--1 ~c..= 0 
(iii) Region £ 
The analysis for this region follows that in §2., 5 .. 2 with the result 
that on the wall J = 1, 
u-5 l (:>) ~ ) -t- h s ( p ' 1) -::=- b ) 
where lr ,: lj t, + l..rp + lr ~ IJ h =he.. + h r + h$ o Thus we have a boundary 
condition on ( ir_s + hs ) at 'l"t. = 1 j which we need for the analysis of 
the secondary boundary layer - region (c )., 
(iv) Secondanz: boundary layer 
In this regi on we assume, as in H & S (92), that: 
"2>/'6e >'> d/a! 
However in this case we make the a dditional assumption that 9 
d la e. ::, > , / e 
2.,6., 1l"" 
(' 1 
or in other words t-i = O(aM""'°2)£.<.r19 the r adius of the inner walL 
Clearly if the eondi tion is satisfied when t = f 1 9 it is also satisfied 
men e = el." Thenj in the layer on the wall at e = e,, ~ if tr5 = --vvve, )(Ji 
and hs= (tv1/e1)hs 1 ~ \r'5 1 and h..5 1 satisfy (2.,25) of H & Sj namely~ 
rtJs 1 + fv\ "a~s' --::::- o, u"'hs 1 + fVI d u~s' - o dQ"- aS 
~~'L ~5- -and the boundary conditions: c 
tl" I - _ .l ~ hs I .:::-- 0 
v6 - . ) - when M ae 
since~hs/oe.)> ~ 9 in virtue of (2.6., 14).. Now we can use the 
solutions for ~: and h~ in §2 of H & S., The result for the velocity 
deficiency on the wall () = e, 9 is 9 
c.. fl \. J 5 tr de- ~5 = ~- .(:+J! 2-~ 
0 - , s c e c -~) t Mo/;? 
- )M1 ) where t ==- Ce:- e1 ) 
and on the wall t ::::: el,.-- g 
::: M (-~)} z_h-/ (<"4-) ! M~2 e ~) . 
Thus r{!-J+~6f)Os~ - 2. IY\ (e,./e>·)+C-~tL~i. ( ~ +~ )+2IV1&.~N j , ~ (~.v.'12.\ (.' c."2. M t' - -1 4.J . This my be r~·writ.ten O 
-t r r p " 
,,.:/"a- ;f o-.. 
Q =- J J ~~J~J_l_- I 
~I -~ 
-
_ J h_(..-J...-,) _\j_ · '1Sb(<2 f: ,) "' _l.1 
- - . 2.'li ~ l MJ-.. Y, '(4 lt1(t".._/~ M 
Now if we integrate (2~6.,5a) across the duct, theng 
y '2. +-a_ J St~J) ch·dl?; :::-2ac.~where 64 is the fall in potential from 
Y-1 -o... 
the wall at r = r 19 to that rjt= r 2 • Thence ·r.._.fJ..o'- ,. z_a.., .6d =- -Bo r"~ u- d~ d ~ + r J..!. J,, d2:-1.P J e J <r" 
'<" , _r;,. 
·r. -~ !c('~ 
. ~ - Ba a + l 1 / c IS' 211 y) dr 
Therefore 6~ = Bo&/2.o~ _: -r_ l·.~<"<'-... /<"1 }("41i01 cr). 2., 6 .,16. 
From (2.,6016) and (2.6 .. 15) we obtain ci.?1 expression for the resistanee o:f: 
the channel Le .. the ratio of b.<$ to I~ 
2.6.3. Experimental results .. 
J .A .. Baylis performed some experiments in the Cambridge Univer::dty 
Engineering Laboratory on electrically driven flows in ~c_ma~ annular 
channels at high Hartmann number (Baylis Ji 966) . Although secondary 
flow was present in all his e:xperlments9 when the field was high enough 
and the current low enough it was negligible o (When secondary flow 
occurs i.D. ~ does not. increase linearly with I 9 as is to be e:x::peeted 
from (2. 6. 17) ~ and is detected in this way),, In order to am..1yze the 
experimental results in the non=linear regime j Baylis plotted the 
'\!'arlable p !)" =IB c, ~v (1' R t1'\ ~a-) as a function of Mll where Q is calculated 
from (2.60 '16).. 'The theoretical va.Jue of P from (2 06. 16) and (2 060 17) 
is g 
~ r , .-- } )(! , '1~fo[2.R_~~ p ~ ~ (lri I_,, ... ~ - M1i(R'2-- 0 ,"·J\n(~-\-a.. /R -c;.) 
- +o . . ... , 
--o... . - \ -R.-o; M-1 (IVl -312. ) J -I 
where R = (r2 + l"i )/2o The experimental results are compared with the 
theoretical in the: table below. 
M R/a 
16031 35 
16037 17 
16074 B 
32037 17 
=32= 
p 
theoro 
1.k'I 
1041 
1.41 
1,,25 
p 
exp o 
1.,4 ± . 1 
1o5 ± o1 
1.4 ± 01 
L 19 ± "04 
M R/a p p theoro ~ --~ 
32oS6 s 1021.i. 1023 ± 004 
64093 8 L15 1o 12 ± .03 
65 .. S 3o5 L,12 1012 ± 003 
129.S 3 t' c) 10062 1 o0J+5 ± 002 
Thus a although. most of the experimental ::-esults differ from the 
theoretical by less 'than the experimental error r they are systematically 
below the theoretical value~ for which there is no ready explanation. =--
. ). We note that the va.lue of M is low enough for the term ' in (M=2 ) 
to be appreciableg so that the theory for the secondary boundary layers 
may be considered to be fairly well testedo Also the values of R/a 
are low enough for the effects of the c:urvaturP of the duct to be 
appreciable 9 so that ~he modif'ication of the ~ & S theory for curved 
ducts may be considered satisfa@tory. 
-33= 
Maggetohydrodynamic flow -in channels of variable cross 
. ' t h .L t 
.. t' .;,, • ld * section w:r ,,.;;.~,,..M ransverse ma ne ic -';.3:-e So 
7 1 'Introdueati on __ ., o 2. 0 0 
-
In section 2o '7 we ©onsdder th e effects of a strong unif"ol'."IJ!. ma,gnetic 
! i eld 9 
£ 6 = ( 0 / 8'<) ! Q) \ 
on steady two=d:imensional flows~ whose 1reloci tfo~ .::,.re given by~ 
_....-) 
v == ( (.; ::t'.. ( :x. , ~) ) u-:j ( )C. I j)) D) 
., f ')-
through ducts with walls at 
1 ~ ~=- f \:_ ( X,) ) :f b ( :x.. ) C\V\d ~ == ~= b 
where we assume b>') (.f1-_-t6 ) (The effects on the flow of boundary layers 
on the walls at z = :±}.'.> are considered negligible). The analysis also 
enables us to exarn.i.ne the now ove, a body placed in such a dtmto Unlike 
a 'Wind tunnel)) a duct for investigating MHD flow over bodies has to be 
placed in a magnet whose gap is usually smalL Consequently the duet 
size is severely limited and 9 for fletw' over a practical size of body 9 
wall effects cannot be ignor!',d even out wide the boundary layers o We 
examine the inviscid regions taking into account the effects of the mll 
and we also examine the boundary layers on the walls. 
Ludfo rd. (196 '1) and Ludford & Singh ('1963) have developed mueh of 
the existing theory for external flows in transverse magnetic fields over 
two and t hree d imensional bodies o They assume that the magnetic field 
is strongj) and that t he ~ondu.ct:ivity is weak enough to ignore the induced 
magnetic fie ld o T'.:-;:L3 is equivalent to assuming that the interaction 
parameter N (~15 _R, :;,: (]\/e U O } >) 19 and that the magnetic Reynolds number 
Rm (:s:::: /A G({J_ o a... )Le_ 19 where~ 0 9 <!' 9 e 9 J"-"- are the flux density of the 
imposed transverse magnetic field and the fluid us density j conductivity 
and magnetic permeability::l1' respectivelyo l). and a are the charac:-teristic 0 
. 
velocity and lengt ho 
Maldng this approximation fo1• twu=dimensional flowj Ludf'ord (1961) 
. I 
I 
I 
I 
* The analysis presented here is the same as that submitted jointly with j I}e,,SoLeibovich in a paper to the Journal of' Fluid Me~hanics o In the I paper the problem of flow over12 dimensio:r1al bodies was considered in greater detail 1 this part of the paper being written by Leibovich,, The work presented in this thesis is my ot,m in all essentials" 
=34= 
round that· in the 0irmer 9 region near the body the inertial terms in 
t he momentum equation are negligible comp;t.red i;o the electromagrietic and 
pressure fot'ces 9 e:xelept in s;i:rigular zones a t t he front and rear of the 
bodies 0 In the 1oute:r, v region sufficiently far in the field 9 OT' y 
direction from the body he f'ov.nd that frJ.ercia forces again beC;arne 
important O He discussed their effect by compressing the y co=ordinate 
to retain the infJnence of :inertia. and found expressions for lift and 
drag 0 The singularities in the inner region wm:'e left and the question 
of how these may affect the flow was not resolved . 
We awly ludfo!'ld ns appro~tions to fl0'"7s in ducts in mfoh we 
also consider viscous effects in boundary layers at the walls. Then the 
region outside the boundary layers where vis cous forces are negligible 
co:t'.l"e@pond@ to Ludfordns niJ.'1ner 1 region and we treat this region in a 
similar manner to Iudf'ord~ thougp 9 unlike Ludford 1 we succeed in. 
analysing the singular zones To!hfoh occur whenever the duct wall eurwature 
is O(N)o We then examine the boundary layers by assuming that ths 
Hartmann number M>) 1 o We are able to exten:i the usual analysis of 
t hese lq'yers (Stewartson 1960) by c:akulating the higher order appr ox= 
irna.tions, wtlich i s possible because of the simplicity of the oo:re flow 
solution (a1r.iay from the si..".tgula.r zones) P for which a.'rJ. expansion in 
imrerse powers of l\T mqy easily be foundo 
The ~ppru,x.:_i1iation used by Iudford has also been used very success= 
fully by Bornhorst ( 1965) to ~alculate the effect of a. magnetic: f"ield on 
the free surface of a. mer~ury flow when N >') 1 o The fact that the 
theory a«:lcu.rat ely predic ted the fr.·ee surface profiles found experimentally 
demonstrates t he usefuJness of the approxuna.t.ion . It i s worth observing 
that 9 in general.9 it is not difficult to devise laboratory experiments 
which satisfy our criteria that N ;;;,) 1 9 Rrn LL. 1 and M >.) i w "While having 
the Reynolds- m.miber large enough for aecurate readings of pressure 9 
vel ocity~ etc to be takeri. With regard to the practical usefulness of 
our approximation 9 our criteria are not satisfied by the nows in moat 
MHD devices at the moment 9 (e., g. in the biggest MHD generators N is only 
O( 1)) o However 9 ss their size an:i their i'i eld strength increase, so 
that N increases~ our appro:xirm.te methods may become increasingly useful 
in examining the flows in MRD pumps 9 generators 1 etc • 
2 7 2 .§t at ement, of the proble m. 0 • 0 -
-
-
'lhe Magnet ohydrodynarrd~ (:MHD) equat i om· f or st eady 9 incompressible now ,men the fluid properti e;s are c on stant , are g 
Cu- . v ) lT ·:=: -.-- i7,, + j l' 'B -1- ::1., \} 4.- lT D -' ~ \. ..... t~ 
,.._ 
L 
"'"' 
\{, :C Q I 
~ (E f ~x ~) , 
0 
J, * V1~ ) 
\7. ~~c ) 
2o'7 0 1 • 
2. 7 0 5; 
2.?.6. 
'Where \.r 9 p 9 j 9 B 9 E are velocity 9 pressure 9 current density 9 -.J ,11"""'1 __..._., magneti~ f lux densi ty and eleljtrilj field respectively. When ~ ..(__<'.... 1 9 we can ignore the induced magnetic field due to J and a.srume that 1 in equations (2. 7. 1) and (2 . ? . 3) 
B .:::: B.::i 
where ~ 1) i s t.c1.e imposed magnetic f i el d. 
If now we ~onsi der a t wo ·~dimensional flow in the x - y plane 1 such that 
0 
and hen~e from (2 .7. i) 
o E-~ (ox ... ::::. ,3) E2;. / e>j :::.. c::i 
Whet he.r th e wallf!§ at .'.:)~{; (x.) )~ (1 ... ) are conduc t ing or not 1 provided I;: Jt, there are no current sources or sinks along these walls9 it may be shown that Ex = EY = Oo (If the electrical boundary c ondit i ons on "the walls at z = ±b vary r apidl y in t he x=di.rection t hen it follows that ~ /az \ 0 and Ex\ O, thus the applicabil i ty of the basic assumptions to real flows must always be caref ully checked. We discuss this point further in the conelusion )o I f the magnet i c field S~ lies in the y=direction and if we r educe the Jnrameters to a non=dimensiona.l form in tem.s of Q, the total flow rat e t hrough the duct per unit depth, B'() 9 and 0\., 9 a ,-...., repr·esentative channel widt h 9 the equations become~ 
I 
I 
I I 
I I 
I 
I I 
I 
I 
I 
I 
I I 
I 
I 
:;:i'52,]_ HIM! ~~IDl'\S _LJ)'IT. { c ), 
baut1 da,Yj )~$ l ~) awi Lud F<d 
\"':'f's ( L) d.; SOASS<J W\ § 2 .'7. 2 
---··-- -- ··" 
I\J ( IA -f {;: O )-t K - I 'V 2. I..\.. ) 
du--
IA -
c)X, 
where 
:-;__) ~ ,i "::' :f_) 'j\r./°') ~\ = c/B<::?d/ecG. Ov'Ad, fR..~e_tlfAlso let +-:4.(i) :;: fl l x)/o-. 
and } 1, (i-) o:c f1J)c}/°'. 
To solve these equations and satisfy the boundary conditions w postulate the existence of various regions in the flowll 'Which we exa.mine in tum o The solutions which a.re found to satisfy the boundary 
conditionsj match each other at the boundaries of the regions and are consistent with the original assurnptionso We will now discuss the approximations to be used (see figure 2o 7) by looking at the general problen of flow ~~er a body placed in a duct with diverging walls. 
Regions C {9ore t_lol[l,. 
In fu ese r egions l1 away from the boundaries 9 velocity gradientw may be assumed to be 0( 1) so that viscous forces are negligible and1 since N..>>1 9 the ele re;tromagnetie for ces are very much greater than the inertial forces" Thus in these regions the electromagnetic force is balanced by the pressure gradl en:t ~ and cons equerrtly the body force 9,,l,<.. ! 9 i~ irrotationaL Since,. as pointed out by Shercliff (1965)!1 the J ;('B 
,..., ---., force only affects the motion of an incompresru.. ble fiuid with no free surfaces when it is rot_ationall) it appears paradorl.ca.1 that !i when no viscous effects are present9 as thej 'I.~ force becomes sw'ficierrtly hrge it becomes .ifrotational o The explanation is that 9 although in the final flow pattern th~ 1-£ force is irrotational, in the setting= Up process the j -f- ~ force'' has to be rotational" Note that 9 w1en the inertial forces are negligible' the velocity is very simply determined by Obm 0s Law and the continuity equation, as sho\\111 in §3., But 9 as Ludford 
=37= 
I 
I I 
has shown 1 this approximation breaks down in the region near a dis= 
continuity in the slope of the boundary walls 9 a.s illustrated by the 
1cink in the stream.lines between C'i 9 c2 a:nd c2 c3 in figure 2,,7,, 
~zj.ons Lo (Ludford Layers) o 
These regions emanate in the .field direction from plaees where 
the slope of the boundary walls char1ges rapidly,, Consequently lr changes 
rapidly in the x direction and therefore in these regions the inertial 
and 'Viscous for oes are appreciableo These are th6 singular regions near 
the f'ront and rear of a body whieh Ludford did not analyseo The struei= 
ture of these regions 1 mich we shall call . ULud.ford i layers 9 . is analysed 
in §4 and is shown to depend on the relative size of M and R., For the 
parameter range of interest it is shown that the thickness of th ese 
layers is o(rr1/3),, Our analysis assumes the slopes of the boundaries 
is always finitej though t he ir rate of change may be infiniteo This 
means we do not analyse the layers emanating from the rear of the body 
in figure 2,,7~ but only from the fror.r:to Howeve:t'9 since du4::lt walls 
usually have finite slopes 9 t.he analysis is valid for most practical 
si tuationso 
Regions }lo 
/71\, t~se reigiorn:i boundary layers are formedo We shall assume that 
their thickness i3 sw.1tll eompared with the size of the duct and that in 
these layers the dominant forces are vis~ms and electromagneti©o 'Ihese 
assumpti ons are &1.awn in §5 to be equivalent to assuming N::::.) 1 and M:::. ) 1 9 
the thickness of tl1e boundary layer being ?(M=u),, In this ana...lysis 'lire 
implicitly assume that 9 as a. result of several experimental and theoretical 
investigations j if N and M are sufficiently l arge there is no sepg,r-ation 
of the boundary laye1"'s o In experimental investigations of the flow over 
cylinders 9 spheres and flat J31ates 9 - (Tsinoberjl 1963 1 and 'rsinober, Shtem 
& Shcherbinin 9 1963) 9 and flow through a diver ging channel 9 (Heiser 9 19641 
it was shown that when the magnetic field is sufficiently great ll it can 
completely suppress the sepe..ration of a bou_ndar-.1 layer 9 while some 
theoretical evidence for this phenomenon has been provided by Moreau (1964) 
who demonstrated that a transverse :rragnetic field can suppress the 
separation of boundary layers on a flat plate and on a cylinder" 
7 3 Core f lQ.,w o 2. 0 • 
As 1&«.dford ( '196'/) has shown 9 in the lir1tl. t N-41' ro P equations 
(2 • 7. 7) reduce to 
. 
(a) df /dX, + ~~ l ~ +l:= D) 
(b) d(f /N)/o~ 
(c) o~fe.i + av-(',YS 
on allowing p t.o grow large with N and assuming that velocity gradients 
are o(N). Equations (2.,7.,8) have the solution 
lJ\. :: - f ( x) ) P ~ N ~ (:l )-x £ c] ) v- :: j f ?;_ ) + j ( i,) · 2., 7. 9., 
Clearly.9 this solution cannot satisfy the no=slip condition u ~ 0 9 v = 0 
at the walls. In fact~ Hartmann layers mst form there j of thi 11kne ss 
o(M=1) to reduce the tangential yelocity of the core flow (2. 7. 9) to zero., (See § 2. 7. 5). We there fore relax the no=slip condition j and require 
* 
onl y that the normal velooity at the walls vanish. 
F'or flow in a duct the top and bott.om walls of which are deser-ibed 
by the equations j = Ft ( :x.-) 9 ir = Fb ( .x,) respecti~rely the boundary 
conditions are satisfied if 
ft-f 'ex-)+ 5C~) = +i' L,t ,. Fh,f 11 + j' ~ Fb1 tA 
or j ( 2,) = .fb F,t 1 - Fi. +b 6 1.-=- f-( 5l jl 
=F'b- +t; 
- j Furthermore v t~:; satisfy the continuity requirement 9 1:c1g _ &, = (-f'th-fb) Q., 
Thus~ 
*ludford=t196"'.f) also deals with the solution ~3o2)o Since he is con.;., cerned with an infinite domain, however, he nmst take f 1 = constant 9 and cannot satisfy boundary conditions at infinity. These are satisf~ed by considering inertial effects for J.Arge y. 
=39= 
may be found in terms of E and Q by integration. 
and for the flow between the body and the bottom wall is~ 
, 1 IT :::-( __ _l__)~ \C b'(3 -h:i )+ ft/ (C t,-3 )l 2o?o 1Jbo V' ;: C-i, -+=b Cb-+i.,/' L J 
The solution for~<J~, and'.x:.. .>.Q"' is unaltered by the presence of the body. Thus the f'lo11r over a body in a duct is identical to the flow in two separate ducts, their walls being the top and bottom walls of the duc:t 9 the dividing streamlines and the t op and bottom walls of the body., 
'Therefore in the following analysi s 9 'Where we only ment ion flows in ducts , we are implicitly treating fl ows over bodies as well. 
Our duct flow solution (2 .. 7 .. 12) (or pseudo duct flow (2.'7.13)) holds whenever the wall slopes a.nd curvatures are finite, since ~ ~.:_ = · N(d-t-nf [V+ll (5-h)+ fb'' (n 5) ~ 
_ 2- ft' - +bi [Et 1 ( 5 - Fb }t Tb1 tri -9 l) . 
...- 11.. +-b - +=6 However j ift-,l b or t-t b = O(N) 9 (2.,7 .8) fails t o hold. As dis-' . ' cussed in. §2 9 we onJ.y deal with the .©ase wheref+_.,l:, = 0(1) , so that we only consider situat ions where (2o7o 12) ±~ails to hoJ.rl owing to the 
curvatur e being O(NL (The solution (2.,.7o n) always fails at the front and rear of a body except i n the unlikely event of the body being casp 8h&ped at these points)o 
The solution (2. 7 e 12) may be regarded as th.e leading terms in an asymptoti c expansion ~ 
_ 1 
_ '2. 
II\ .-::: v.. t. + N I.A 1 + N l,\2- + . 
lit' -- lr~ + N-1 u-1 + N-2 u-~ +,, i " N ( ft; -t- 1,r 1 f i t - . . , ) J 
To consider the higher approximations and still ignore viscous effects 
M has to be sufficiently- large. But since we are only interested in 
tb3 first or second order- a.pproximati ons, 
effects in the core • To find L.\.., ,. u-, 
into (2 0 7.7) and equate terms of O(i ) . 
valid. 
2. 7. t• 1}.e ludf ord layer egua ti ons. 
if H;:..) N we can ignore ·viscous 
, f, we substi tut('! (2 .. 7 .1 4) 
2.? .. 16. 
In regions where the wall curvature is O(N) the inertial forces 
cannot be neglected r;;n d the solution f"or the core f'l.ow, (2. 7 .. 12), is no 
longer v·alid. [ktppose su eh a region exists at x = 0, then we see from 
(2. 7 o 12) that u -o has an O ( 1 ) jump while U 
O 
is c orrtinuous a.t this point. 
(Note that . lJ i ~ (A1 and higher order terms are, in general, all 
discontinuou::: . See the example of §2.,7.,6). Now let us assume that 
the Width of this region in the streamwise direction, [ , is very much 
less than the ·vtldth of -the channel.9 Le.[; zz 1, and that the region 
appears to be a discontinuity i n the limit N~~o Then the proplem is 
t o show that such a layer can exist by finding a solution . for U. l) Lr in 
the layer which :rratches U. 0 9 tr O in the core (See figure 2 .. 8) .. 
We first stretch the x=co-ordinate according to the rule~ 
2.7 .. 17. 
layer 
Th_, since lJ0 ha.s an 0(1) jump in the layer , if the change in the .... , r is.6u /:J u = o( 0 ) o Ac?ordingly, in t he layer put 
lA = {/ h Co) t S' ( \_) ()(; g)) ) p ~ P/S 
'fbere h ( X- ) = Ft ( i ) ~ F1;( i, ) is the channel width at station .. Also let h(o) = h
0
.. (In the following analysis we assume that 6 .>) c (N -') 
2 .. 7 .. 19;, in order that we can ignore higher order a ppro.ximtions to the core flow) • 
...... 
I ' 
I 
In terms of U, v, P, X, ~ , e_quations (2. 7. 7) are 
'2. ( ) 1... d Ll -t- au 2JJ + S1.r1)LA_ = _J_ o t3 - r NU.- N[E o-tl )+l d ~) I' a ho dx 'eJ'f, Oj dLc)){ ht) t&~,< I 
(b) L 6 tr +SUd tr +- 6tr~ = -~p +..1 ~, 1 ho d X ot< 0.9' ag RE E,X J 2" 70200 
(c) dlL/d)( + dV"'/dj - 0 
where terms of O(R~1) have been neglected comp9.red to those of 16 4 R°:1) All unknown quantities appearing in. (2 .. 7.20) and their derivatives, are 
assumed to be O(~): Equation (2.,7.20a) is thep 2. ) .·. ·~·, 
'2>P ::.. - ,2> 3 NU - 62.N(Eb+J_ )t- « 6 /+6(Jt i) oX h0 2 .. 7 .. 21 .. 
while (2., 7 .20b) is 
1- dO- -=- - ~ ~.~r + -i- Th + <{_6) h6 d x '?,: :) R.~ 0 x 'G. 
On eliminating · the pressure and the core value of the j X f ignoring the error terms, one obtains the equation ,,..., 
";tlr f h 6 ~SN 67.~ h n d+tr ::::- 0 
1 o 1':. ci g 2. R & c) )(t 
which is als o satisfied by U. 
force~ and 
Depending on the values of N and R .four possible situations may &rise, leading to different values of S : 
-42-
(a.) 
(b) 
(c) 
(d) 
cf 
-d.- '2. Electromametic=yj_scous balance g -6 NR., =- b M ::=-\ 9 and [; R._ <<. I e Thus i) = M=1z j and ]IL.,> R2 is the requirement for 
the existen ce of such a laye r o (This condition satisfies the 
criterion (1, .• 3) f or ignoring higher order terms)" 
Inertfa.l=viscous balanc e g ~) = R=1 and ~ 3N>) 1, which holds =- I , 
i.f 1-«-. R2 ...(_<'.._ M4' R2o (To satisfy (2o7 o19) this condition must 
be altered to 14'.. R « M <::<-. R2 ) o l 
Inertial=viscous=electrornagnet4,c_ bali3,n~ ~ 2> 3 I'\! -.::. L-:::._ \ t{ I)) 6-==- R -
which holds if M = KR2 " (This also satisfies 2,7 o 19\ 
Inertial=electromag,netic bal.a.nce g 2) = (h /N) 1 /3 <<. 1 • 
- ,:- ~ ' R 2>>> 1 9 which holds if 1 « R2 ~ M <<- R • (Since 6 = 
O(N = 1/3) 9 S.:>) O(N=1) and (2o'7o19) is satisfied)o 
We now concentrate on the type oi' layer which occurs when M,R and N ha-,re 
typical experiment al Talues) e o go M = 500 9 R = 5000 j N = 50 o Thus we can ignore situations (a) and (iC ) 9 but we have to consider both the situations (b) and (d) since they may both occur in the same range of M and R.o 
Howeverll t..here is no solution to (2o 7 023) which satisfies the required 
boundary condition as X-;.. ;± co 9 if the electromagnetic term is neglected 
arrl a balance of the ine!l}tial and v·i.scous forces is supposed to exist o 
Therefore we must c onsider the very nruch thicker layer \\hieh occurs in 
situation (d) vihere 2> = O(N=1/3) o We call this layer the Ludford layer in recognition o.f the similarities between this work and that of his 1961 paper and assume its structure to be governed by the equation 
c;x:,v .,· i,..! ';_;7- d -...1[ :::  0 
~;,.:. . dq 
We find that it is pos sible to ..... construct a solution to this equation 
satisfying the boundary conditions and therefore we conlude that the 
errors, due to neglect ing the higher order terms in (2o7o23) of O(N=1/3), 
an:i due to neglecti ng the viscous tenns of O(M/R2 ) 2/3 9 do not affect the 
solution to this order of' approximationo 
It i s important to note t,hat, with this length scale, the boundary 
curvature s till tends to infinity with N, in fact it is O(N=1/3) in (X 1 :r) space, Thus~ the walJ. still has an abrupt change of slope a.t X = Oo 
Since the problan. is linear j we mey break it into two partso The \ 
core flow, from (207,,12), may be written as v
0 
= v1 + v2 where 
I 
I 
I 
I 
I I 
,,.1 = Ft 8 (j - +--i-J/ ~:- 9 v2 = Fbfl(Ft. =S )/ \,-/:.- v and separate sol utions 
for the I.udfcr.!"d leyer may r,e :tourxd whicll. mat~h with v1 and ,,2 as X--:a> ±;o0 Thus j without loss c.f g,ane::t'~l.ity 9 we may ass-ame that it- is the bottom 
wall w. hich en r~.;~s ab{u·r};. 11; ). I(f f m~td e11~ '(o~;t)y J~e problem of mat©h.~ng with ~;' ') 0 Let ~ - -:c..~...:::. t ! .(,j G~t... I h-J, r{+ ·- -.f'tt • r O ;;,t I~), am. put Ft ( 0) = L l = f., ~ -o,. ;r=-.,.,0+ 
· J 
,- ·:=-'·. ( -·Y o ,. - •, \. 
The lower wall i s then g:1.ven by Y "" 0 in the layt"',r o The boundar0y 
eondit.ions on v(X 9Y) in the layer are then 
u--(r<ltJ - 0 I 
'1J- (X 1 u) R 
1..... x.' :> 0 
- r;,;. + 1 
( : ... y ) ') " x -> - <._,,0 , I l,.,,,..r,.. 
in order to :rna:tc:h with v,, o 
"'-
The problem for v may be further di vided by wr-iting 
1- \ :f: 
r<"" 1 ;_r 
f 
where ·L. I I 
\. 
-" I \T 
and 
-
.• ;li• 
,. \ <.,· l, 
-+ --· a )i, ~:, a "('- , 
The solution to (2o 7 o 31 ) may be found by separating the variables 
and us :ing a Fourier series solution first making the further si mplicati on~ 
.i.-
·- Y + li ' for X > 0 
'\ 
for X < 0 I I 
I 
D,'6 - 'I- c, ,2.. 
/ 
I (~, 
I 
I 
I 
f ( 
bf 3(o.) .'{9,(~(\,h0t tf_:_ ..u: _~lt, X ,~ i\Je, 
-· Lud.,~J_J~\,~~(1_cJ VO[~DV1$_ v.olMes_ u ~- '( . 
I 
! 11 
' I 
I 
I I 
I 
I 
, I 
' 
I I 
I 
i 
I 
I Ii 
I 
I 
I I: 
11 
, I 
l 
I 
X I 
• b-1, 
~'I 
-, 
3 5 I 
I 
I 
I I 
I I 
* +-Then these t wo solutions for IT) and LT -+ are found by matching If) ,f.. o"- u-"' 
2. ~ · and ~
-;_· at X = 0. The s elution is : d-;... (.) . 
·::~"~ tor X > 0 :- ). 
v~= 1-Y- 1 '--
- n~ l 
tor X < 0: lc,<::, f )n+I (fl1TY,6 )< \ \-! 
• ( r. l ) J. :.- g_ _::__;;:- e. ~ li ll l l -Y ) IT j l"H\ 
"" .... , + we see that these series for ~\r and 2)2...j" /a--/~ ~re divergent at X = o, d~ 
' 
and therefore our method involves matching divergent serieso However 9 the series are convergent forl XI = G for (if:: as small as we like., , Therefore if we equate ?s'J'""'/cYt-.'l- at x = = t to d;,...Li~X"vat x = + 6 ~ our method is legitimate. 
These series for u-+ may also be deriv~.d using a Fourier Integral, as Leibovich has shown, (Hunt & Leibovieh ( 1967)) and then the difficulty ot matching diverging series is avoided. Graphs of' \.r.y and f-Y- t.r~ against X are plotted in figure 2o9-; the discussion of the graphs is left to the conclusion of §207• 
2. 7. 5. Hartmann boundary 11!1.yers. 
We turn now to the boundary 11!1.yers, B, and shCM how to calculate tM !low there to the same order of accuracy as in the core. Consider the non=dimensional equations . (2,,7,,7) written in terms of the s, n, z 
-
co-ordinate shown in figure 2.10 so that U.. is parallel to the wall and lr is normal to the wall, and the maerietic field is at an angle ex. to the norma:}. of the walL We have; 
LA. d \A I u-- d :c-. -- - d p - N OS:;. o( c E O + ~ os-sd- - u- ~/\, .)_) t Rl Q?.. IA, -r -=-r- 207.33. d& C)'V\ c>e, 
~ -)f - "'l~ o1.[E0 + 0:.~o<:- u--iw.d. )+l \1 2 er 2 .. 7.31i, .. dS . f<: r 
I 
dij2Jo_ UohJ:lfilLfc1\fe, b=d~ !"ju ~ _W),_ j2,.7,5 , 
i/ ////// 1 
J, 
t Bo 
~ 
. ~ 
I [ / 0 -r . I (, ll H-1) ( 0 14 0 
L! ! l l/ 1 11 1 
----1~ -- i---;o/ 
Ti I / I I II I / I /)-s 
/~ 
, -('- , --,v 
I 
II (_aj ___ ~!j~~~ _dy·~b~.u~~rL 
\'t _ ~ '(U~'l)'tu lM . 0... 
'jhl .. 'f't.!\JleJ dM ct __ o± _ ;;('.,.= Q) J. =-4§ ~ 110/ 1//I 
,1 
These equations rr:e.y be simplified by ignoring terms of order, [, , 
where~ ( .t:..<.. 1), ii the non-dimensiona~bot1::.da}7 layer thicknesso 
Then if' we wri t13 J = n/ £ 9 and V = lY' .i ,) ~ , the above equations 
become : I.Ad~+ V ~ __ ·-· ~ _ N C\\S ~ (Ee ,+~). Ui c</.. ) 
dj c) J o5 ~ ;_- b t-1 --·~ + b( 1 N) $iR c>\:2 
I -' 
0 .=~ -~ () (~ 1 N) + 6 ( R- j 
- c)j 
~+~ - 0 
os c)~ 
As with the 1Iudforc:f I layer, the structure of this boundary layer also 
depends pn the relative sizes of M and R. In this case if N(~( /R)>) 1 
there is only one possible type of 'boundary layer i.e. one in which the 
electromagnetic and viscous terms are very much greater than the inertial 
tenns and balance each other o H~ce it follows from (2o 7 .,36) that 
! <" 2. 0) -1 
N== olo, ''" 
( )·-' or f; 
1 
= 6 N R. :i 2., 7 ,,37. 
For the boundary layer thickness to be small compared with the duct 
width , D1 must be small, or M >> 1 .. 
With this approximation md, using (2. 7 .37) we can obtain the 
zeroth order solution for LA , 'CAD, wiich satisfies 
l 1-f - c.-s.t.(E. 0 l-;,. o <'..<!;.() t ~~f o. 
No o)~ 6 = 
and the no slip condition at the wall. The solution is 
:::. \.A (1-.e-j~.,,z) 
O c,,<) \.A t.::, 
where I.A. 0 = is the component of the zeroth order core velocity parallel 
to the wall (Stewartson 1960)0 
The higher order approximations to \A , u-· and I depend on the 
relative ma.gni tu de of M and N, (~~ /R) , since IA , u- and t may be 
~1 =1 
expressed as asymptotic expansions in M , N , or in a comb nation of 
powers of M-1 and N-1 • Let us consider the · two limiting cases when 
N-1 -1 -1 . -1 >'.> M and M ->> N and the expansion may be written~ 
1/\ ;::. LA. 0 
\J" = tro 
f ::c f,J 
r . ( N-1 ;-, , , ; _z. . ) ,.. -t 
l f + \·· , -t- I~ \-~ i " . --1--- / M a , i · l D 1V\ :::J. ,f'\ 1 . I W\ ~ I · -
where the expansion in either the square or round brackets vanish in the 
Th . th f. t . N=·J .,f=i th · two cases. · en, in e irs case, i .. e. ..;>) £( i> e expansion can 
only proceeed until N-r '""'' M-1 for some r, at which point it must either 
..;r =S be terminated or a new mixed expansion of the form N M must be 
considered. We may note that in this case M and R have the same relative 
magnitudes as in our ~alysis of the Ludford layer, Leo 
R2 -<.(<.. M ~ Rz. 
' 
which is a. condition satisfied in many experiments. Also in the first 
case it is important to realise that the higher order approximations may 
be matched to those in the core. 
= 'l =1 In the second case, i.e. M · ~':> N , M ...:;,':. R, t.11.e expansion is carried 
out in terms of ~c 1 , or equivalently sl which means that at the wall t he 
core velocity is not regarded as parallel to it. 'i'herefor e the core 
v-elocity also has to be expressed as a series in 6 1 and has to be m~tched 
to the bounda:r7 layer solution in such a way t hat the core velocity 
ce!:ses to be independent of the boundary layer now. We i gnore this 
expansion siri.ce it :i..s of no pract:i_ci!i,l use and concentrate on. the first 
case. 
We .fi rst 1:i nd L-lt.. . , using 
- .11'1. 
the zeroth order solution (2.7 039) fl 
LA 1"' s~tisfie~ / 5 _ ) ~ l> ~V-c + i>L.)~O 1-f ~ &.) 
OS ~ ,j \ O OS 
) 
To find ~'"°and V. e1o0 we use the results of §20 7 o3. noting that 
Li~· .::. u O errs .. ,,,:t.. + 1.1b 1:v-J11. cl... , 
--A i'°": - 1 , ~ .,(_ + \j'\ .(JA,t-._ ,/... > &!JU V'- I ..e - V\ I .. 
where ~ .J = Fb1 and \AD , u-0 ,, Up Vp a.re a.s defined in §2~ 7 o3o 
In principle, higher order terms in the expression f or U may be 
found, since only linear equations need be solved. The algebra is 
compli ca:t, ed, however. 
2. 7 .6,. Example g flow through a straight-'W'alled converging and 
diverging ducts. 
We now eonsider the flow through a simple duct a.s an example. 
Let i t have walls at: 
,....,, ;,-..,. <!'.. "' ( -.., ) '""" ~ y = ± 1 tor x -;, 0 and y = ± 1 + x ta.n o( for x <. Ov 
where the upper inequality is appropriate for the diverging duct and the 
lower for the conv-erg:tng. 
Then the zeroth order solution outside the Ludford layers is 
"' '.C.. 
x > o core; tlL.o'- / u-'I) = a l 
op0 / 02- = - N(l+~e), J 
boundary layer; \.\A,
0 
= l - .,e.. - j 
i ~ 0 core~ \J\ = I - \ lr o -:=: ~ ~ L ~) "0 
. ~ 2.-( I+ x. 4-c-,"V> ~) L-l H ~c C 
Oi),_;~;'-, - - N r-~-) + E.Dl J Iv v ..,._, - • L2.{ I+ ~ 11'MII tl-.. ~ 
~ ·~·. 
JJJ-1 I I I I I I Ll I I 
1·;·11/'J 
__ I ; ,. ----· . 
-1 1Tll l 77 I I 77111 
~ 2 . II. (b)~_y eJ~(~ _di,,J;;,b "'ki, ivi ik _ cme. 4~ ~" ~''.:9-
~.w .IV\ (A &~_#-l ... ~ue.& _ 4vJ:Xat._.X ~-- a c1_ = -45 b 
Us C.<~-= ±a O JO ) 
ol::=. ±+s O 
3 . 4 
Usl )t>Y :x:..:::. -a >J. ::::. --45 o 
112-· YJ,~~-4-_us,, __ a,nl _y$,1o_~- ~ l<'lju -
Ntier(. ___ .x,.~ _ ± a ~A-=- _ :±4-s () ~-( b.l~ __ _fuat .. u 5 ~ __ U_s_c:, +_J~:A.: ~-Us-r_t.:uY\,,J . --- . 
--!.1.-Lwkre.._ n jsifliG __ _ 0:,- Qrd_~ __ n_ru_mal__jE __ $f_. __ .t~_:_)__~ 
""I - . 
where f or j .> ( \ -' ·)C, :~ () 
for " ' ~ <'.:.._ () 
--t 
.,,,:,. y· 
..t~ 0 ·-~._...,. 
for -, :> :.1 ::-, ...... ~ 
,,..,, 
,;;, 
I .::L < 0 
for !",,. 
-.( D t; I 
.-.,, 
"' 
. 0 ,._1",.., .: ... _ 
When x ~ 0 the .first order solution is~ 
r-.-. I -~ 1,... i·Ji,;. - c1,- - C),l;;> ,-\£ - 0 
.\.~~ - 1 - ,, - -
(AD. higher orders are also zero). 
When x ~ 0 the first order solution is~ 
I '3 ( - , ,. c , -- \' ') 
• 1 ::::. - ~ /\ ~- , - ' i ···, ' I J-- , , i ' ,. ', ""'. J .' ,:·, ." . \.i.. f --/I ;> · . \ 
'-'\ I - -. I ( '' -::~ l . .-; ... j \ ::-- ) \ .- - ~· .. --~··:~ - ' ..._. .._,s-:-_.:_ __ - r J ._,, f 
..:, c,:,;-::, _,,,.... i + _,_ '- L'·", -- ~ ,,1 i , 1 t < . , - 1 1 · , ... , ... ";). , "' t , • t I. \ ·A ~- '~ :;JIV,.- v ,:-t_ ; 
In figure 2o 11 we show velocity profiles in the core for flow in 
diverging and converging ducts 9 i. e . positive and negative c,l. 9 and in 
figure 2~ 12 we show velocity profiles for the components of velocity 
parallel to the wall~ 1.A ~ in the boundary layerso 
a) Q_gre and boundary la;yers. 
The e.xample presented in §2. 7. 6 reveals s ome of the effects of 
considering highe r order terms in the core and boundary layer flows o 
Although the zeroth order approximations for the core flow are identical 
in converging and di verging ducts (except for direction 9 of course) 51 the 
first order approxi.rmtions di1'.fer f and in a ourprising way~ in that" for 
a given value of x 9 the core velocity in a straight w&Lled dive~ng duct, 
such as that considered in §2., 7. 6 is greatez- near the walls and least in 
the centre." whereas for flow in a converging duct the reverse is t:rueo 
=49= 
S eems to be no obvious physical explanation fo r trds effect .. which There 
, 
only occurs in ~l~rtain types of duct since 9 if i:11 e duet width is proper-
/ ( ~ "" '\ ·~ <ii . N O +·h 1 ° !., 0 t t O tl t tional to 1 1 =:it , i or , > x:::,, · ~ _;_ e ve rre:1."L,y is grea es in 1e een re 
for a diverging duct and 1e1;~st in a co:rrverging ducto Thus we cone lude 
that the first and 9 presumably 9 higher approximations to the velocity 
profiles are very sensitive t o the rate of' change of the duct width with 
distance along it. 
It i s of interest to compare the values o.f , ,\ c and U I in our 
example of §2. 7. 6 in order to calculate the value of N which enables the 
required condition.11 N=\....i, << u (f to b? satisfied . For example, when 
cJ.-- = 45° 1 x = 1 and y = 0 9 & = Yt2 9 so that ll even if N is as low as 5, I.Ao the condit i ons for the ana l ysis of t he core wuld be well satisfied. 
On the other hand 9 for the analysis of the Ludford layers we must satisfy 
the condit ion that N1/ 3 >.> 1 so that in an experiment where N ~ 10p sa.y P 
the experimental core flow would be adequately des cribed by our theory 
but no t the experimental Ludford layerso 
Figure 2. 12 indicates how inertial effects be·come apparent in the 
H&rtmann boundary layers when the first order approximation is considered, 
so that 9 when the core flow is decelerating as in a. divergl.ng duct 9 
t here is a. slight tendency f'or back flow to develop near the wallj 
'-lb areas when the eore f'low is accelerating P the flOW' near the wall is 
f&:stero It is interesting that the tendency for back flow to develop 
in a diverging duct is very mu©h grea.ter when l/1. trfa L.. c , as i n a duct 
whose 'Width is pro:p,5rtiona.1 to 1/(1=5t) 9 than when u1 )o , as in the <>c en..~le of §20 7. 6 ·?-rliich indicates that the f i rst order a.pproxima tion of 
the core flow has an important effect on the boundary :!Ayer flowo 
b) ~.dford la.yers and the relation between~ . Ludford I s solution 
and the duct f l ow problems. 
We have considered the structure of the Lud±~ord layer when the core 
flow is continuous in U
0 and 1'fuen the predominant forces are pressu~e 9 
inertial and electro=magnetic !i the criteria to be satisfied by M and R 
for ou r analysis being 
2o'7 049 .. 
The key to 
a physical tmdersta..riding of t he le;9'er lies i!~ the 1•ole of the pressure 
gradients, the JJ'.i:''SSS'l.11"9 gradient. in the y=direction 9 ~/ay.9 is O(:l'r1) in 
·1 ° th T",.:it~ -d - - · •t •. O(N1/ 3) b •t. th the core mi e L."1. ·- e .1..,u~_._c,r. _Lay~r 1 is . j ecause :i is e 
pressure gradient which accelera.tes the f.luid in the y=direction 9 not.9 
ot c:c~1r,se 9 t he electromagnetic force. Since tfu.e pressure varies in the 
· 2/3 y-cli.rectionjl there must be a. c omponent of dp/cbc of O(N ) which also 
varies in the y=directionj) Le. different from the core value of dp/dx = 
o(N) and this secondary component of dp/dx is balanced by thej-,(B force 
prod~ced by a perturbation velocity U of O(N=i/3). The practical signi-
ficance of the pressure gradient is that 1 since pressures are measured 
more easily than velocities 9 probably the best way to confirm the exist= 
ence of Ludford layers is to check vhether the pressure difference across 
an as:Yl!!!J!:etric cbarmel a.t a point where the wall slope changes suddenly is 
O(N173): 
Note -that the graphs of t.rt- and ( I = Y = t/' ) shown :in figure 206 0 can 
be interpreted directly since U-,.._i (= U-/ ~)_ is proportional to v when 
)t_ .... 0 9 that is 9 f'or a straight duct joining a diverging duct, and 
( I = Y = v-i (g,,e lF/ ~J is proportional to v when R+= 0 j that is 9 for a 
converging duet joining a straight ducto F'rom v-\nd ( I ="1' = Lr*) we cm 
calculate v for the general case in which k- + and k. _ a.re both non=zeroo 
Also riote that the damped wa.ve 9 for which there is no obvious explan~tion, 
, !lways oic:curs do1'mstrerun of any change in the duct wall o 
Our 1:111alycJis has been for two=dimensionu flows, but since 
experiments have to be performed in ±inite sized ducts the effects of the 
side walls parallt=)l to the fields must. be considered o Also it is only 
by considering the side walls that we can determine Ezo These walls 
may be non-conducting 9 or~ if eonducting 1 they may be split up into 
segments. They may also diverge in the z=d:irect.iono In these cases 
Ez may vary in the :x: di :rection and Ex is likely to be non=zero 9 in 
which case secondary flows may result.9 and our analysis will not hold 
e:x:cept perhaps in the centr~ of the duct away from the side wallso 
However 9 our analysis is expected to be most applicable in l'i. duct wit,h 
continuous conducting walls pa!!allel to B0 since then Ez will be uniform 
in the core and Ex = O. Ez will then be determined by considering 
the external eleetrical circuit am. the total current leaving the duct.. 
Even in this case the analysis 1,dll fai..l where the conducting-electrode 
walls end at t ... h.e edge of the power extraction or injection region. 
A preliminary analysis of the effect on compressible duct flows 
of very strong magnetic fields was given a.t the conference on MHD Power 
Generation at Salzburg. (Hunt, 1966b). 
-52-
3. Some electrically driven flows in :magnetohydrodyna.micso 
3~ 1. Introduction am surrunacr . 
A corrimon feature of many magnetohydrodynamic nows where the 
magnetic field strength is very high is the existence of narrow regions 
extending in the direction of the magnetic field across which cH..scont-
inuities in vel.ocity, electric potential at' current density occur. The 
universality of such regions was first hinted at by Braginski5- (1960) in 
examining the Jl'lHD equations, since when many specific flows have been 
analysed in which such regions have been foo.nd to occur e.g. the various 
•wakes I which occur in the flow over bodies placed in transverse and 
para],lel m!.f:'1letic fields, (Hasimoto, 1960, Ludford, 1961, and Childress, 
1963) . Although Braginskii himself outlined the possibility of such 
regions being caused by sudden qha11ge~ in the electrical boundary 
comitions, he did not ana.J,.yse any particular physical situation so ~s 
to conclusively dem~mst.rate the existence of such a. layer. However, 
var:i.ou~ analyses have recently been made of such situations and since 
they are not widely }mown it is pertinent to briefly describe them. 
Yakubenko (1963) examined the pressure-driven, laminar, incompres-
f',l- \/ sible i'low of a unifonnly conducting T luid in a. rectangular duct whose >. 
walls perpendicular to the magnetic field, BB, a.re ver:v much longer than 
the walls parallel to the magnetic field AA, and whose walls BB are perfect'.cy-
conducting for x<: 0 and non-conducting for x> o. Then, when the 
Hartmann number M :;:,) 1 , the veloei ty e:x:pre ssed in terms of the pressure 
gradient is O(r.f'2 ) when x .::::_ 0 and O(M'""'1) when x > O, so that some shear 
layer IlD.lSt exist near x = O. Although Yakubenko obtained an ex.a.et 
solution to t}ds problem by means of the Wiener-Hopf technique he did not 
interpret the result physically nor did he produce any numerical data .. 
Waechte (1966) has recently analysed the flow in the same long d\lct 
in which there is no pressure gradient, the walls AA a.re non-conducting, 
the wall B at y=a, for x < 0 is perfectly conducting and held at a. 
potential cp 0 , the wall B at y = -a for x.c::::: O is also perfectly concmcting 
but held a.t a potential - rp
0
, a.nd both the walls BB are non-conducting 
for x > o. In this case there is no now in the core men x< 0 and 
therefore no discontinuity in the velocity. However, there is a 
11 
discontinuity at .x = 0 in d. q:> ;'ay 9 'Whi©h necessitates the existence of a 
1.ayer at x = 0 i n 11,ihi it::h the velo('.: ity is non=ze:r-o o Such a layer" was 
first discussed by Moffatt ('1 961-1,,) who exa.m::1.r.ti?- d the case 1'here the wall at 
1 == a is perfectly conducting a...ri d held at a potmti aJ. (po for x > 0 and 
for x <: O i. s also perfectly conducting but held at zero potentiaL There 
has te be an infinitely s mall -insulatin.g segment of wall at x :::,, 0., The 
wall at y = =a is perfectly conducting ,and held ~t '3ero potenticd; Again 
in this case there is a layer at x = 0 9 through which d <p fC):y is disconti n= 
uous and in which the veloc:j'..ty is non=zeroo Moffatt discussed in detail 
the physics of ruch a layer 9 t~ugh wh{cj.1 there is a discontinuity in 
the eleet:ric field parallel to it, so that we now have a. clear physica.l 
picture of mat to expect ...hen such a discontinuity occurs. 
However 1 there were some anomalies in his mathematical solution 
which Wa.eehte ( 1966) has nCIW' eJarif'ied-o 
Alty (1966) examined a..'1 altogether more difficult problem, he ~ 
undertook ltheo:retical a,nd exPeriJr1enta.J-. investigation of the pressure 
driven flow in a square duct~ two of whose walls are highly eonducting 
arrl two non=eonduc:ting 9 when a unifonn :magnetic field is imposed at &i 
arbitrary angle to the walls o By only cot:1sidering the flew wh.en M:::,> 1 
by dividing the: .flow up into various regions 9 which he investigated in 
tum 9 arid by using some of the resultw of' Moffatt 1s (196.li.) analysis he 
was able to provide an approxi:rrate asymptotie analysis in which he dis= 
covered the e.rlstence of thin layers emanating from the corners of the 
duct in the dirsction. o.f the magpetic fieldo In these layers the vel= 
ocity and ele ~t1:·ic field changed d:iscontinuou sl y v in a similar way to 
the layers of' Yakubenko and Moffatt o The existence of these layers was 
confirmed by the experiments 9 though indirectly from pressure and 
electric potential measurements at the walls 9 no probes being inserted 
into the flow O 
The main i nterest in these studies has been on the curious layers wi.~ ,.h 
•l.l.cc. emanate in the direction . of the magnetic field from the places 
where the eonductivity changes. In ea©h case dif'ferent layers are found; 
yetv de&pi.te their similarities 9 a complete analysis and description of 
these layers in pressure or electrically driven flows is still awaitedo 
=54= 
I 
I 
11 
II I 
II 
11 
I 
i 
11 
The mathematical difficul~y is similar to that of amlysing MHD duct 
nows in that t,!:!"_ <r:~oupled linear partial dit°fe:r,'='ntial equations of second 
order nn1st be solv0d (eqµations (2 o4) and (2o 5) of H & S) o These 
equations may be decoupled by increasing their order as shown by 
B~ginskii ( i 960 \ though in that case the boundary conditions for the 
various parameters Y e,, go ·ve loo ity and potential ~ then beeo:rre coupled, 
this being the method of Moffatt and Waechteo This method is only 
suitable for the simplest boundary conditions and '.:ypes of boundary. 
The other approach to solving the cou,pled equations is to add and 
subtract them~ as originally performed by Shercliff (1953)l1 and as we 
did in H & So Then, provided the current distribution along the 
boundary .is spedfied 11 we can obtain a soluti-onll the problem becoming the 
transformation of the current bou:ri:Elary condition to that requiredll e .. g .. 
the specifi~ati.on of the potential 9 or matching to a finitely conducting 
ele~trode 9 which in general requires the solution of an integral 
equation 9 one S"cwh be:L.'lg that solved in §2 of H & So The great 
advantage of this met hod 9 particularly when M~ 1 9 is that one is dealing 
ld. th an ellipti c second order equation whose asymptotic properties are 
fairly well understood.a In t his chapter we adopt the latter approach 
(suggested to me by Prnfessor Shercliff) to examine the flow produced by 
various electrode eonfigurations. 
I."1 ~3o2 we examine the simplest situation in mich two line 
electrodes are plarced opposite and parallel to each -0ther in parallel 
non=e:ondu.c ting planesI an electric current travels between the 
electrodes a.nd a magnetiic f'ield is applied perpendicular to the planes. 
Assuming that the flow is laminar 9 uniform and incompressible we find an 
exact solution for a rbitrary values of M a.1d an asymptotic solution when 
M ~ 1 o · We show these are identical when M >> 1 9 and how the results may 
be interpreted in physical. termso We then analyse the now when the 
electrodes are displaced relative to each other, the magnetic field 
remaining i n the same direction; this flow is similar to that discussed 
by Alty ( 1966) .. 
In §3.,3 we ari..alyse the flows da.e to circular electrodeso We first 
analys e the flow due to point electrodes plaeed in non=conducting planes 
OpPosite each other, be.fore analysing the flow due to finite circular 
. . \ (3o.) I J ,.5i 
--4-
1 
I (1) 
I l-(2.b)_ I 
I /fob)\~ I 
/ /// // ~l/7/77 
I 
I 
I 
electrodes. We consider two cases mere the current distribution at the 
electrodes is constant and where the potential ('.)f the electrodes is 
constantf our anal:v-sis in the latter case not being complete. Many of' 
the salient physical phenomena f'ound from this analysis were shown to 
exist i n the experl.ments described in chapter 7 o 
,:i 2 'I\..r0=dimensio1:IB.l electrode configurations. ;;o o 
--
-
We conm. der the steady now of an incompressible fluid wi tb uniform 
properties driven by the interaction of imposed electric currents and a 
unifc,rm 9 tran sverse magnetic field. In ·this section we consider two= 
dimensional situations, in which all the physical variables j including 
pressure~ and the boundary condit ions are functions of x and y only . 
Therefore any external circuit connected to the conducting walls of the 
duet is continuous a:nd unvarying in the z direction. ( 'I'his condition may 
be relaxed if the magnetic field due to the applied currents is small 
compared to the imposed magnetic field.) We can apply the uniqueness 
theoran of §203 to this situationj the only difference being that dp/dz 
,:, Oo Therefore P if we can construct a solution consistent with the 
boundary ©tmd:itionsj it. :Ls the correct one. We will assume that there 
is only one compon~t of velocity (in the z=direct~on) and since this 
assumption provides a solution we are justified in making it.. Then 9 
using the axes define d. in figure 3 .. 1 9 the equations descr•ibing such flows 
are the same a s these of MHD duct flow but with dp/dz = o. Jx.- == CS ( _, ?>r;;>~x.. - v-z! Bo) ) j ~ == 6" ( - drj,(a}j) t 
~jx./dJC- t djj/Q_j =- Q I 
d++~/dJ j ~ == - a·l-t~/dx.. 
3.2 .. ·1 ,, 
o - -~ (F + f~HJ~) 
(:) .:)(.. 
O " -; ( f -t JA k.,}2.) 
~ . ~ + ~ (c/uz;:; + d.~lrr-.) 0 - J )C.. to c. '?;:c..,. d_~ 2.- 3.2.6. 
I 
11 
I 
I 
11 
11 
11 
We can ignore equati ons (3o2o.4) and (3o2o5) since we do not consider 
free surfaces and ·we can r ewrite the r~st of th. ':' equations to give two 
coupl ed second order parlial dH'fe:remial equations in V-..?= and Hz o By 
normalising in terms of some reference value of H ll H1 say, such thatv 
v- ... u-~ I ( H I ,/~-;l ) I z 
h = ++cl:- I ~ I I 
and 5 = x/a 9 "'L = y/a, where a is some charact eristic length, 
then the governing equations become~ 
62- v- ·+ d z._ l.T + f\/1 ?.>Yl 
6 J ?. d -.le- -~) '"1~ 
+ 2<,h ~ Md~: __ 
~ ·(';:-
C"-. ., 
0 
1 
where M = B
0
a( ts"' / 1_, )2 ,. is the Hartmann number 9 We can rewrite these 
equations in terms of' Xll = 1;-+ h_ 1 and Y ~ = IT = k , by adding and sub-
tracting them as follows g 
(£ . + 6 .l- )X t M ?)'A ~g~ a~ ~i 
-~ -+ ~ Wt M dt ( ?:/... "a'- ) and dj )- oi , I , ~ 
3o2o2o Alined line electrodeso 
= oj) 
= Oo 
We now anal;rze the flow between bx, walls at y = ± a induced by a 
current I per unit. length in the z- direction entering the fluid at a 
line electrode (ioeo one of vanishingly small width in the x- direction) 
at x = 0 9 y = + a 9 and leaving the flµid at x = O, y = -ao A magneti~ 
field is imposed in the y=dire ctiono Let H1 = I/2 ~ then the boundary 
condi tions are g 
'.r+ °',) l :~ ~ (::> µ~ .:.. -H x. > () \' 
- 0 h - I 
~> 0 1 
~- ~I J 3.,2" 11 0 l ;~ ;: 01 4-i!- -=- - H ,)L< 0 1 J-=--°'-1 I 
- 0 , ~ =- -I ) !< 0\, ~ ~ ·1- I 
We can~ these boundary conditions in terms of X as 
=57-
1' = I j > 0 
5> 0 
1_, =· -t-
¥1_ 
-t 
-
This solut.i.on lo r X is convergent when 5 = 0 and therefore . we match 
convergent series a:t ~ = Oo Also the solution is valid for all values of M. 
Asymptotic solution for J.arga Mo 
As M ~ oo ~ ·the flow may be_ examined separately in certain regions 
(see fig o3 .2) o We examine these regions in turn making approximations 
in each . Showing ·that a solution exists consistent with the approxi"" 
mations and the boundary conditions justifies the approximations. In 
this case we can also show that the asymptotic solut:i. on is equal to the 
exact solution for large values of M by comparing values of X computed 
for various points. 
c= -t-
-J. 
r'l_ I -
J. j = (j + t j ii ) ~j; jt ) Aj 2 ::~ t!\'64 1-·°'j ~ , ~ J 2- = J\/l 24-t ~ ~ 
This solution :to r X is convergent when 5 = 0 and therefore . we match 
convergent series at ~ = O.. Also the solution is valid for all values of M., 
~pt otic solution for larzy M., 
· As M ~ a.o , the flow may be_ examined separately in certain regions 
(see figo3.2)., We examine these regions i:n tu.m making approximations 
in each. Showing that a solution exists consistent ·with the approxi.aa 
mations and the boundary conditions justifies the approximations . In 
this case we can also show that the asymptotic solut:i. on is equal to the 
exact solution for large values of M by comparing values of X computed 
for various points. 
I 
ii 
I 
II 
11 I 
\r t 
·~ 
,3 
-2 -I 
i 1,-
I 
--·2. 
I 
I -·4 
J 
--·s 
\ 
~3.3. A\i1\~ l'1V\e., ~\ectr-ode.s : _______ _ 
'le-lo~§- pi-o~~-jn_ ,~i~_~J_ al; ~: ·5 
onJ_fld_(o'A-ts:de ~ ksuund°'-!j \~e.tj 
T~_Ji<iltid li"e. ·,_V\~co.,te& _ h ()~--i~. 
v_do~~~r.i~ ~ ~3'611\ ~) ___ _ 
where t,.t-i e suffix refers t,o the value of X in region ( 1). 
Region~ (2a) and (2bL 
- Here o/a-rt = O(M) and «1fe5 = O(M~).. Hence ~/o-ri_~a/a,~ and equati.ans 
(3 .2. 9) become g 
?}- X. -t {V\ d 'A. =- 0 ~ dyt 
The boundary conditions for (2a) are~ 
X = 1 for X > 0 9 = 1 
X = = 1 fo:r> X < 0 9 = i 
x~ X i(vi= 1) as ;(1- ~ )M~d:O 
Hence X2a = 1 for X> O 
= =1 for X<O 
since X1 satis.fie s the boundary condi tionrs of x2a . at 1_ = 1 • 
The boundary c:o:ndi t ions :tor ( 2b) are ~ 
X = 1 for X> 0,.41 = =1 
X ,.,, =1 for X< 0 9 vt = =1 
X ~ x1 (u; = =1' ast(1 + ii )~~co. 
Hence XZb = e-MLl+ 1\ rf (~ D~-e.-Nl (i+1 for X> o, 
=-e.-"('"'"~frF(.~)]t-e-M(i+"')J for x<o. 3.2.17, 
.E?zions (3a.) and (3Ql 
We e an see that both x2a and x2b are discontinuous when J = 0 9 
though x1 is not. Hence there must be a region we call (3) near the 
ele ctr,odes in which~ rv ~ and henee the full equation· (3.2. 9) must be 
')3 c)1_, 
=59= 
11 
.1:'/1, 
z~ 
GIA'ITe.rlt I 1\JI\ 
> .I < // , .... / /r/ // / 
_ __/1 1~ 
_ _lo(t1-1) 
\ ' 
~ ]': ~ ~lMn-Ot£ St;(etJ.ttllAJ#!.S b~e.,e.,vt 
loot da..6-ode..s Nh.u M ~) I 
1""/ 
-+ 2., 
I L_L-:~~ ~~/: 
/ / 
-- L--r L 
t . I / --L-. 
--1 --, . 
-~(3) I ' 
k ),j C..... o(M-t) I I , I 
I o(M.,.Jt2J I ~ .x.) ~ I . (/ b) (4) I (foJ I 
~3) 
(2.b) I + il_;o(M-1) I 
I t-
2-b 1~----· 
tj J ,5 . 1)11\Qc.en .JinCJ __ cie.c.ttbdes_sh_llW~--a::JOlft:;te, _r~rcilS 
considered rather thari (3 o2o 15).. Since regions (Ja) and (3b ) a.re 
iJDbedded in (2) .9 bt.."1', do not extend into (i) 9 ~,ltl.ri, = O(M)o Hence 
a(iJ. ! = O(M) and thus these regions extend a dist1;i.nce O(:r.C1) r ound 
the points 3 = 0 9 -Yi_ = ± 'l" Since these regions are small compared to (1) and (2) and do not exert any cont rolling influence, we can ignore 
thetno 
If we compare the values of' X computed fr::i:rr1 the exact solution at 
M = 20jl 40 and those taken from the aeymptotic solution we find that t he 
agreement is clearly good enough to show that these two solutions are 
iderrUcal vm en M~OQ*., In fig., (3.,3) the velocity profiles for two 
value s of 'Yl_ are shown and in fig., (3.,4) the current line s are shown 
schematically o 
The best way of understanding the physical reasons for the 
distribution of velocity and current is by considering what happens to 
the current arrl the velocity when the magnetic field is turned ono When 
there is no magnetic field there is no velocity and a current passes 
between the electrodes the current Spt'eads out from the top electrode 
at least a distance of order 2a before curving back to the bottom 
electrode o Let us consi der the quadrantj> Oi> °1> Omen the magneti~ 
field is applied; the large component o~ ~ ~ aeeele :rates the fluid in 
the -1"?. direction o Howevero as ll= increases u- B increases and thus ' = r o · Jx ~ ~reai3ee " Then 9 sinc~x. B0 decr~ases 9 the acceleration of u-~ 
dec:reases o This process continues untilj is reduced to a value X-
!llfffoient for th~},;;,. B0 force to balance the VJ.scous stresses produced by 
\J~ • 
Thus as we see from figs(3o3) and (3o4) in t he regions (2a) and 
(2b) where t he viscous stresses are greatest 9 Le. O(JY.12 ) i> there is a large 
component of current perpndicular to the magnetic field such that JX~ = 0(~) 
In region (-J) 9 however~ t he viscous stresses are much less, L e ., O(M) i> 
and c:onsequ.ently the current has a smaller component per pendicular to B0 ., 
It is perhaps wrth noting that we can construct a s olution for 
* More recently Pr'Of o Williams has sho·wn analytically that t he asymptotic and exact solutions are identical as M ~oo o (Hunt & Williams» 1967) ., 
electrodes mich have a finite thiekness 9 b, where f; <<. a9 provided 
119 specify the ©tl!'rent distribution on the electrod. eso Then it is e 
easily shown that as c:n·O~ th e sol ution becomes that of the line 
electrodes . 'llierefo:re our solution is a limiting solution of the 
elect r0de thickness ten:iing to zero. 
3.2o'.h Displaced line electrodeso 
We now analyze the flow between tvro walls when the electrodes are 
displaced sideways by a distance 1 2b. See figure (3o5).. If b/a =f.. 
ani H1 = I/2 9 the boundary conditions ar e~ 
v-==C ) h =X= I 5.:::.R.l 
lr:. 0 I 
~ >- '-, 
e 5.i({ I 
s<-t) 
:1=0. , 
j-=-"'-, 
._:j .::. 0. I 
3 .. 20 1 s .. 
there is 
little interest i n doing so o 
solution . 
We move straight on to the asymptotic 
We now conside'.l'." the solution ,ilhen M~ o::.o ., We will assume that 1 Mis large enough t o satisfy the condition that aJ.C2L._<.. bo 'l'hen in 
this situati on there is one new type of region not found in the alined 
ele et rode lroblan . This is the region j (.4) ( see fig., 3 o 5) 1 where 
. - c( wi - ~~) > s > - .l + 0 ( M Y2-J I 
f. - b(IV\-') >"l > - I + 0( i·vr··I) 
1 
in oth er -oords this region lies ~etween the Hartmann lqy-ers on the walls 
and the layers of thickness O(M=2 ) emanating from the electrodeso The 
solut ion in this region is simply 9 
X = =1 
i I 
I 
11 
I I 
O M-~) Lr 
_J.,,~-J----t-------~ " 
;/ 
"' I r 
·f rf~s \ :i I I ---.+-------1-
f~-t 
\:~ 3.6~. VJou~ frof"ks fr £ow- b!.,~ 
tlispl<lc.~d ~ne., dwtrode.s ~ re~{,) ond l 4 )- ~otks ore. sche.VY\ot'c, {!ta>>I) 
~g.(:,b_ C:Uaent stx~s fr £ow ~+wev\. _ 
~Jooui. ltn~_ ~ec-trodes ~htn_ M>> I ~OJekct~  
It\ Y-!j?nL~ is_ aJso __ JtbwV\ J _ 
I 
I I Ii 11 1 
I 
1/ 
/1 
:1 
I 
I I , 
I 
Therefore \.J =- - I 
The solution f or region ( 1a) is 
. r, r1t'.·'5 - e.)' IMJ 1- - ,U'J l2. tiZ 
and in (1b) g 
X = =1 • 
j 
'lhus X does not change in (1b) which is to be expected since X = -1 in 
(4) and X 7 = 1 as j-?) -~ o 
The solution for (2a) is much the same as for the alined line 
electrodes' ioeo 
X=1,5>l o 
- x = =1 .5 ~- l O 
and the solution 
s~-
for ( l -5) JM ~) I 
t t} ~s~- M (1+j uf (LlE-))1- Q,-M(i+,ill 
IT - j 
this becomes g 
X = =1 + 2e=M('l~ 
when 5.c:~R. ~ x = =1o 
. =1) Thus we aga:in must have two regions (3) with thicknei5s O{M near 
the electrodes i n whit:h a/~~ :i.s of the same order as d/<i:b--i o 
, '" t)nJ 1ij l 2 · 6u\.) 
We see from (3o2o19) and (3o2o20)Jtiiat the major difference between 
this case and the aU.J."ied electrode situation is that a net flow is 
inducedo Thfo is simply calculated to be: 
,r-o<:i. ,l.1 f f crd ~ d'l = - 4 R: (1- i;,); 
-;; -· £ . r rlrcJ:c,dj=-2.~ - (1-- ~) . 
'Ih ·- d':J - O'- 11,, e reason for this net flow is that~ since the current must pass between 
the electrodes, and since there can be no current i n the inviscid core 
(region (4)) because the flow is steady and there is no pressure 
gradient 1 all the current has to pass along Hartmann boundary layer's on 
.Se~ f.j l · 6(M 
the two walls 9 a current I/2 along eachoJ ,Now Shercliff (1965) has 
shown that the relation between the total current flowing along a 
Hartmann boundary layer (I/2 in our case) and the velocity outside it 
I 
I 
11 
2o. 
7 / 7 
I l~ I 
l 
~~) __ 
/ / / 
z, s J-, 9 ~ -r-, e 
CV rrent t .I ovt 
r ~,s 
·- ~ 
~e I 
(4) 
(I) 
J 
I I (2b\ 
1-/r3b~I 
2.b 
c
5
) Bo 
__£§_3 !-_'[ _______ ~.!.~!l~ ., ~ -'-'~-~ -<A~ _ -~O<:!e.s 1/\Jlw; qtS ~ ~ pt!>-li~~IO~ 
:I 
I 
I 
iT = ' I/(2 F1) 9 
wtience we can obtain the first term in our . expression for Q., It is iJllp<>rta.nt to note that the first term (3 .. 2022) is independent of the 
value of B
0
;i though if the electrodes were finite such that there was a finite pot ential difference between themj..6. <p , it would be found that the relation between Q and.b f depends on B0 o This result was to be expected from § 3 of H & S where we . examined the flow in a rectangular duct with perfectly conducting walls parallel to the field and showed that for such a flow when there is no pressure gradient the first term in the Q = I relation is independent of B0 i, whereas the Q = Lip relation is 
not. It is also worth noting that the f.i..rst term in (3 .. 2021) , is the 
same as that of the Q = I relation for a rectangular duct with sides 2a 
and 2b which is t o be e:x.-pected since Sherell.ff ffs Hartmann laye:r relation 
shows that the distribul:',ion of current along the lines X = ± b does not 
affect the first term in (302.22) .. 
3. 3. Circular ~~ctrode[. 
~o 11'\h--r,du. tfil>~ 
The disadvantage of studying flows due to Jine electrodes is that 
such flows are difficult to produce exper-imentally. Inevitably at the 
end of the container enc.losing the fluid some recirculation occurs which 
may upset the f.low elsevm.ere. However if circular flows are used there 
are no such end e±'fects, although the flows, being more unstable 9 entail other problems " In this section we examine the theory of flows produced by circular electrodes and thence predict some of the effects found 
experimentally, as shown in chapter 7. 
3. 3.2. !1!ried point electrodes. 
We consider the axisymmetric flow induced between two point 
electrodes set in insulating planes opposite to each other (see fig.3.l)o 11e discoverd in §20 6 how such flmrn induced radial pressure gradients 
which in turn induce radial flow and why, if the magnetic field is strong 
enough, t hese effects may be ignored. We make the same assumption again, 
' I 
1: 
onlY considering 'the azimuthal or sir;d.rl component of velocity f and the 
radial and axial 00:m:ponents of current. T.heY!. 9 in terms of \f;E, and f+-o 9 
the azimuthal compone:-;t.s of velocit y and induced rmgnetic field 9 the 
governing equ.a.tS.ons 
'ciH e 
f =- Bo · -
-' - d:2:_ 
6 -= 13c d 1.re- .+ L (d~ + J.. .cl. - ..L .1-,_~:>- )µ e. d~ 0 c, y :l- t' d 'r )-l- 024 
Let the current entering the electrode on the wall at ~ = =a and 
leaving the electrode on the wall at z = +a, 9 be I 1 then the boundary 
conditions are~ 
-r' l+B ~ T/ 2:1'i 
tr~ =- 0 
U-~ and H9 are continuou1:J. 
at z = ±. a g 
We now non=di.mens:ionalize in terms of I~ 
- a 
- 0 
The solution of (3. 3 .5) subject to the boundary condition (3. 3.6) 
u- + ~ = t-'l ~ fc-1V ol j =h0".12J k ,(AjeJe.- iv;0tiJ 
J~ o L J(J 3. 3. 7. 
(-1)j ~J ~ 11,(i·,11/~ K i(t-0e) e.--\v~ S ~J~.j3J 
1'fuere }A j 
~ - (j + k) rr 1 ~J, = J 11 ) Aj =- ol j 1- fvl 21,q__ ) r j 2- ~ ~ J 2. + M 21t . 
=64= 
I 
I 
11 1 
As we have found before, the asymptotic solution is simple r and 
physically clearer o Dividing the flow into :3 r egions, as in (fig . 3. 2), 
with region (1) l ying between t he electrodes , regions (2a) and (2b ) 
lying on the t wo walls , an d r egions (3) extending a distance O(M""'1) r ound 
the el ecp,:roctes. Then in regions (2a) and (2b), (3.3. 5) becomes: c;?L --,~ f\l\cf )( e (o-+ h)) :- 0 3.3. 8° 
and th e solution in ~2a) is : 
e (u-- + ~l) =- I 
In r egion (i), cl,/oj= o(i\, ~ ~ O(MtJ and e ~ 6 lfvf~) and therefore 'c)e_ (3 0 30 5) becomesg 
(M ~ r -t ~ ( ! ~-) \I e ( u- -t-h)) :::::- o 3.3. 9. , OJ 6 ~ · '- 0 e A- JJ The boundary conditions are: 
and '\\hen j = 1 
Thence 
e( u-+~ ), =;_ I - ~1 1~ \- M t 4/4(i-J)] 
In r egion (2b), the solution to equation (2.3. 7) is: 
t (1i- +ht1o = 1- .e..- MeY(4-(1- $J)f - .e. -Mt,+ .ij 
By considerip.g the symmetry of the flow we see that in (1 ), 
.· ·_· 1_ [ - Me_"'/4(1+ 5') -M eYC4a-sJ'J l.r' = 
.£ 
_ e. 1 3.3.12 .. Ze 
L J_ ( n _ . ~MtY~ (i- f J) -Mrt/46-3)] and " . 2e L ,._ ,:_ 
-e. 3.3.13. 
Though the form of the velocity profile is sim:i.lar to that for the line 
electrode , the important difference is tha.t in this case 
1 V = O(M2), 
I 
I 
I 
I 
,I 
Profo Williams has again sho-wn analytical ly that, as M-.;;,. <:>a ~ this fl asymptotic solution is equivalent to the exact solution (3.3.7L II I Hunt & Williams, 1967). 
I I 
I
I II 
~65-
I I 
I 11 1 
I I 
: I; 
wtiereas !or a line elect:rcx:l e, 
lr =0(1). 
We can also deduce this ~esult by an order of magnitude argument: 
In the region ( 1 ), / B u-: · :::::= arp 2>r O 
e ·= o@> r = o;tJ, where l = O(aM~) is the 
thielmess of region (1) and, s;i.nce at r = . O, ef;. =, O(Ia/U""'il<b2-) 
= O(JJJI/a a ) , 
ITe = o r!M ?<. L-cr-°" 
= Q r- J:: iV\ },~ J 
L-.~ J lr t\.. 
For a line electrode, at x = 0, 
{p = 0 ~ ~fa h] 
atrl thence tr -6 = O [! /Ji,ri_ ] 
where I in this ease is the eurr1;3nt passing between the electrode~ per 
unit length. Thus the different values of v-8 in terms of M result from the much higher potential W1i.ch occurs in the point electrode ease than 
the line electrode case. 
3.3.3. Asymptotic analysis of alined circular electro::ies. 
I 
In this section we examine the flow- . induced by current passing 
between the t~ electrodes show-n in fig. (3.f) men M .>) 1. 'The 
ana~si~ presented is not complete, but even in this incomplete fo:m,. it 
is useful in interpreting the experimental results presented in chapter 
7. We will only exam:Lne the flows when the Hartmann number is :!Arge, 
since the interesting physical effects are then seen most easily both. 
analytically and experiment ally. 
We first analyse the flow when t .he electrodes ~re perfectly 
comucting using the sa.ine non-dimensionaliu:~d parameters as in ~3.3.2. ~ we consider the current I to be given and the electric potential, 
C::i. (p , on the elect~odes to be a depen:lent variable. We have one 
tu.rth~l." parameter, ;if the radius of the electrodes is b and ,l = b/a then 
- 1 
I 
I 
I 
I 
I 
the boundary conditions in non=dimensionalized form are~ 
e - -t I b < 12.. d°h . - 6 L- ==- o ' J- -- JC J -::;:r - I (_ .. .JI 
l' = >e e __::::, 0 ) r :::. o, 
el ~ =, h ::: ye (J -::> 0 
Now in order to solve this problem, following the methcxi used to examine 
the boundary layers on walls 11..A in H & S 9 we specify the value of h on 
the electrodes and then find an equation which this distribution of h 
must satisfy in order thatoh/a! = 0 en the ele~trcxieo Let h =f({:) 
on s = ± 19 e < e .9 where f = 0 '\\hen e = 0 andf = 1/ f when t = t 
The latter condition follows from the r1ecessi ty of h being continuous on 
the wallo 
As before we divide the flow into various regions as shown in figo 
30 S o For this analysis we assume t ~at M is large enough to satisfy 
the two criteriai M>) 1. and Q.~ O(M=2 )o 
Region (!J_o 
! f ( ( c· {) -l ) l ;v· h.1. '\ 6dj= o,), dcc ~o A. _«.ol 1) 
'lheref ore ( 3 /3 o 5 ) becomes ~ 
. . \ 
an::l. therefore 
Therefore 9 
~Je_c _~~ 
oJ 
tr -
and h ~ 
=O, 
0 
f Ce) ~ 
~~i ~ ~ ~(en 
and since fQ is constant at ) = ± 1, it follows ·that~ 
~ ?~ ( e_ 1) =- 2 /\ t l 
-where .A, is somee_oonstant to be dete1111inedo Therefore~ 
Re • 
_gion 
l In this region ( e = f_ ) ~). O(M°"'2 ) and I U- = 0 and h = ~ o We 
{ , as calculated in see that though lr and h mey be continuous at e = 
regions (4) and (5)' dh/c)e . andc)~/ d 5 are discontinuous. Therefore 
regions must exist in wiiehc>.ffoj changes from its value in (4) to zero in (5), these being (1) and (2). 
Regions (_2a) .and (2b). 
- We treat these regions as in §303 .. 2. 2ifeJ = O(M) ,"?J/ae_ = O(Iv~) 
and therefore d/o j >) d/aeso that (3.3.5) becomes: ( }j + Md~ )( o-+ic)-- 0 . 
Thence, in (2a), 
V- -+~ = flt) .fw- t <_ £ I 
== 
1/e e>.e. . 3.3.16. 
Region , ( 1) o 
3.3.18. 
ReJci:on (2b,l. 
-
Using the so lution fo r ( 1) we can now wr ite do1rm the solution for 
where (v + h ) 1 (f = _ 1) is the value of (v _+.! h) 1 when J = -1. Note that 9 
except when f = 09 o (c1r4 .&,.), ff',:;-1)/a~= O(M2) compared withd(o-~A)/cij = o(M) 9 and therefore our solution for this region is consistent with our 
assumption c 
To find f (t:) o 
We now find ED. eqa.ation for 'Whieh . f (t) mu13t be a solution in order 
that e>h /o J ~ 0 at i = ± 1 when e < t O J 
Now ?>h (e , j) =- du u-+kXe1 s)] 
c) f C.) 
and theref'ore 9 since in (2a) 9 
cil~·t l-)/ dj = 0 
and since c;1h/o J = 09 "W.en 5 = ± 1 je-<. e.l) 
ldITv+~XeJ' .~ I 0J l J -=- ·- i =O 
In order to sat,i sfy (3 o3 o20) 9 we use our soli ... 'rtion for region (2b)o 
Thence 9 men p< €
 or ~
1 
<( O 9 
~. 
,~[(~+~h +i )] =-~ rter,')- JIJ~, ff (t)0xrf~-etM/~dr dJ J=-1 l LJ2ci _ 
·~-
-,;(I 
- M;,~-jr t::i er;f l[t-e) ~.!)/~-/de + ofvi-1~ 1 3.30210 2 r 2:if . .c"l. c ~ J Th O U:!19 to f ind f ~)9 we have to solve this integral equation (3o 3. 21) 9 to do whfoh we first need to know the boundary conditions o 
However 9 before determining the boundary conditions on f in the 
I 
regions (2a) and (2b\ we must consider the regions (3) & Sinee , ... hen (< 0 
c>h/oJ := 0 and ·when t'I >O 9J~(): a discontinuity exists at t' = Oj 
which implies that some regio11~ (3a) and (3b) must exi.st in which ~ 
== O ( 0/oe) = O(M). 
i'Tow in the regions (3a) and (3b) for line electrodes, (§3.2. l) 9 h 
changes i'rom +1 to = 1 , so that w:e have to determine whether the change 
in (3) is comparable with that in (2). We now sh.cw that this must be 
the case for t he integral equation (3.3.21) to be satisfied. 
The equation (3.3.21) speeifies :f(e) such that, v.fuen e'-< a in 
regi.on (1), v + h has the same value atj = =1 as at j' = +1. If t he 
change of fin region (3) is negligible, when in region (2) f ='(J+-h\ . 
= Yt when t' = o. l' "~·0 
Therefore the maximum value of (v + h) in region (2) whene < I} and J = 1 i s the :maxi.mum value arzywhere in the plane J = 1 a Since in 
region (1), (v + h) satisfies the heat conduction equation (3.3 .. 17) 9 
the rre.ximum value of v + h . at the boundary between regions ( 1) and (2b) 
is less than that at J = 1. Therefore it is impossible to satisfy the j 
equation (3o 3.21) and t he boundary conditionf = 'lfl when t} = O. 
However9 if the change of f in region (3) is comparable with that in (2), 11 
SO that t he bOTu"idary condition for f in region (2) at e = 0 is~ f = J'A/ t 9 where 111.fi. <.f it follows that it my be 
possible to find values of f'A a:.11d ;\ as well as a solution for f i~ 
region (2) by means (3 o3o21)., 'This result implies that c,f/djis O(M2 ) 
in
1 
(3 ) and ~hus 9 1~ea1J = 1 when e_' = O(M=i), 'o~/axis O(M2 ), mereas for 
e = =O (M~ ) 9°a~/ax. = 0 near J = J. We bel ieve that we_ can find fA. and 
I ~ by invoking t he mi nimum d:s:iipa-tion theorem mentioned by Moffatt ( 1964) 0 
Since the rraterials used for electrodes in experiments are not 
perfect ly conducti ng it wruld be of interest to examine the effects of 
finitel y conducting walls. But, since no analysis has been developed 
for this situation 9 as a first approximation to a finitely conducting 
electrode we make the assumption that the current distribution is uniform 
O'lrer th e electrode. Then 
_f Ce) = e/fl 2i 
an~ dividing up the flow into regions as before,.. we find that in region (4) 
0 
j. 
0 
I 
""-0 
I 
D 
0 ----- · . 
-
0 
!J 
c 
w 
0 
,t .. -
~-
5 
... --
c 
w ~-
/ 
/ 
// 
I 
I 
J 
I 
I 
I / 
. 
0 
0 11 
r:----: ~ (~ I, 
£ 
<; § 
(T 
Lr= o, 
in region (5) 
lT = o, 
in region ( 1) 
h= / l 2-e I 
h= 
- ~ ,~ 
utll\, t 'J.. Lt' 
where ~ e I J/Vl I (2. II-] ) 
(v+ h) for region (2b) can be calculated as beforeo 
It is interesting to see how the electric potential varies along 
the ele ctrode when we make the assumption that the current distribution 
is constant o Using the solution for region (2b) Leo 
V-tt_:; t-+ e~ ~ -1\1\(1+~ I I.&, ___ ,e,r-fe--::!-
€.r~rfe251[1-LM(,+j)7 
{2- t L-i?, efif J J 
where e = t l J~~/ri, 
h./ " p we find o '/ aj at J = ± 1 and hence, 
- di I ==- ~ I ~ J; f ·J J=-l '• 
-== fv'\ fel:~! -~ (t -~e-lur l-r)F 
Therefore!) . t ,t- e e -'J 
l - ,21 ~ c::. ·~_:.. rz _, + (i ?_+ ± Xi +R-1f E:1 + i t e: e t -, j J~, 1 ,(2• l,... ii'~ _ 3o3o23o 
A graph o.(f1 .. ;-L.,.~ .. /:C ~ against t' M! is plotted in fig . (3o 9) and we see 
that the potmtia.l rises from its constant value in region (4) to half 
that value when e = i or t' = Oo Note thatj on j = 1, ~ /s negative 
and th at this change in potentia:i;' occurs in._ a,, distance O (M"""2) , so that, as /lei);(' ,te.. eJ<(<. "1' n,:e a,viC, M increases I the potential across the di.s c becomes more nearly unif o~o 
Thus for the potential across the electrode to be constant it follows 
that the current density near e.= ~ must increase, which we found when 
examining perfectly conducting electrodes. It also shors ~Jhy in a finite 
conductor the distribution of the current density on the electrode or 
i s likely to depend on the value of M. As regards the flow1 the chief 
significance of 'this potfntial rise is that H; i mplies the existence of 
a current of deniity O(M2) p~,rallel t.o the electrode and consequently a. 
velocity of O(M=2 ) outside th e Hartm8Im boundary on the electrode o 
When the potential of the electrode i s constant., although a velocity of 1 
. o(rif'2) exists outside the boundary l ayer on the i nsulator where ~ 1 ~ Og 
there is no current parallel to the dise whene' <!..O and consequently 
there the velocity is zero 9 to first ortler. 
We now show that iff Ce)is a f'ur1ction of e'JM in region (2)9 then i.>i region ( 1) the distribution of ~ for a given valu e of J is similar 
for all values of M, ;:.) 1 . If we let f (t) =- t + FMt :J (~fM) I 
then in region (1) . 0 ( JM)) 
u- + lh - M.:_ [ f ( l + + J 3 i: M P.Jx r (- (t-ei)~/V\ kte 2.Jtr(1-f) 
-t>O t.. tM-.. 4-(1- jJJ 
+ J~ 8 &f. r·· [--(t.-e')L1V\ J d.tJ-
0 {_')., 4(1- f) 
whencep u- -= I 'r,:;;·:.;& [ r~ (z J8-t ) /bfr' (- (l - E-)2) dt 4-L .. 1iA>1 J J 
~ 0 ~ 
-I 9 (ifitl t ) ~r {:-(~ -t+Y) dJ: -f (:z, sq, l ) ,.,,r ('- (t, '-~1)d.t 
... gc(l c;?6 ~ 
+ ( (2~'J ) Mp (- (t -r+Y°) dtJ , where '"'"" I 
e _ ~ e' JM /(2.lT-='j ) , e + ~ e' JM 1/(2 .fi+J) . 
'lhus f'or given j i) trJM is a function of e'JM onlyo (Note that the case 
of uniform current distribution is covered by this analysis)o tJow in 
region ( 1) ~ since'dhfor = O(M-!) and Mir = O(M!)' it follows that~z M 11"' (3.3o24) as WM first ~oted by Mcffatt (196li.)o Thence 
-
I ~ : l~ (M ~av-Jd ( e1 IM ) I 
and t.herefore ~ is a function of (f.fM) only9 for a given value of J 
=72= 
,, 
I 
I 
We use the converse of th~ result in chapter 7 to ~ educe from our 
a,g>arimental resu.l.ts thatJ-(t} is a function of e' JM. 
We now 5how how t he resistance between the two discs may be 
calculated .for an arbitrary distribution of current and why the resistance 
JIIU~ ahmys ~end to the same limiting value as M--?ex>o We define the 
resistance~ Rj' between the two discs as follows. 
R = 2 .64/I 
"1ere b +Cl\, 
2--6(/,= - 11' b,. i [<f> ( "D_°' 2 '"-rd\"'J 
ie the mean value of the potential difference between the tw:> discs. 
Since, 
which may be re;tt·: in .;:. non-d[ima;i5l.1° ~n,aliz•~tsrm a~~ 
1f ob e;,t 3.3026. 
rr h = f Ce) at J = ± 1, then Jn ~~ion c1) near e = 1, we can use the same 
rormu1a t o find c~" + h) in terms or f Ce) , which we rrund in our 
investigati on of perfectly conducting electrodes, namely, 
lJ t- A, = {V\ t (ff (d ""pi+- e'. )"' /,1\,4 ( i - J' ~ d r 
e.f«(I- J) l-~ ' L 
. T 1-~ e-irt"~~~/4(1-.fDdr]. 
Since f = l/ f... when p = £. ,. &e are only considering region ( 1) and not 
comd.der ing any distinction between regions (2) and (3), as in our dis= 
cussion of a perfectly conducting electrode), it follows that when M~ 
v+h = \/{. when t = J... o 
I 
I 
11 
Then, since v is antisymmetric in f , 
+I r' J, ltl~ct ,:1 J = -!., I b+ilt~.i. d J _ 
and from (3.3 . 26) it therefore follows that 
R = R = ?o.. ~ 1r~b' 3.3.27. 
Note tn.at this result only depends on the condition that f =\/(when e = t . 
of R vdll be 
It does not depend on the form of f and therefore the value 
the same whether the electrodes em.it current at a constant 
potential or vd th a constant ~urrent density. 
To calculate R when the current deS?1ity is constant we use (3.3.26) 
whence (t, Q.. J' l /1 ~ 2ri:J )JJ -- -1fob l \ J;r{ M~ I 
-, 
or R_ 
- 'R-~ (1- J.:..ill. r ' ~ ' ) 3.3.215. ,,.t JV\1.--
Thus the form of the current distribution .on the electrodes will only be 
indicated by how the resietance varies with M and not by its ultimate 
value. We expect the form of the variation of R with M to be similar 
for all ty-pes of electroq.e becauee the thickness of the region (1) i! 1 
alwaye O(M""'z) and t..1-ierefore the current ,;iensi ty in the z direction, for 
given I ,, is reduced by 0( f Mk)-1• This implies tha.t
1 
(~/clz) and 
consequently the resistance are also reduced by 0( 1. M2 ) o 
-71.v-
I 
I 
I, 
11 
I 
I 
4• Qn the use of pitot tubes and electric 12.9tential probes in MHD flow~e 
4
0
1 0 Introduction a.'l1.d S1!l!F.l!fil".:Zo 
In fluid. mechanics and MHD W1.en a flow can be theoretically analysed, e.g. Poiseu.ille flow in a tubejl then measurements of presmre 9 electric potential 9 eletltric current etc 9 taken a t the boundaries of the flow, e.g. on the walls of a duct or on the surface of an aerofoil 9 can be compared with those theoretically predicted and 9 i f they are in agreement, the analysis is considered to be verified. However? if no sueh analysi3 ha:!!1 been ma.de 9 then external measurements often give little indication a3 to the nature of the flow J> and in that ease direct, internal 9 measure:rrent5 of velocity J> electric potential etc 9 become necessary. To take such mea~urements we need to use pi tot tubes 9 static pressure probes, electric potential probes}' and hot wire anemometers to name a few. 
In fluid rrechanics we measu:re total and static pressure and velocity wherea!I in MHD we can measure several more variables j e.g. the electric potential and induced magnetic field (Ahl~trom 1964) j so that in principle we have a further check on our measurement a., However.? this advantage is more appnrrent than real in that the d:ii'ficultie s o:f measuring all the:se variables are very much greater in MHD flows, in particular a magnetic field changes the r elation between the velocity and the total pressure measured by a pi"tot tube 9 the velocity and reading of a hot=wire anemo= JEter 9 and the electric potmtial and that measured by an electric potential (eoP•) prohe o In t.lii~ chapter we analyse the way in which these relations are affected for P1tct"and eopo p~obeso(D.G.Malcolm of the University ofWarwick is imrestigating the behaviour of hot=wi:re anemometers) o 
Most of the internal measurements of MHD .flows have been in gases 
'llhere the ma gneti t~ field has not been strong -enough to induce errors in the probesJ> but where the errors 9 particularly with e.p~ probesJ have resulted from electronic and ionic phenomena. 'I'he only investigations or the continuum errors have been made by those interested in liquid metal flowsjl (as we are). Lecocq (1964) developed some of the baaic theory of pi.tot and e 0 p 0 probes whicll we develop further 9 however hi~ eJCpe.riments were sueh that he c:onld not tes·t his theory since the rragnetie 
=75= 
~·-~ .,/... . /i 
_____ ,;_ 
13 4.,,\ . 
-n' j --+z. 
t~~ZJ+-a ttV\ .(1:1v, t1hl -1 11.bc.s l - \ t:u 
f'JL,, h a.J pvcbes ~ 
i 
c; 
I; 
(1: ,1\d 
> 
I 
!"I 
I I 
I : 
I 
1 11 
field effects were so ~mallo He measured yelocity profiles by means of" 
Pi.tot and e oPo pr obe s 1 but did not correlate hi~ results between the 
probes and)l since t he .f'.lows we:r'e turbulent so that no theory for the 
'9locity profiles exists y t he experimental results could not be correlated 
ltith th ooretical resultso East (196li.) performed some experiment~ on 
pitot tubes to determine the MHD errors involved in measuring unifonn 
flows, he did not use the pitot to exanine shear f lows or flows with 
elect:ric currents presento Moreau (1966) has used a, pitot tube to 
examine turbulent velocity profiles mere the MHD probe errors are 
negligibleo 
In §402 we consider Pt.tot tubes in uniform flows 1,here the magnetic 
am electrl,c field!!5 are perpendicular to the flow and analyse in detail 
various 5Pedal probes 9 finding that the MHD error is highly dependent on 
the probe 1:hapeo We then give a physical explanation for the prominent 
effects and discuss some of th e-pl°actical conseqL.ences of our analysi~ o 
In §4o3o we consider e.,.po probe:s~ though in this case we do consider 
the e:tTors of tll!ing sucll probes in shear no~.. We first find that with 
uniform flows there is UQ. probe error 9 owing to the syrrnnetry of the flow 9 
but when t he flows a:re non=umf'orm certain interesting effect~ occuro 
We examine . these analytically and then discuss the physical reason~ for 
the error ~ o 
I:n ~hap ters 7 and 8 we make use of the results of this chapter in 
the int er pretation of our experimental results " 
4o2o Pitot tubes 
4o2o 1,, .N,n=dimen~ional equations. boundary conditionso 
In this section we consider the relation between the pressure at 
the tip of a blurrt body, Leo a Pitot tube, and the velocity of the uni-
form flow impi{r1g on it 9 and ex.amine how this relation is affected by 
the application of a rm.gnetic field to the flowo The three major 
assumptions we make are~ (1) Rm« 1, (2) the velocity to be measured 
is un:i:ro:rm~ (3) the shape of the probe near its tip is symmetric about 
the planes y = o and z :a: o, with the point 0 9 x = y = z = o, being the 
cent:re of the total pressure measuring aperture (See f'igoli-o 1 )o If for 
example the probe is supported by a stem well downstream of the tip 9 we assume its effect. on the :f'.1.ow at th@ tip is ne glig:i.bleo 
In order to non=dimene:ionalize the equations we use a characterl:!!ltic length, d~ 1,11hicl1 rray fo,r exa11ple be equal to a 9 b or c., Then the 
ro11o·11.1ng forms are the most suitable re:;: the non=dimensional val'.'iables g lJ = C--/iL =- ( u}tr!w-)/Ll I § = k /R,,, 60 1 J = j /<,Ld)0 , j; ~ µ/,,U/· I .-1< -...t ,.... I I I I__ f ;: ({:>; d tt U O ~ I "9 , "2: -= X I j I~/::\ l ~ =~/I~~} 4o 2 0 1 e 
,.., 
-
and 
, ~.,~ ~ t:' t" LL:--1 1 where \,\.. is a characteristic velocity (e.,g., 
= 0 ) 0 Then the equations (2.,2 .. 1) to (2 .. 2 .. 6) 
the vaJne of t.l00 at y = z 
become g 
(~ -V)~ - -v'f + N(q-)<J) ~ V""~ , (o\; 
r =J .. /v " k1 ( J) I ~ V, ~ ~ 0 I { e)/ 
\mere v =. (J/dx. > 2>/aj) a/2).,~ (1i'J'e assume the flow 13 steady) 0 
\<.l.. 1 we deduc e from (4.,2 .. 2c) to (4 .. 2 .. 2e) that 
Since 
In specifying the boundary conditions in the free stream we consider l!IOme typical flow:!:I in which pitot tubes are likely to be ueed 9 which include the flows ire ax:amined experimentallyt described in chapters 7 
and 8.. As regards boundary conditions on the probe we will consider two kinds of probe ~ (a) those 'Which are non-conducting or 9 mich amounts to the same thingj probes mose surface contact resistance is very Jarge 9 (b) tho~e 11Jhose conductivity ie large compared to that of the nuido 
In the free stream ( or JSc/ 1 )-g} , ) i J~ao) ~ 
Li-- = ~ DC = ( uo0 , o t> ) > Ca.) } 
\7 ~ = \j cp ( Q J O _, d t<XJ) I ( b) 
""" ~ 
=77= 
'b) Highly oonducting probe 
0 
J (b) ---
where: ~ and ?_, are vectors normal and tangential to the probe surface. ,...,, 
We will assume that the highly eond-oo ting probe is vral.13 are electrically 
insulat ed from the fluid inside the probe.Ji lo r,..1miJ li'--~e. sovJ'"'l..€ e{-error.(~~+c!_q64p 
4,.2.2. The genera.lproblem cl!ld . some ~~ial cases. 
In order to use the probe as a ~itot tube we have to calculate the 
pr"essure at the point 0 9 ( f O ) / · ell11er relative to the pressure far 
from the pi tot or~ if a pressure gradient ex:Lst:s in the free stream 9 
relative to the p!"6SSU:re at O~trti.,~e (.J;;~(.@ Jtli:e. pm6'e.-L, ( r~ )o 
However» we first deduce an important t>ymmetry eondition relating to the 
general flow over a symmetric body 9 subject to the boundary c onditions 
(4.2.4) and (li.. 2. 7)., 
N 
IV It we consider the governing equations written in terms of J,C k M thatg (E- -v) ~ = - VP' +~NAM([-< \?~t) .+ 
(c) 
(d) 
Then we see t hat if j = -J the equations and boundary condi tion(S are 
and 
con!istent with the following resul't ~ 
[_i.'. tr', w-'. b:. 'c ~ . b~ J ==- EA )-,>' w- ' -bx.J ~ ,-bz-J, 
,mere the prime ref'ers to t he valu e of the variable at the same value of ,_ ~ ...., ""' --i,and 2; y but at Lt, = - ~ ., I f f& = ... ~ we have: 
b. .d__ ....- ,v r-/ 11>/1 I 1"v 1 ' r·,1 '. l , l ~ ·, lr I [.;J I b )(_ \ j J O L ' ::=_ ~, Lr- , - l"-r-, :::> ::x:. J lj j 1-t) z_ J J 
where the prime now re.fer~ to the value .ot the variable at the same value ,...., (#'""' 
of .x. and j P but at ~ -= - 2 o 'Thence we eonclude that for the 
symmetric probes we consider, on the line y = z = o 9 
U-- = W- :)b:Jfai~ t2; = 0 I 
whatever the value of the ma.gpetic field., 
We now integrate (l~.,2.,2a) from;;__ = - L to X- = 0 alol)g the line 
1 = z ': 0 1 where L (>) o./ b) i~ such tbatJ.{~_ i .(~~ - L) ; J~ 1 where~ ~(ti'.> Q~)'?,:.h,) 'Then 9 using (402.,2.,~> and L .~e result or our symmetry 
condition (4.,2.,9) we ha:vei ::, 0 p, - fL c1- = ~'::, + N 1· (O.f/cl~""- e<)dX 
:X:.=O I'll. x.=-L 
+ l .r. (4. - 'o'.2-u_ ) ctz' R_ j c\U + ~--:ic. =-L J o-z In thi~ equati on we have ignored the e:r.ror due to the velocity not being 
zero at the aperture in the probe at x = o9 which is generally considered 
negligibleo Now 9 when R > 100,, even though the probe is used in a 
shear :flow it has be en 31:J.own experiaerrtally that the error due to the 
vi~cous term i;s negligibe ( < 1%) 9 (Rosenhead 9 1964)o (This assumes, \J 
ot course 1 that the pro~ is not U:!led in shear flow3 whose characterii,tic 
l~h iis much Je:s15 than d, though it may be of the sane order) o If' dP«>/lYi \ c'A BO 9 so that apc:><O (ox \o 9 we are i nterested in 
calculat ing 6f 9 where 
-- ""< , bf --po - f o 
u...:ng (1 ) = ~o2o5 and assuming R > 100 1 we have 
+- N r ( fi(i-f .. ) I- ([.,-"-))di · 
~ :.co 
=79= 
In order to calculate the extra elecrt;romagnetie term 9 (the error) 1 in 
general we need to sohe t.h e set of equatiorrt>< 01-0202) 9 using the 
boundary condi tion:s U,o2o~.) and (4o2o 7) o (We could concentr~te solely 
rv .--.-r I"" 
on p ~ ~ and 3 p but with less physic al imdghtL HRwever :rueh 
a task is in general beyond us at the present time, so we concentrate 
on certain simplified si tuationso 
(i) R 4S...J.,0 
In many instances the value of N is 10!1' c~nough for ~ to mke the 
- ....., "' approxims.tion that N 0-1 and that~ 9 i' 9 <p eteo f may be e:xpanded 
a:!! a series in ascending powers of' Nll eogo 
i = ( u., u-, tru) = (u 0 ,1.rb, w-.,}+ N( u, u-, 1t.r,)+ N~{tAL,v-2. ,w:i.)4~ 4:2 .. 13: 
,...,. 
Thi:!! approximation has often been ueed to examine the .flow over bodies 
with va.riou~ types of magneti c field and hM been shown by .Ludford to 
have certain singularities associated with it (Ludfordi 1963L However, 
if we confine ourselves to examining t'he first order approximation only 
we a.void such d'.ifficultieso We will ignore boundary layer effects 
since R >) 1 and . N ..:'.'...<. 1 9 which means that we must reconsider the bou..ndary 
condition5 C4o2o7)o 
and 
Using the expansion (4o2 o 11) and matching powers o:f N~ 
12_ 
~ 
.ea 
- IJ1u,,- '<o)-(~1-/or-.ii.jaz~JX, 4.2.14. 
whe:re ~o is foun d f:tom t.he potential flow solution over the pitot 9 and 
6~6/ CY'z;: i:5 ca.l~nla:ted from 
-... 
eince \..i~ is irrotationaL The boundary conditions on ~o are~ 
In the free stremru V<po - ( (), 0 6</{.t:l/02: )} 
outside the probe boundary layer~ 4o2o 16., 
(o..) (-\17>" + 9' ;< lrl> ) . ~ =- 0 1 {b) V(f o ;~ = 0 
Si:q.ce the:r"e is no gene'ral solution to (4o2o 15) subject to the boundary 
conditions (4o2oi6)~ we cannot write down a more easily calculable 
axpression tha~ (li-o2o 14) .. However, there are two special cases 'Where 
\49 can: 
a) N <<- i J).., \lr:p Qi3' = 0 9 _ highly con~ting. 
In this case,, if \i71, ,k} = O, the only solution to (4.,2., 15) subject to 
the boundary condition in the highly corn:lucting probe Le .. (4,,2., 16b) is: 
-V</:> a - 0 , 
whmce it follows that (4.,2·. 14) becomes simplY'~ 
. £=~ * x bp, = .. [ ( ~~- u. 0)d ~ 4.,2., 17., _ L-
x..--l A..., Knowing the potm tial solution-, lA.
0 9 enables us to calculate ~ f !I directly._ Note that in this cease there is a static pressure gradient in the free stream and that to find 6 .,...,r one would normally ne.ke t'WO ~ ~ ~ -, measurements off o - r~:::..-L and p:>i.;..:.L Po , the first with the probe 
and the other in its absence. (In th,6jsituationd ~a; -= d f~/~fi = 0 
and we are assuming that static pressure can be measured error free at 
the boundary of the flow 9 e.,g. pressure tappings on the wall of a duct)., 
b) N L..L.. 1 9 a <,~ b. 
When the pi tot tubes shape is such that a << b, then near the centre 
of the probe the .f"low -,rill be such that d~!J'>'>c0/c}~ Then it follows fran the fact th <.:/ : ( 1..t1·l E-::tt.hat 
r, 
dE~ .._ ciEi!- ~ 0 , d~-- - ~ -
!10 that we can regard f-2; a~ approximately constant in the flow round 
the pitot. Then (1~o2.14) becomes~ 
.... _ 0 
~ f, "" J C uo<)- u. l d x , 4.2.18. 
x.,.:::-l 
so that, in this case, knowing i..,\. o. : enables us to calculate 6. p,o 011tensibly this result is the same whatever the conductivity of the probe, but clearly if ~ = 0 in the probe, unless the probe short circuits the 
external f1cw 9 our result is not valid and therefore the conductivity of' 
the probe determines the ratio of a to b for the approximation to be 
Valid. The application of the result (4 .. 2.1 S) to a particular probe 
* Lecocq ('1964) appeared to be groping for thi~ resultj but hi~ anaJ.ysi s contains so many ob!lcuri ties as to be unintelligible. 
-81-
l!lbOUld be nade with careo 
As an example we now evaluate Ap ~ for uniform flow over a two= 
diJnen15ional probe 1rd th a square endo Using the potential flow solution 
of Milne Thompson ( 1962 9 pp 2'73=275) 9 we f~nd. that 6p, _ Lto<> a/2.. , 
d~ 
. 
80 that() - p, = l_ e u~ (1+ (i. 502..0.., \ 0 • {~ 4o2o19 .. 
ro D 2. '1fpuA) I It is important to realise also fhat the degree to which the flow j 
ie two dimensional depends on the value of N.. ii! 
(ii) a :>). b 
When the pitot tubes shape is such that a.>) bi, then we can aesume 
that ?lo ...'.1 L.< 6/dt in the centre of the probe j :50 that 
~....... -El :::. ~ ~ o, 
d !:f a::? 
and therefore the ii.ow over. the pitot tube is unaffected by the magnetic 
field (Shercliff, 1965).. . . Then we can express the non=dimemiional j ":t B 
!orce(-NJ2 1 b I NJ~)as(-cbj/d~ > O ,-db~fo~) so that 
- r-. 
~ - ~ - ~ +i - p = u~ - N bu ..... + N b l"'x..== o x.--=- L ~ .J~-=-t> YJC..~-L 
Now in the free stream 06:1/dX-:: d~oiq /d-:c. -::: -depA/d; -t' t.f ~ I 
and by definition b~~~-4 (~~(.:ic..~-L), ~o that 
6 -p :. t.{ ~ -+ N (b u: - bl,.t_ ) 2 ~:X..;:: 0 J~t..::.C} 0 
Now it al30 follow·~ f'rom (4o 2020.) that (4o2o S) become~~ 
0 :;: 9 2. b 4o2o23o 
- ' whence we can calculate b:J 9 given the boundary eondi tion:!!! o We will 
again coneider two special cases for which simple solutions exist o 
......., 
a) a .:>) b 8 j o0 = 0 
If J7 = 0:::; 9 v~ t = 0 whence, ueiing (4,,2023) and the boundary cia 
-oa 
conditione on the probe (~2 .. 7), Le., actually on the probe ~ at the 
edge of the boundary layer~ it follow~ that '7xb=O everywhereo 
Therefore (40 2 022) becomesg "" 
* Th.is result may be c ompared with example L,.,, 13 in Shercliff 1s book 
where he finds the error for flow round a circular cylinder to be Tr times as great a~ t.his" II 
lA ~ 
... -...... ,. """""°'--
This important result :is va1:ld for· all values of N, again its limitation is that WI:! do not know !I except by mor e detailed calculation 9 what the ratio 0./fr needs to be for this result ·t,o be a© (:ttrate . (The conducti vity of the probe does not affec:t this !''!?.Sult) . ,, 
' b) _a ·-'/ ·!2,.._ a .';.:) 
If now· we consider the -orobe to be supported at y = ± c2; 9 its length C being comparable with b 1 and if' we also use the probe as a 
m:_tot=statio tube 9 the~ we can show that such a probe ls error f ree 
'Nhatever the value of J d,:) and Ng provided it is non-conducting. (This result ha~ already been given by Lecocq,, ( 1964) L 
Let the probe be c onstructed as shown in 
"""' 
pressure tapping (S) on the flat face at ':C.,:.. t'9 
figt.U"e 4o2 l1 with a static 
"'"' "'\-50 that c,,,.. ·-- D1/. .. c .. / ... -= -J o I ,;;, I ,,, ·· -- · ..;~ :: - ef:2 Now if the probe is non=conducting the boundary condition (4c2.7) leads to ob'j/c~ = 0 along the surface of the probe whence the value of 61 at C i~ the same as that at So F'rom t he conditions at infinity it follows that 
~-
: l,··...:..' 
f_c.· - ;_ and thence from C4o2 c2'l ) that 
D. f: =· . ;.,'.'.'.'/:-.. /-z_ 
This result may be extended to compressible flows in that t he electromagnetie e:r.'T·or term i3 stJLzero~ The us e of a conventional pitot-stati© tubs may incur large errors in 5uch flow~ due to the Hall effect and therefore this result may be of · some practical use in tho:se circumstance:s ~ (Hunt 1 196fo) o 
c:) S:, :::,..> b9 a _:..~> c. 
We examine a simple but unrealistic kind of :Qitot tube in order to Calculate the er ror if T , ~ ''.:> c We consider a non-conducting probe .• ,. ,<,; ~·~ ~ 
~ 
nu.1.\,(1 ii, a cir <,"U lar cylinder of diameter non=dimensional 1-,L 1 who~e centre i~ at ";i:::;;.{2 9 ,_-,.::::. c ~Z= (). 0 Then to find the error)' we have to llolve C4o2 c23) wt th boundary c0~d itions ~ 
Free ~treamg b ~·}A: R~:,(i.-cvJ~on the probeob~/Js =- O ., 
f 
J 
I 
/ 
i 
\/ 
\ 
\ 
---------
-~ 
Fibt ~~~ . 
l\ol\- cond/\J\.0\-:5 
J 
,iiich ehows how in some circun~tances the error ean be n!:!gative. 
4 2 3 A physical discussion of pi tot tube errors. O O O - - - -
In this section we consider the physical reasons for the MHD error 
in pitot tubesi where possible comparing our conclusion~ with the analysis 
ot §1,..2. 2. 
We examine two linrl.ting situations for simplicity. 
(i) LkGO :=_ o<p.R. /dr. 
In this case)~ = D ~ but 9 owing to the fact that 1A., has to 
reduce to zero at the :obe tip and thatdi/)/di:.. does not. decrease propor-
tionally;, currents are induced in the vicinity of the probe. If the 
probe ie non=e onducting then t he currents flow in the =Z direction infront 
of the probe and return in the regions -where the fluid moves faster (fig. 
4.3a). If the probe is c onducting then the electric field induces 
currents in the (=z) direction through. the probe., Some of these currents 
return near the probe tip (as in fig.4.3b)o Thus in the fo:rmer caM we 
expect that near the tip Jz.. Bo /'0 9 and in the latter casejziBo < o. In 
(4.2. 18) we showed that ..6r,>owhen a << b '\tkiich agrees with our conclu~ion 
for a non=eonducting probe. We noted in §4. 2 "2 that (4.,2. 1 S) is not a 
good approximation for a finite probe which was highly conducting in 
agreement with ou.r phy·sical reasoning which ie also :!'lupported by the 
results of East (1964) -who found experimentally that the error term was 
~gative when the circular probe he used WM highly conducting. 
(ii) '7~- = () 
In thi s casejii!<"')== 6u..0B0 and dffax.:: - cn~~o that near the probe 
tip wnere u.. decreases 9 Jz decreases and dp/c);x_ becomes less than its 
free stream value. Then, due to thie effect, although the pressure at. 
the probe tip may be less than the pressure far from the probe upstream~ 
the pressure :Ls greater t han at the point x = y = z = o if no probe were 
I 
I 
present 9 as demonstrated by C.4 o2o 17L 
' When J.cc-0 \ () there i~ another effect 9 namely that cci;µsed by the 
obstruction of the f ree stream current by the ~itot ,,.\t1beo We showed 
in §4 0 20 2 how t}'iis can lead to an incr~ase inJ ?- infront of the jito\ 
so th <'-.tjkB""):Jr:1oand therefore the error may be negativeo F,-d !.1 -3(C1· 
Finally9 we have t o mention that9 though the veloeity distribution 
is altered a5 the magnetic field is incr eased, thEl only a ffect of thi:5 
is t 9 alter the term N fl,, ~0 ({ *-},7_) · Ji: and U2.i the dynamic pressure ·· '1- J_-: . -L 2:,,,.: .e;; term 9 t4.~ 9 on account ~f the general result (4.,2.9) of §4.,2.2., 
~ 
4.2.4. Conclusion. 
The most significant result of .our analysis of §4.,202 is that 9 if 
. -we design a probe for -which a >) b 9 then if jc,i,;).:::. 0 .. we can use it as an 
error free pi tot tube and ifj~ O we c.an use it as an error .free 
. -~ pitot static tubeo 'I'herefore we expect that the shape of a pitot tube 
will have a significant effect on its behaviour -when a magnetic field is 
present., It is important to note that one cannot imply from our results 
that the greater the ratio a/ b the less the error because the velocity 
near the end of a conventional cir cular pitot tube increases as (1=r=3) 
compared to a more t-wo=dimensionally shaped pitot ..mere it increases as . 9 1 (1=r=2 )., Therefore y in the former case l (L!:,o- ll
0
)di,m.aybe less than in 
the latter . Clea't'ly t he fact that no mirr}nts, are induced as a/b-?00 
,......_, .,. finally reduces the probe error t o zero whenj ·= O 9 but for f:i.nite 
...-.,<>f> ·-· ·-,----values of' a/b it is an open question as to which effect dominateso 
. While increasing the ratio a/b lead to a reduction -i~- p:robe error'J 
clearly such a flattened tube is unpractical for examining Hartmann 
bomi.dary , layers but i s very $uitable f'or those on walls parallel to 
Thus increasing a/b is not always possible. 
. .. 
It would seem that using conducting pitot tubes can lead to negative 
errors while non:..Conducting tubes have positive errorso However JI any 
attempt to find that value of con due ti vi ty 1 0-p , -which le ads to zero M1m 
error would be · u.-se/~ ' ' · j since the error would only be zero in 
one or two situations. It is unlikely that a value of op exists which 
giv-es no error in all circumstanceso 
I 
I 
11 
I 
I I 
We note that if the p:ro be er ror is thought to be appreciable in a g1ven MHD flow 9 then Dp shou1d be plotted a gainst u1,oa and if the resulting curve is not linear 9 then some MHD error i s presento We use this principle in t he experiments described in chapters 7 and g to detect and measure the }1:HD error term" 
We have not investigated the effects of a nagnetic field on the behaviour of a pitot tube in a shear' flow 9 but since for two=dimensional shear flow round a cylinder there is no effect on u.. along the line ~ :: 2:=- () it would appear that the MHD error term is not greatly affected" However 1 this problem needs further imrestigationo 
4
0 30 Electric potential p:robeso 
40 3.1. On the use of such probes" 
Electric potential probes wer~ first used by Lecocq (1964) to measure the turbulent flow through a square duct. He was interested in comparing t..11 e results from t hose probes with those from pi tot tubes and therefore had to c onvert his potential readings to velocities. From Ohm's Lawp (2o2.3) 9 and (2 ,,2 08) g 
V 2 cp :=-. ])D (vx ~), 
mch may be integrated to find V-.f: in a fully developed duct f'low~ -:£, 
u-tl ,c_, y) - L S: ( a:t + !)) dx_. 4.3.2. 
where the duct 0s walls are at x =±bi> y = ± a 1 and the magnetie field is in the y=<iirect.ion. Using this equation one should be able 9 in such a high velocity flow as u:icocq 8s 51 to deduce the value of U-~ in the boundary layers on the walls !2_arallel to the rnagpetic field 9 and in the central region 9 but ll£t:. in the boundary layers on the walls perpendicular to the field mich entails measuring <p very accurately in the @omers which cannot easily be doneo Therefore we begin this section by noting that electric potential probes are most suitable for measuring electric P<>tential i q> P but if they are to be used to calculate the velocity then it is important to realise this involves even greater errors than those in measuring (/) " 
=86-
4•3.2. _!he general :m,:9blem and_ some s:oos~i.§:.1..~a~§_,. 
We shall ,conside:r"" a probe whose extern~.~- shape is similar to that 
o! the pi tot tubia (figure 1....2). The dif'±'Hrenc e is that we measure 
electri c potential at O. x~ lj=-=:l =-(}instead of pressure. Since the probe 
hB.S t :., t :ransmi.t its-·information of the potential at Oto an instrument 
outside the fluid there has to be an electrically isolated region inside 
the probe j along its length 9 whfoh may be an el&c+:,:ric wire or simply the 
conducting flui d. (Por a. pracrtical ex.ample of the ~rpe of probe used 
see figur e 7 ol). Since this Ufnformation transmitting region u,frthe probe 
needs to be supported~ (an electric wire being too weak) g the probe has 
walls v-ihich n,.ay be electrically conducting or non-conducting. Thus th~ 
probe=fluid boundary conditions are the same as those for a pitot probe 9 
i.e. (4.2.?)o As regards the free stream boundary conditions ·they are 
the same as (4.2o l~) only we now consider that d~~f 2>3 and d c{:;cRJ / 02:. 
may be funct,ions of y and Zo 
Using the same notation as in §402.2 9 the X=component of Ohmgs law 
may be w.ci tten g 
' ~ '""/ 
-.I J - --- dcp c)x - w-,r.l .)C., -
Now if<{> is to be measured in a flow determine~y the free stream 
conditions (4o3.h) then we need to calculate Gf{) Y where 
~~ -= ~~ q5~~-L. 
0 0 
/jJjJ'"' - JJ._d &: - J..W-di 
As i n 84 o2o2o we examine Yarious spe cial eases which can 
analyse in gr eate:r deta:iL 
(i) Uniform free=Stream fiow and electric fieldo 
. If the probe is synrrnetric about y = o and z = o and if o~<.R> /a j: O 
and'ofJ-o/ d~ C$ _ uniform then we· can use the result (.li-o2o 9) 9 'Which 
showed that c,n the line 9: i. "'o9 Lr =IAr =b:;1 =b2; = 6 l) to deduce from 
(40304) that 
6~ ~ 0 4o3o5o 
. This r esult was first prmred by I...ecocq 9 but for a restricted situation o 
Our result only assumes uniformity of the free stream conditions and of 
the probe shape9 the val~ or,.y being arbitrary. (tJ~oiJJJs~o~ 2if~tf;o_,. 
~-3 -~ I~~u ~olk R,J ~f ~/~ =:;=~4. ofl'/jjf'b.." ~ ~ ~o I ~ ?.s'jJ~~0 . 
>> 
(ii) a « bo Non=lmif orm f r ee stream flow. 
In this an d the next s:p3cial case ·i,.re consider the situation found 
in boundary layers parall el fo the magnet i c field in MHD duct flows and 
in the curious 1f"ree a shear layers analysed in chapter 39 where, when M >) 1, 
o'if>~/ az: ==- uoQ (2) . 4.3. 6. 
. l ~ fox. 11 'eJri/ oj II and l ~ are O (M~) and may be i gnored, when M >) 1 0 We 
0; attempt to estimate the probe error when the probe size is comparable 
with the dist.ance9 b 9 in which Uo0 variess, since such situations arose 
in our experiments. 
We rmke the asS11mption that a>) b ~ so that~ ~< ~i!:- and (4c2. 8) 
- d~ -becornes 'C7l. ~.:: 0 .. Then, since we assume that -'J<tO-== C , whether the 
probe is conducting or noa=conducting, .it follows that J = o .. Thence 
(403.1;.) becomes6<b ~ -J. uf di: . I-f we eonside°r as an example a 
probe which is a circuJ.ar\zy-linder wi. th its axis x..~ -thl' z. =- O and 
radius f:/2. p and if we assume that du~ ;a;_ is a constant, then9 
using the standard result f'or an unconstrained shear flow over a cylinder , 
we find that 
6~ - (auoa/dz)(s 7'1-) 
Tims for a general shape o f probe we expect that 
6fi ~ o[(a~~/ai)a1 , 
or, in normal wr:i.ables, L'.'.cf> = e:.F" e.., (,,Uci:; /<>z) 
In general of course W- is less in a three=dimensional now than a two= 
dimensional fl ow so that the estimate (~ .• 30$) "lill tend to be high. 
(iii) U.), be= Boundary layer f'low=probe on the }fallo 
(ft.e 1/ When the probe is on the 1.'V·aJ.l 9 t:;: - .6/z. 9 in a boundary layer flow, Ll..o-=Z.~o0k) where du.,o ('aZ:::. is a constant, then, of course, I.Ar 
is .12,ositi-ve in contrast to the unrestrained shear flow where it · is 
De,g~__ive. For a t vJO=dimensional probe square ended probe of width by // 
considering the continuity of flow we easily deduce that I 
6 f =- - °' ~ ( duoG fa2:)/ ~ v 
so th at for a general flow we expect t hat 
6<p =-o[Bo (<}~~fa~)°'] 4o3o9 .. 
40 30 30 Physical~q_i,§_c:ussion and conclu sion. 
We mentior1 here briefly the physical reasons fort he errors which 
'tDBY' be made in using elect.rl e potentiaLpr obes . The first point to be 
made about such a probe is that i t is like a static pressure probe in 
that
1 
unlike a pi tot tube, it does not necessar ily have to disturb the 
!loW to perform its ftL.'1Ction. It does have to disturb the f'low9 however, 
in transmitting the information o.f the potential 8.t a point to a 
measuring instrument outside the flow and the e rrot)ar e incurred by the 
disturbance to the flow of the 1i nformation transmitting region' of the 
probe. 
If an electri c potential probe is used in a fluid in which there 
is no flow but only electric currents, an error may be caused by the 
obstruction of these currents. We rave shown in i1~.3.2 that this effect 
only affects </J,x.. = 0 i f the eurrents are non-unif'orm or are parallel to 
the x=a:xis of the probe 9 (in our notation of fi g ,.,4,. 1) o In general the 
error in (/;) .x... '=- C> is easily seen to 'be O Q Jc()/ d / 6 J 
The second cause of error i s due to the disturbance of the n ow 
velooityo In factj as we showed in §.4.3o2 9 this only affects </>z ::. () if 
the flow is non~unif <Y!'mo 
We rre.y conclude our exam:ina tion of these probes by saying that for 
most situati.ons the powerful symmetry condition (l~o3.5) ensures no probe 
error. But~ i f s ome non=uniformity of t he flow or current exl..stsj we 
can estimate the direction and the magnitude of the error. Finally we 
note th at we have ouly considered a symmetric probe 9 whereas in fact 
most probes are supported downstream of their t ip by a stem whi ch only 
°""" . exists i n one directi on (fig. 7 o3 ). This may be"cause of error, e.go it 
could short. circuit the flow 11 but if it is far enough downstream should 
have negligible effect on the probe reading. 
I I 
I 
11 
, . On the stability of. J..ncomp1:.essible .flm-.rs, in ID..q,J!Detohydrodynamics. 
5.1.1. ~· 
The third part of this dissertation rias tl'rr'ee main aspects. 
The ·first is an examination of the effect or a uniform magnetic 
[ieldv Q_{;;i=('B 0 0:,s<p 1 0JB0 sil"l~)on the stability of parallel shear flows 
described by~ 
tr - (uo(:1),c ,o) 
The main result of our examination is that a magnetic field can never 
completely stabilize a parallel fiow ~ i.e. stabilize it for all values of 
the Reynolds number • 
The second is a new physical interpretation for the stability of 
MHD !lows~ thereby explaining some of the effects found in this disser= 
tation as well as those .found previously by others. 
The third aspect is a danonstra:tion how the results of investigating 
t he stabilit,y of parallel fl ows can be used to examine the stability of 
the boundary layers fo'tIDd in the MHD flow in rectangular ducts)> and the 
shear layers fou.nd in our study of electrically driven flows in part twoe 
Since still very little is known about the stability of viscous 
incompressible flov,rs im.en there is no magnetic field present 9 it might 
be thought bold to attempt to understa'!'ld the stability· of MHD fiows. 
However j in a i'ew simple cases of rotating flows 9 the theory has been 
found adequate to predict the effect of a magnetic field on the onset 
ot secondary flow~ (Chandrasekhar 9 1961) 9 though there have been no 
experiments to confirm the theoretical predici tions for the onset of 
instability in parallel shear flows. 
Despite the lack of experimental confirrmtion of the existing 
thoory in rarallel shear f'lowsj we intend to develop it .further, . first, 
though, co rrecting some erroneous cone lusions of previous workers. The 
reason for our interSst in these flow is that we want to lmow9 firstly, 
hc:Mi if at all~ the magnetic field stabilizes the various flows Which it 
indices in rectangular ducts 9 as shown in ps.rt one~ or in electrically 
(iriven flows, a s i n part tiv0 o Our second reason is that it is of some 
practical importance to understand the stability of the flows which are 
round in experiments used for examining very hot gases 9 eog., in thermo-
nuclear fusion research, where jets of highly conducting gas are injected 
along the magnetic field lines in t.he area where t he magnetic field contains 
the plasma ll as in the rr cusp a geometry 9 or where the gas is ejected along 
the r:t<?.l:d lines 9 as in the 1·theta pinch 1 (Taylor and WE;f~n 9 1965L 
5o1o3o Contents o 
The major part of this chapter consists of the paper 10n the 
stability of parallel flows with parallel magnetic fieldsn, Hunt (1965a) 
which is an examination into the effects of a tmiform parallel rrag1etic 
field on the stability of plane paralle 1 flows of fluids with finite 
viscosity and cionducti vity~ uni.form pro~ties, a-id no free surfaceso 
We p1:ove that 9 when a unifonn magneti c f'ield is parallel to the flow and 
sufficiently large 9 the wave number vector of t he most unstable distur= 
banee is not 9 in generalj parallel to the flowi Le., i t is a three= 
dimen sional disturbance" Our result invalidates the conclusion o:f 
Michael (-1953) and Stuart (1954) who asserted that the wave number vector 
cf the most unstabl e disturbance was parallel to the flow~ Leo a fa~= 
dimensional dist~ft)ance o Using this erroneous assumptioni Stuart .('1954), 
Velikhov (i959) 9 an.d Tarasov (1960) 9 examined the stability of plane 
Poiseuille f'low with a parallel magp.eti~ field., Drazin (1 960) has 
examined some general aspects of the stabilizing influence of a parallel 
magne tic field on a plane parallel flow 9 al so considering only two-
dimensional disturbanceso Wooler (1961) has examined the stability of 
a plane parallel flow when Rm<:<. 1 and when the . :rmgnetic field lies in 
the plane of the flow but is not parallel to it o He fourrl that three-
dimensional disturban~es can be the most unst able, however, he did not 
point out that 9 even when the magietic field is parallel to the flow 9 
three=dimensional disturbances can still be the most unstableo For all 
'llalues of' Rm we show hO'W' results obtained for two=<iimensional disturban= 
ces can be modified to take into account three=dimensional disturbances 
and thence draw some general conclusions ab out the stabilizing influence 
of a parallel magnetic: field on a plane parallel .flow. In particular 
we prove that a parallel magnetic field can never completely stabilize a I !I.OW$ Le. stabilize it for a ll values of" the R;\Yllo~s ni.:pnber. ( o...r . 
1 (11"'-lw, ionJ A(M~it. ,-.,.&,,,,J~ µLA fN>R41 ~ lt~ {Lt/66) k- "'11&~9wf-~ /t:j, 1-lal,tt; lft><~f In our paper we also discuss the stability of MHD floWE 1"rom a new JJ.ta,,)} 
physical point of view so as to demonstrate tha t the effect of a magnetic . 1 
field on small disturbances in an MHD t'.low :is different from that in a. 
static situation, usual discus si.om make no different iation between 
these two situations. We first examine the currents induced by a 
disturbance 1 pointing out the important difference between electro-
magnetic and .mec;hanical distnbanc es, arid then examine the electromagnetic i"' B force. By considering the rotationality of thej '1-- i force we 
find under what conditions the magnetic fie1d affects the for ces acting 
on the disturban<deo We pg.rticularly concentrate on the importance of 
the relative directions of the mean flow 9 the rragnetic field)' and the 
line along which the disturbance travelso 
In ~5o2 we examine the same situation as Wooler (1961)f reaching 
the same conc lusion when R <<. 1 . We extend his analysis to include the m 
case of Rm being arbitrary and th<:n show that the stabilizing effect of 
a magnetic field whieh is not parallel to the now is less than one which 
is parallel. 
I:n §5 03• we discuss the stability of the boundary layers found in 
ottr' examination of MHD flows in rectangular ducts in chapter 2 9 and the 
layers fotmd in our examination of electrically driven fiows in chapter 3. 
We conclude that the magnetic field has no effect on the disturbances 
whfoh determine t.h.,e onset of instability of these layers and there.fore 
that the osriillatory velocity profiles found in Hu.t1t (1965) and §3.,2., are 
extremely unstable. 
Finally we present an Appendix in which we expand on the derivation 
of the equations in S3o2 of Hunt ( 1965a). 
5.2o J:,ar0al1.el f101i-.'S with coplaner but non=parallel magnetic fields. 
We now cons'ider the stability of plane parallel f.lows (t1-"'-(U(iJ)JO,o~ 
·with :magnetic fields which lie in the plane of the 
=92= 

f].oW but are not parallel t? it 9 80 :::. (f:>o ees 9' 1 0 1 B() ~(})) j (see 
figtll"e 5.t: ) . Wooler ( 1961) has deduced the equa tions of a disturbance 
travelling at an angle 9 to the mai.n flow" Using the same notation 
as in §3 o.f Hunt (1965a) the equat i on,s/i~ A.\ ( 11 \ 1 ~ ) (U - e::'j.1J 11- ,\~w) -~ Lt· U." .._ ~~~~,l~ tlJ -I ~. 
(: Q ecs~ . 5.2.1. 
:. -l~ (u-N--Q.,1\~Lfll+,,t\4 ;,, ) Al I 
and ( U-c)~ _ B_p~~lli~ -· -__!:::__ ( lp '' -J\12. (f)) . 5.2 . 2. a C.0$ e ,,,\ R_.IV\ 
WhEll R <:.< 1 these equations :may be written as (u-cX tr''-Xu-)- v-U" + ~)\ 'ii' V- ;, -~ ( t/' .. 2. t J'+-A+ lr) 5.2.3 • 
.,;\ Q 
When 1\i is arbitrary the equations mroJ be rewritten in terms of 
~u-c.)'--A'-.JL' ](tf-X")~ +2u' (U-c )tµ 1 
) \ ~ n,. 
= -L l( ~~:$:'.:.:) l/!J + (~tc)(}:>,.__ 1\"-}Z QJ l 
,, \ R ;\R:M J 
·+ ( 1.) 6 - )\:a. r~ tJJ 
where 
~o,_ .,.;\ ",\.. ~ Q.\<)\.. .} 
A = ~~~:,i§:J) JA e CL~ C,.t)S'l..C' • 
We f'irs't consider the case where Rm 4::-. 11 then it follows trom 
(5 . 2 • .4 ) that 9 given f\/2. .> (/) )> () 9 q,- is least when e < 0 (in 
particulal' 4 = O 'When f{>-e = 41/2 ) and for given'G(~6)and q09 ~ is less than its value 'When (/;> = Oj) since 
~ o ~2.c G-re.) < 
O>~e 
9(0 Ct:G Q iy f'l/2. ) ~ :> 0 
_93= ar1al 9 < 0 
11 
I 
! 
-Then, ii' we assume tha:t. Rc:rit
6
incre_ases as ~ increasesi, the lowest 
value of R •t for given q when"\\ ''i>n,,>()wi ll occur when t)<-O cri. o ,~ T. 
and from the pr eviou:s l"'esult 5.t 
than the lowest value of R . t CI'l o 
i"ollows that this value of R •t is less cri . . 
for the same value of ~ when f> = Oo 
Thus 9 as Wooler pointed cut 9 for a sufficiently large value of the 
ina~etic field the most unstable disturbances travel at an angle { e/ > () 
when the magnetic .field is not aligned wi.th the flow 9 Le. { <p} >o o 
Woole:r did not realisep however 9. that the most unstable disturbances 
also tend to travel at an angle{ej>Owhen (p = O. Also we see that 
when/ ~f ~ (), the a = R it . . curve 1'1.i:ill lie below the curve deduced 
"-0 er ., ,min. 
tor (fJ = Oo. In other IDrds a non-aligned magnetic field is less 
stabilizing than an aligned one, though we cannot find a simple construc-
tion in this case f or determining the lowest value of R •t 
cri o 
When \i is arbitrary and we have to consider the higher order 
differential equation (5o2., 5) the problem is more complic atedo liJhen </) = 0 
th~ Alfv:n number for the equivalent tv>.0=dimensional disturbance, A = B0 ( ( U Ij. (>) is the same for all values of 9 , ~ereas when I <P/ > o the 
Alfv~n number of the distrubance i A.,,,_ Bc/tlJ ~ e) varies -with <9 .. 
Given fi/2.) ({l) Oand e- < 0 9 then A < A O Alsop given e j) R and R are 
m 
independent of q> o 'Yhenj) if R " ... increases as A increases j) for 
given R given e ( < ei) and ·i it c~~ows that R . t and hence R . t are m9 m9 cri • en o · 
lowe:r when (/:t )09 s ince t hen A<.A., In other wordsll if the effect of 
increasing the conductivity (or R) a.rid the applied magnetic field (or _A) :rn. 
is to stabilize the llow9 then a magnetic field which is not aligned 
with the flow will ba ve reduced stabiJ:lzing influence., Clearly, if 
the effect of inc reasing the conductivity or the rragnetic field is to 
ruistabiliz~ t he flow then we cannot predict whether an aligned or non-
aligned fie l d will have a more destabilizing influence. 
We can now gene ral:i. se the results of §3., 3. and 4 o:f Hunt ( 196,a) 
and state that 9 when a :magnetic field is coplanar with a parallel nowj) 
in the s ense we have defined, there is always some finite value of the 
Reynolds number for which the flow is unstable. 
fi'J:-1e other conclusion to be drawn from these . equations is that when f=· 1fl~. and 9 =--0 the equation for lr (5.,2o 1 \ has no electromagnetic 
tel111• Therefore, in this case tm velocity perturbation is unaffected 
bY the magnetic: f:l.eld and sine e this particmlar disturbance is the most 
un9Jiable when t he re is no ma.gnetic field,. it follows 'that th e stability 
o! the f'l.ow is unaffected by the rragneti c :field when it is perpendicular 
to ito 
On the stability o:f MHD flow in rect_anruJar.:. ducts. I 
Using the results of §3 of Hunt (1965) and §3.2 we now examine some 
general asp ect s of the stability of MHD flows in r~etanguJ..ar ducts. The 
conclusions we reach have already been used in the detailed discussion 
of the duct f lows in chapt er 2 (§2.4.2). 
When there is no applied mgnetic field 9 the velocity in a 
rectangular duct varies a,qross the duct in the x and y directions . (We 
use the notation of fig.t.J)o Very little theory has yet been developed 
tor examining the stability of flows whi ch vary in more than one 
dimensi.on and none for the stability of laminar flow in a rectangular 
duct. 
When B
0 
is such that M~ 1 ~ the problem is simpler since a unif"orm 
core flow develops :i.n the centre and boundary layers form on t he walls. 
Let us consider these boundar'y layers and the effect of the magnetic 
field on their stabilit-;9.9 concentrating on the practical situations 
where R // ·1 m--. • 
~9undery layers of thEi walls SR,. These boundary layers always have the 
same j'eloci.ty profile -whatever the various wall conductivities 9 the 
thickness of these layers being O(M""'1). In examining the stability of 
these layers we must co nsider the induced magnetic fie l d~ B 9 a s well as z 
the applied field 9 B • Althouf)l the induced field 9 B , has no effect O Z 
on the mean flow it may influence the . stability of the flowo It is 
easily seen that the maximmn value of B = O (R ) B , when R << 1., z m o m Therefore before ignoring this term it is advisable to calculate the 
'Value of q
0 
= oB 
O 
2 ~ / (? V c 9 where b is the boundary layer thickness 
and Ve is the core ve~ocity . Thus ±'or the effects of Bz on the stability 
of boundarJr laye rs on fBJB to be neg1igible 1 it is necessary that 
M Rm 
2 
/R <<. 1 " 
The stability of these layers was first examined by Lock (,955) mo 
assumed Rn..<< 1 and that Bz was negligible . He .f'ound that the rragnetic 
field has a negligi ble effect compared to that of viscosity on the vel= 
ocitY perturbations .9 even though the interaction parameter 1 q0 .9 for the boundary layerj ~ O(M/R\ is of the same order in Mand R as the 
reciprocal o:f the local Reynolds number 1 also O(M/R). (In the case of 
the ilows we exami.ned mere B
0 was parallel to ll., q0 and R-
1 
were not 
related since B
0 
and !! could be varied separately. ) Thus the rmgnetic 
field tends to make such boundary layers stable by affecting the mean 
velocity profile rather than the actual disturbances. Lock 1s conclu;;;; 
sion was that the now is unstable when R :> 50., 000M. 
Boundary layers on the walls Mo The velocity profile in the boundary 
layers on the walls M depends • . on the conducti vi t,w of the various walls, 
but there is one property in rc.41:mmo-11 . .r)f all such boundary layers which .. , 
is that their thickness is O{M-2 )o Therefore, though the velocity 
varies in the y=direction 9 in these layers~)) t and at any value of 
y we can consider the velocity profile to be a ±'unction of x only~ Leo 
![ = j O j O 9 U (x))., 
I':, If the induced fieldito be negligible, 
14.3/2. R /R<-< 1 
me ' 
where R :i.s based on the core velocity and R is based on a and the me 
ma.timum velocity in the boundat"y layer, and then we need only consider 
the ef'f ect of 
!3.._b- = (O, _B0 , 0) . 
This combinati1:m of mean flow and magnetic field is the same as that 
menti.oned in §S:2 ;E'or <p = 11/z. men we shcJwed that in this cas~ the 
lnaflletic field has no effect on the most unstable disturbances j Le. 
those trav·e1llng parallel to g. The physical reason for this was made 
clear in §2 of Hunt ( 1965) where we showed that if the vorticity of a. 
disturbance is parallel to B it is unaffectedo Therefore we can examine 0 
disturbances of the form~ ~ \) r ~ 
lA -= (u (:t) 0r E ( x z: --fot)J) 0 1 lA.T ( X) <!Jv·r L L ( Kc - jS lll; 
by the usual Or.r=Somerfeld equation, and conclude that the onset of 
instabi lity is unaffected by the rra.gnetic field. However, since it has 
been sho"i'm experimentally and theoretical ly tl:"J.B. t at large gro'Wth rates 
the disturbances growing most rapidly are of'ten t hree=dimensional 9 it 
seems that the magnetic field must affect t he ul t imate t ransiti on t o 
turbulence of the boundary l qy-er flow" 
~mensional shear lays;rs. 
- I t is interest ing to . note that in the t,.,,n-dimensional electrically 
driven flows exrurrl.ned in S3o2ll the vorticity of t he shear layers away 
rrom the walls is parallel to the magnetic field 9 B0 and consequently 
the most unstable distCl711bances, whoo e vorticity is par allel to that of 
t oo mean .f.lowj are unaffected by B . . This shows that the flows 0 
predicted are likely to te h:i.ghlyunstable. 
Appendix to c!}apter 5. 
L"l this appendix: we discuss the approx:i.mati ons made in deducing 
equation (3o5) from (3.4) in §3 of Hunt (1965a). This approximation 
'l'd1foh is the same as that made by Stuart (19%) 1 has been criticised for 
its reasoning by both Hains ( 1965) and 'ratstuni ( 1962) j though they 1:;,oth 
agreed with Stuart O s results. Put simply1 their objection was that 9 if )£1 _ o[l ~I ] 
- - l.lJ be. I ..., 
then t he correct approx:unation for ~in (3.,4) 1rihen Rm<<. 1 is ~ 
t'~;\'L4> = ~ol _R~ ~ 
What th i s irnplies phy Hi cai ly is that the oscillations in k induced by an 
oscillati ng field out s ide the fluid are much largetr than these induced by 
the veloci t y perturbations. However the ©S©illations thereby produced 
in ~ are of such a. high frequency that they cannot be amplified by the 
If fa is low enough for mean flow and are damped by viscosity. 
y> "- J\1- lp =- 0 
then there is no electromagnetie force term in (3.3) and thereforej when 
oscillat ions in k occur of this magnJ~ they riave no effects on the 
stability of the flow. 
However' 9 if I ~J /t0 := 6 [i3 ~ ~ f ~ J //~I , 
anct if only the oscillations or wavelength o[L] anct frequency 0~, /~ , 
=97= 
I I 
I 
I 
are considered 9 (3e 4) becomes~ 
( \'2. \ 'l1 l'-)\ lf) _ Boll 
- · 0 
This approodlnation must be justified9 a posteriori, in any particular 
ease using the values of ol... and C found in the analysis 9 bb.t in all 
known cases this approximation has been f ound to be valid for 
disturbances of interest o 
6• §2£Perjplental armara:l:;us., 
6. 1. Introduction., ~ = 
6.1.1. ~~rpose of experi.ments. 
The imin object of the e:lq)eriments described in chapters 7 and 8, 
was to investigate certain .flows by means of d.i rect measurements of' 
velocity and electric potential (eop), using Pitot and eop. probes. 
niere were four main reasons for the se imrestigations" 'rhe first was 
to conf'irmll at least qu.alitat:vely~ the existence of some of the remark= 
able phenomena predicted by the theory of chapters 2 and 31 which occur 
when M » 1 . The second was to see 'vJhether the results of the two probes 
could be correlated against each other. The third was to use the pro bes 
to measure the velocity an.d potential in some simple!) well understood 1 
MHD duct flows and c ompare the rei;m.lts 'With those calculated from 
erle:rnal pressure measurements in order to calculate . the probe errors 
induced by a magne'ti4'j field, we hoped 1 thereby j to confirm some of the 
ideas of eh apter 4o The fourth reason was to show that the use of such 
probes can indicate how unstable and turbulent flows develop 9 in a wav 
that external measurements (;a,rmoto 
Although th ere are still some ex-periments which need doing~ the 
stu.dy of' :tJlHD dt.1ct;, f'lcws 9 turbul ent and lam.inar 9 by means o:f externa l 
reasu.rements of static pressttr'e and electr.fo potential has now provided 
"Virtually as much inforrration as it can about the internal mechani cs of 
the f'lowo Owing to the pau~ity of the theory of UMtable and turbulent 
MOO duct f1.mrn 9 we ~an only ext.end our under.standing by internal flow i!Lµ.~.\-low 
measurementso (For the latest review of thepi.terature see Branover~ 
et al (1966)), This was the reason for Branover & LielaU$is {1962\ 
~cocq (1964), and Moreau (1966) using probes in mercury flows 9 and 
clearly rmny more ,,R:>rkers will have t o in the futureo ThP-refore we 
hoped that an imrelBltigation into the use of su ch probes 11,ll..,known or 
&~ro:xi:mately understood f'lows would enable future experimentalists to 
have greater confidence in the use of these probeso 
60 L2o ~rem.ents of the appa:r,,a:~¥"o 
The apparatus 'vttit~h was needed for our e.xperimmts had to have 
five main elements . 
t11ese ~ 
We outline here the main requirements for each of 
(i) The electro.a.ma.gn..Q.t had to sa.tisy three criteria., the first 
,,as that the magnetic flux density 9 B0 ii and t...he gap between the pole 
pieces, 2a~ were greft enough to satisfy the condition that M >> 1 9 
wtiere M = Bc/:1.( i:s /i., )2 9 because we were interested in the curious flow 
whi ch then occur., the second was that ·t:he gap be g:r•eat enough for the 
boundary layers and shear layers which occur when M..),"> 1 to be imre:sti.-
gated by probesjl the smallest of which were about 002511 in diameter; 
the third was that the length of the magnet 9 t be great enough for 
].aminarl' andjl if possible 9 turbulentjl flows to become fully developed 
when M~ 1 . 'I'he criterion for the development of laminar flews in a~ 
non=conducting pipe is that i ~ aR/M 9 io.d1ere R is t he Rsynolds nmnbe-i-t\off{!qs6, 
t,ha.t 9 for given M, J.,was determined by the mirrl.mu·n va l ue 
of R for accurate 
enough readings to be taken with the pitot and e, p aprobeo 1"fe do not he,ve 
a criterion for the development of MHD turbulent flows, this needs 
finding out . 
(ii) A re.£_tanruar du._ct was needed in which we could :imrestigate 
the .flows analysed in S2o.4 and 9 if posru. ble 9 many oth er kinds of flow· as 
well. The requirements of the duct we designed were thesei firstly 
that its walls perpendi.cular to the magnetic field ( BB) should be su~h 
as to have no contact resistance with mercury and to have a ver;y much 
lower resistance than that of the volume of mercury in the ducty i.,e o 
J.A (:::: 0-w vvj « ~ ~) \ 9 where CSw is the wall n s condur: ti "Vi ty and VJ its 
thickness, secondly that w <<~ so that the distance in the fl1xid 
parallel to the nagnetic :fieJd should be ma..,um..i..sed, thirdly the walls of 
the du.et pa:?4llel to the rmgnetic field (AA) had to be non=eonducting. 
We also hoped to design the duct so that the oorrlucti vi ty of i ts walls 
could be cl"!a:l'l.ged a..nd that its walls could be ?"emoved to insert smaller 
ducts inside, to place bodies in the flow or to place grids in the flow 
!or turbulence e:xperiments 1 in other t'K)rds design the duct as a flexible 
1
mere:u.ry-tunne l n • 
(iii) A probe rrechanism was needed to move the probes in the plane 
perpendicular to the axis of the ductj) Leo x=y plane. 
(iv) !_flow circuit was required fo pump mercury through the duct 
(. 
\ 
.,;--0 
' 
o,I 
i~t O 
'· 
' 
- ·o 
-
0 
N 
at rates varying from approximat ely 00011- litres/sec to e05 li·~res/sec 9 these flow rates corresponding to Reynolds nu.rnbers of about 250 to about 
300(), Le. the range in which Alty found the duct f'.Low, described in 
12.4, unstable. Also a f.l.OIT meter was required to measure these flow 
rates. 
(v) Inst:rummts~ we needed a manometer to rreasure as low pressure differences as possible ( '.:::: o020rn me:ft0 ~ consistent with a reasonable 
time to take a reading and with the fast flows to be measured wcu l d not 
all be completely steady. We also needed a sensitive potentiometer. 
6. 2. The electromagnet. 
Jfaving considered the requirements for the magnet and its cost , we 
ordered an electromagnet f'rom Ll.ntott Engineering. The magnet has an 
iron cor e and is energized by wate!rl>-eooled copper coils which are wound 
80 as to be flush w:ith the pole faces of' the :rmgnet, the coils can take 
up to 60 volts at 1000 amps 9 .DoC" The pole face area is 60 11 x 911 and 
the gap between the poles is 311 o 
We calibrated the magnet by measuring the flux density, B0 ~ ir1 the centre of the gap wt th a search coil and Cambridge flu:xmeter 1 as a 
tunttion of' the potential across a shunt placed in series with the magnet. W~ measured the h;fsteresis affect on B09 by.t no discernible effect was 
found (fig.,601)0 We then reasured the flux density near the edge of t he pole fa ces to exami.ne the uniformity of the fieldy finding that when 
B0 = · 7~ 111-J~ ~the field was un:Lform to within f% over a volume 60" x 311 x 3" but when B = 'I o:24 wb/rvi .... the field was uniform to within }% S11 in f:rom 0 
the edges of the pole facesi 1%9 2 11 in1 and 6% over the 60 11 x 311 :x: 311 
volume O 
The cooling water rate required for the magnet is about 3! gopom. a pressure switch setting off an alarm if the flow dropped below this rate. 
'lhe magnet pole faces tend to move together as the current is increased 
so spaces are needed to prevent t his distortion 9 lengths of 111 brass rod 
were found to be suffiC'J..ently strong. 
6e3o The rectangular ducto 
To satiBfy the requirements that dA )> 1 9 w << °'-9 and tha.t no ccnta.et 
E:io l\s},, ~..J o4 _ _ G,c.k , r,r , 
5e-OJvvf!.,, J\ V\'(g ~~ .Si~e,d - /<· . b!eed,'.j orf 0.11"' 
I /I
 / .· Joii'\ls L '' boll hbles h,'t" 1 1 ,. 'D\ I C 
1 1. , 1 \\ ' I I 
U'?i USe..cl lui 
J&" A se, "'.:) 5"1de, ~ s d '-·-J hcl'e. A. ...--· ~ 1 ~---·1 1 11 11 l n--, ·1 ,r-J. 
~
--1J__'._ - ~pc--'"-,.,_ - r • --c-· -------- \ 1 0 H --·--- )- __ ....!. ' 1 't·' " I f I w - I j cJ i 
,1t 
~l.i I i i I I , [ , I I I I I ( at) , , · . · si 
I /~ 0 0---:-:_ - _ "'.6 \ )--~--- --·-----~.- ·----- 'l \ ~--·.,+-, ~---'.· \} f-6 --·I-T- - (;, "'c'J Jo,ol: . ~ i 
-- ······-·-- ., I r -·-··E --~-·-·----·-- t--L-r--- -- ~ o(b) ' 
7" 11 -·---~j·-f{tf':::: :::·~~~:·~=-'f' I r==,-Ho\e. E, ·----~ , - , ~~ A I _ _i \ I i( )- c: ~:;,,~ed j 
- - '......... ~ ,~\" robe. ==re'\d\e_ I 5. -...____ = ;. - - . , 0 ! 
- - 1- I 3·010 .. <= . 1- ,- - - - =i..-=, - - - ~ 11,) !-- · l. - -·--· . - -····· ~ --·· - - --'j- - - - --· - -- -- !)",.'n of - !1- - - l-1 __ -, ---!r ~, 1-------1 1, ---1 x. }bw / , r- _._ -·· r- - -·- _ _{) ~' 
I \ cb. sh · ,, 1 1 • 1 . l . -J.1. \ - ~ - - .~.~~i.__,,. .. - , / t-~.,,;.~ k.,~::1 !) I f' bcire_ _e , 
-
1 
1 (46) ~. v.c. ~k i_ 
'6 ' 
.,.'.::' 3roc,ve, 
~ o I 1 . o o ~- r ~_k y ~ ):\ o i I &) S 
r- u=~ J r= _ _ -~--1_- -_ · ____ _ l ~ I - ; ~ ~ \ 
q,zs" -;) ~ i 
A <E I i 
(°'-)_Si~e.- ~1evS ~I~ §f{-" ~'Je 1-g,[(s Yevv,oveJ,_'.5-~o"".''j ~ov-_ -fu-ro 1 botkw. {ck1J:o)<br~\ ~rf21 (~) cw1d r 
s~S (_ · '111'\e:,~_s - _ _. 'o' VI~ 'lfYOo'JQ.. ::. I 
t -~-~ l . - ~ 1) r.::..-_:-·--=1=- ·-·- 1-1 -1· ,_ 1
1
- , ,~ -- . .:..._-~L . ~ 1 
- -/!- · ov'I : - -- - - - - --·- , 11 --:::. _...,. _ _ --.....,, 
_ o,,,,.1f1 18.1\ J. - \ i ., o b -; 
" ·-::..::::r- - - \ \ ' ' - ,-~7s, l l\- / 8 , I = - T-" - ..,,.._ - . - - - --.i . s:'I z,q'6'D 1' / ' • : - - - - - - i · - 1 
( I I \ \ i O i L . ·,) ~ , - - I I 1 : \ \ ' 1::· - -:. - • - - - ·, ::;;;:-, 
-·- -"-... l \ I - - - - I -0,--'L _ )- ___ 1\ __ . J_ -:::::_ i ..J_........:-_'(o ,i 
- - -· - -- - - - \ , -,, - \ 1- - · _T:~=-=-- ~1 
<E i 20· 75" .__!} 
"'-'' 
••- -. sfs r 4~ -crlas1I: -...:.111111 c 11•< .;..a« 1'.i z-Me - ~~ 
t 
<E- -- ~II 
,-,...., 
- ----- u y-:, 
- ~
0Jt:tls 8 8 
< 
J;j _( .2 ( c) Se.c..hb<'\ A A Ii aW,:3 c:luJ: 
-5urrcte.d on ~f-l o.J1"j dianneL. 
resistance should ,be present in the w:--:u.ls (AA) 9 the only~ suitable 
-,tal is eoppero To satisiy the reqmrement that the walls (AA) be non-
eon:fucting9 we c;ould either have/. used rretal walls and covered them with 
a non.,.conducting layery e.,go sellotape,, or 1med a non=conducting material 
suc:h as 8Perspex 8 o 'Tue problem lay in se,J,ing the dueto The f'irst 
Point to estalili$h on a small scale duct wa.s 1/l!hether it was possible to 
construct a duct in which all 6 constituent parts, L e o the 4 walls and 
the 2 end pieces for connecting the duct to the flow circuit 9 could be 
assembled without permanently joining any of them, eogo by glueing or 
welding1 but by bolting them together and mechanically sealing them e og" 
by rubber sheets &""ld 80 8 r:ingso . All attempts at th:i.s 9 the ideal solution, 
!ailed 9 because leaks developedo There were t hen two remaining solutions_; 
we could ei.ther make t.'b. e 2 walls AA and -the 2 end pieces out of one 
piece of copperl) as Alty (1966) didl) or we could weld or glue the pieces 
togetherp in either case sealing the hole t hus formed with the waJ.ls AA,. 
This solution ~'Es rejectedll firstly because to rraclline a hole 6on x 3n x 
2!11 out of a piece of' copper 6611 x 311 x 311 is wa.steful and very difficult ll 
and secondly be ~ause all our attempts at welding copper fa:i.ledo (Glueing 
was out. of the qu.e.stion because of the way in 'Which mercury amalgamates 
with copper 9 (Alty 9 1966 )} 
The second solution was ei.ther to make the walls AA and the end 
pietes out of one pie~e ~of copper or Perspex,9 o!' again ·to permanently 
join themo After om· previous e:xperienee with welding oopper 9 the 
simplest solution was to make the walls AA and the end pieces from 
Perspex and to glue them together a Then the walls BB would consist of 2 
standard copper ,trn stripSbolted to each other throu.gi the Perspex9 the 
seal bei:ng made by an uoa ring sea.ted in a continuous groove cut into the 
side of the waUs AA and the end=pieees., A small scale duct was 
constru(;ted to this design 9 filled with merC1.1ry 9 and found not to leako (T.his small scale duc"t was later used for the electrically driven flow 
exr.ie:riments described in cJ-.ia.pter 7j and is shmm in figo7oi)o 
We then designed a large s~ale d:uctj which is shom in figure 602 0 
The Perspex part of the duet was made up f':rom 6 separate pieiees Ja belled 
1aD 1b 9 2 9 3 9 .4a. and 4b 9 the various parts being glued together as ahown. 
in the drawingo Great care -was taken in the .fabrication of the duct to 
see that parts were glued together only after they had been stress 
relieved by a~";l31ing, . also the v-d.ri ous parts were always annealed after 
rough machining[before the final :rmchining was do:ne. (Where Perspex 1,ras 
used in other parts of the now circuit and not annealed before glu.eing 9 
the joints sometine s tended to leak, where Perspex was machined 9 but not 
annealed, it tended to craze under stresa)o 
The rnachirdng of such a long piece of Perspex was diffi~ult because 
of the tight tolerances required i particularly on the 10~ ring groove, 
and also beeause of the consid erable thermal expansion of Perspexo The 
511 x i" copper strips which were used for the walls AA were ground and 
polished to remove surface blemishes arrl enable a smooth amalgam to be 
formed~ They were thm drilled and tapped to take the Aluminium Bronze 
retaining bolts o 
We now mention some further details of the designo Perhaps the 
most striking point about the duct to a fluid dynamicist is the fact that 
the flow entering the duct div-e~es ra:ther t han c:o:rnrerges 9 so that the 
now would not settle down by the time it reached the end of the du.eto 
'lbe reason f'or our design is that a rmgnetic field forces a flow to settle 
down in a sho1,,...0er length and tends to suppress the eddying motion 
eharacteri.stic of rapidly di verging flovrao Clearly 1 j_f f1.ryv1s are to be 
examined 1itlen the ratio 1/t'/R is very small then a new design will be 
l'equiredo The tappings were made along the centre li.ne of the Perspex 
wall using the method of Alty (1966\ and some were made along a line at 
right angles ·to the wall under the rear probe positiono (The reasons for 
their positions wi.11 be given in chapters '7 and 8)o These tappings 
could be used for either pressure or potential measurementso T'ne very 
na.rrow9 (oo62" i.,_do 9 .,Q,5 11 o .,d.) ~ Po'V.C., tubes which were glued into the 
tan,ings were joined to wi.der tubing about an inch downstream of the 
tapping in the c avity under the duct, this tubing in turn being joined 
to ! Ii i.d o tubing outside the du.et; the method of jointing these tubes 
was again the same as that of Alty. Note the presence of two air bleeds 
at the extreme downstream end of the wide porition of the duct and in the 
Plug used to block the hole A~ "vfuich was not used in these experiments, 
they were used for e:xpelling all the ai r from the duct when filling it 
and enabling the duct to be emptied. 
=103= 
,<) 
~ 
J) 
"() 
t ~ 
J 
.l,) ::;'.; Jd 
. ! -.I 
JJ 
-~ 
.., 
-
v 
I 
~ 
u 
. ,,. , 
~ 
r· 
LSl . --0 
- '"i 
l ~ 8--- ('> 
""--, <" 0 
' ('[") ~tf-(;:-'\\ t:t 
-
"\ 
IJ\ 3 :..-,) cl ia--
er S c --0 'v;--- -61'' ~} r:;-- ,-e, ~ 1 61 e,U 
f ~-h ('> .. r- · 
< 
t ~ i !J": ~ ~ (\ c_] ::,- ~ 9-- r~ ,. 
--.:, ~ ~ ~ ~ 0 <57 ;F u- ~ v (~ - ('I 
l ~ ~0 n-
plane 
The difficu1ties in designing a rre chanism to move a probe in the 
perpendicular to the duc t axis~ the X=Y plane i> (see figo6o2) are Ctllt..se4. 
~ · the fact that one c.an only move ~he pro oo s via t he top walls 9 (AA.) ~ 
utic.a:use the magnet O s pole pieces obstruct the wa.11:s BB ,and also the 
necessity of avoiding leak.so 
Two ingenious solutions to the problem have been f'ound by Moreau 
(1963 9 1966) ., In t.he first h e L.a.de the top of his duct a, continuous 
movable steel ribbon through which the probe prouected» a volume above 
the duct being full of mercury enclosed by t he ribbon and side walls o 
The imin disadvantages of this method are its ext,ra1ragance w1.th mercury 
arrl the lack of really accurate positioning required for boundary layer 
traverses a His second solution was more e conomical of mercury but agait1 
not roitable for boundary layer work~ 'Which Elid nor, worry Moreau since he 
was investigating jets o 
We adopted the met hod of Lecocq (1964) who had used his probe 
mechani sm to investigate boun.dary layers (see figs o6o'.'3 and 7,.1) ., The 
prlnedpl e of his method was to move the probe in the y=di:rection by 
plac:L"'lg its stern~ f Y in a vertical hole drilled at a radius R i n a 
cireular cylinder 9 t he spindle S 9 so thatj by rotating S jl whose arls is 
WJ:rtical 9 the probe stem moves in a circle Jradius R; i.n the y=z planeo 
'Ihei immediate adv-antage of this method is the ease ·with whieh the . spindle 
and the probe ma;y be sea1ed o There are three main disa.dvantages o The 
first is tha t the probe is moved in the z=direction as well as the y= 
direction, this does not matter j of cour se , in a fully developed duct 
flow 9 but does matter in our electric.ally driven flows described in 
chapter 7 o The seeond is that 9 if the probe is fixed to the spindle, . 
its orientation relathre to the flow chariges as the spindle is :rota:t ed; 
to mrer©ome this difficulty the probe must be free to rotate in S and 
its orientation must 'be fixed by some other me ru1s o The thil"d di sadvant-
age is that; for our type of duct design~ t he spindle cannot be as wide as 
the <fa ct so that the probes cannot be pointing straight into the flow and 
touch the walls BB o 
The final desigr. we adopted is mown i n figure 6030 In order to 
keep the probe pointing straight into the flow when examining duct flows 9 
118 used Lecocq's method 9 'Which entails clamping the probe to a crank bar, 
The crank bar B is constrained to be parallel to the duct 1s axis as B. 
each point on it traverses a circle 9 radius R9 by means of 2 cranks S 
,drl.ch rest on the Pt!1;spex -block which supports the whole meehanism 0 
p].aced on the crankJ.is a micrometer screw to p:r-ovide the vertical move= 
!DBnt of the probe. Then the exact position of the probe should be 
determined by the read:ings of the mic:rometer sealell Mll and of the 
angular scale)) A, attached to the spindle. However, there are some 
errors in this system which have to be corrected, the first being caused 
by the slight 0slop' in the joints between C and B which makes the 
position of B to . a limited extent , ( r--, 20°) 9 indeterminate 1rihen the cranks 
are in line, i.eo 1top dead centre'. (Lecocq 1s apparatus did not 9 
apparently 9 su.ffer from this defect because roller and ball bearings 
were used throughout rather than the simple fitting joints we used)o The 
other fault of the S'Jstem is caused by the 10 1 ring seal.9 s j vmichjl if' 
tight enough t0 be ef'fective 9 can cause twisting of the probe stem so 
tm.t, even if B is parallel to the axis 9 the probe tip may not be. 
In order to avoid the first of these faults we attached a vertical plate 9 
W9 to the probe mechanism block . This plate is accurately parallel to 
the duet axis so that II by ensuring B to be paralle 1 to this plate 9 we 
could ensure the probe to be correctly aligned. 
The plate, W9 was also used in conjunction ·with th e ~d:wnmy probe 1 , 
D, an attachment to the probe stem which gl ves the posit ion of the probe 
in the y=z plane , by measuring the distance between D and W we could 
check on the movement of the pro be in the y=di re et ion and by taking into 
account the value of A we c ouldr~~l~ulate the movement in the Z= 
direction; we used the latter method in the experiments on elec:tl;rlcal.ly 
driven flows, as described in chapter 7.. The distance moved in the z= 
direction could be checked further by measur:ing the distance between D 
and a set square clamped to the plateo We used th e plate Win the duct 
flow experiments for clamping on a idummy duct 1 , the purpose of which 
was to indicate "When the pl'."Obe was near the walls BB and also to enable 
us to measure accurately the probe ~s position in the y=direction by 
Using slip gauges between the durrany duct wall and Do (see figo6o3a) .. 
These extra measurements still did not elirnir.ate the cause of error due 
-105= 
t o the probe 's twist., However we found when using the probe that this 
error was too small for measurement. 
An incidental advantage of our deslign of the probe mechanism was 
t lll.t to change the probe we could lift the whole perspex block off the 
duct after simply undoing the retd.ining m.rt.s, N (figo7 .. 1 ).. In Lecocq's 
duct , the w.pporti.ng block was an integral part of the duct so that the 
spindle S had to be withdrawn .9 and the cranks ranoved, whi eh was far more 
· cCIIIPlieated. 
Further details on the use of the probe mechanism is left to chapters 
7 and S. 
6.5.. The .flow circuit and its components .. 
Before describing the new eircui t as a whole we first describe its 
mm.n components, the pump a..11d the now meter. 
6., 5o 1,, Pumpo 
The main considerations in choosing a pwnp was that it shdu:}.d not 
leak when used wi. th rrercury9 and be fairly steady in . its operationo The 
Orbital Lobe pump used by Alty (196,) could not deliver a sufficiently 
high flOW' rate 9 while the electromagnet ic pump which Shercliff (1955) 
used was not very steady and tended to heat the mercury; t he problem of 
using a conventional centrifugal pump seermd to be that of avoiding leaks .. 
Jn t he end we chose to buy a Watson=Marlow Flow Inducer, for what now 
a.wears to be very inadequa.t.e reasons.. The pump operates by rotating a 
di sc about its central axis, perpendicular to the disc face are rigidl,y 
fixed three equally spaced spindles, on '!rbieh a.re mounted freely revolving 
rollerso As the disc rotates the rollers squeeze a. pipe placed between 
them end the fixed traek 9 which is concentric with the diso axis~ (See 
fig. 6 o 6) o Thus the pump us motion is that of direct disp la.cement and Y 
il,bnsequence is necessarily pulsat,ingo However no leaks occur 
(~rovided the pipe does not burst) 9 the speed of the now can be changed, 
and, very important :1 the flow can be reversed by the flick of a switch. 
It soon transpired that we muld need to oper ate the pump at rrnt~h 
higher pressures jl up t o 43 Po Solo or 7811 Hg 1 than its nakers recommend 9 
20 PoSoL Therefore we had to inffstigate the limitations { the pump 1s 
I I 
I 
()_ 
~ 
? 
.a..... 
I 
I 
--
performance ,m these more extreme conditions. We have found tha t 11 while 
the stated 1.aximum fiCM" rate of the ptunp used with a. nfn dirun<rter pipe at 
nominal head is .. 15 litres/sec, the ma.:id.mum flow rate at about 6oti 
difference in head is about .06 litres/seco (It slightly depends on 
thettubing used)o The other limitation, which was more serious .from our 
paint of view, was that the working life the tubing decreases veey sharply 
at the se higher pressures, so that an elasti c P .. v .. c. pipe of } 1'1 diameter 
sprang a leak after 30 minutes of running at 43 PoSoL as compared to the 
quoted lifetime of 30 hours at p ressures less than 20 p .. s.,L By trying 
out various different tubing we found that reinforced rubber tul;:>ing was 
best, eogo Dunlop, Neoprene, 3 wrap reinforced tubing 'I'ype 420B 1 but even 
then hal t o be changed after 5 hours intermittent running, at these 
pressures .. 
6.5.2. lfil,.eetronagnetic flowmetero 
To measure the rather low rlOWTates in our experlmer.rts we decided 
to use an electror:ra.gnetic flow meter and designed the first one similar 
to that of Sher el.i.f f ( 19 5 5 , fig o 1 9) 9 the main di ff er enc e being that we 
used a Swift=levick pennanent magnet~ whieh hag ,"l. greater field strength(i'Oo~) 
than that of Shercliff 1s magnetron magne~(~MAL" U JOO gauss) The flow= 9 
meter was calibrated in a slightly different ,1a;r from that of Alty (1966) 
and Shercliff (1955) ~ in that we recorded the time taken for a given 
volume to be filled at a given fiowrate rather than t h e weight of mercury 
passed in a. given time.at a. given fiowrateo The apparatus used is shown 
schematically in fig o6o k... 'l'he header tank w"as s:ilnila.r to ·that of 
Sherclif'f, except that the pipe inside the tank had a urose v on its end 
to make the bubbles smaller a. nd flow smoothero The weigh tank had two 
openings 9 at the top and the bottom; the 'Perspex 1 tube inside the tank 
had three insulated stainless steel wires, x, y., zl) leading down its 
exterior which were bent hori~::>nta.lly through the tube wall and bared 
inside the tube at the points X.? Y, z.. The process of calibration 
Consisted, after first filling the weigh tank (wot) with mercury, of 
opening the on=off cock A and pumping the mercury into i;.he ,header tank 
(h. t) g th e air vent 1 V, being open o Then the air vent 9 V 1 . and A "w"ere 
closed1 the adjustable cock 1 C 1 was varied approprla tely and the eock B 
Was opened.. Then the mercury filled the woto a:nd when the mercury level 
=107= 
I 
( CA-} .. 5£.Ght)t'\ ~A 
~__.,,,_> B 
) i 
! 
res.ched Yg the wires x and y became connected which triggered off a relay 
t o either start an eleetrie counter or light an electric bulb. In 
general two readings of the potentiometer were taken be fore the level 
reached Zp 'When the wires z 1 y, and x were connected which stopped the 
count er or extinguished the bulb. We found that the flickering contact 
111&de at Y and Z led tot he timing of the electric bulb by a stop watch 
1,eing more accurate than using the counter. With the shortest time 
measured being 4 seconds we found that the variation in the measured flow 
ra.te
9 
i.e. P x 8 9 i..fu.ere P was the me an potentiometer reading and S the 
ntl111be:r of seconds the light was ong was about .,5%. To measure the 
'°lume used, we put the w.t on some scales and, a s we filled it by hand, 
measured the i ncrease in weight while the light -was on. (The tank was 
out of the ei reuit 9 being disconne ~ted at the top and to the left of the 
cocll': 9 A). The error in weighing was about .,1% so that our final cali= 
was accurate to ,,6%~ in the formula Q is the volume flovv.rate in litres/ 
sec and V is the potentiometer reading in m Volts. 
To avoid the complication of s creening the nowmeter from the field 
of the main eleetro:rragnet9 we .de.cided to plac:e the fiowmeter immediately 
under the header tank (see §6.5.3 and .fig.6.6) and therefore su::l:'fieiently 
far from the ma@",1,e t to cause no error o H0 wever 9 we found t hat this 
caused bubbles of air to appear in the diverging part of' the tlowmeter 
where it changes section from a slit, .035 11 wide to a 1 n diameter tube. 
(The 111 section then decreases t o the ! 11 • tubing). 'rhis phenomenon 
'Which only occurred at flow r ates abrnre .03 litres/sec led to bubbles 
near the .flowmeter electrodes and erratic r eadings from the f'lowmeter. 
We then had to make a new flowmeter (Mark II) for these higher flowrates 1 
shown in fig.605 9 in 'ffhich the transition from the wider gap (e12ou) to 
the f 11 tubing was carefully me,de by filingo Even this flowmeter suffers 
from the same entrainment defect 9 but fortunately at flowrates above 
.05 li t.res/ sec . 
6. 5.3. The !low eircuito 
We now describe the now circuit used for the experiments in t he 
=108= 

r"ii::x=J:- ======== 
. C',,:::7-., :t:=:::,,.. 
' ---
Ii 
I 
I 
ina,:tn duet and the calibration duet which are described in chapter 80 The 
pr.tnciples af the circuit a:re the same as that used by Alty ( 196i) 9 Lecocq 
(1964), and Moreau (1963) 9 in that the pump is separated .in the c-lrcuit 
rrom the exp:3r:imental duct by two weirs ($ee figure 606), the mercury 
is pumped from the lower weir tank (Lwo t) to the upper weir tank (u.wo t), 
'tlhere a constant height of mercury is maintained by the mercury falling 
over the weir and returning to the do"l'mstrea...m side of the Lw.tovia the 
over.flow pipe. That which does not overflow returns to the Lw.t. via 
the .f'lowmeter and the experimental duct. The main dii"ferenees between 
rur c:i.rcuit and Alty 9s a.reg a) the larger scale of the circuit resulting 
from the higher f'.l ow rates and larger magnet used,. b) the f'lowmeter being 
far from the ms.gnet 9 as already mentioned, and c. ) the addition of an 
extra pipe for draining off the min duct. This extra pipe was used in 
the following way~ first the tap (5) was closed and the taps (3) and (g) 
were openedf (the taps (1) 9 a.rid (4) 9 and (?\ being already open 9 (2) and 
(6) being elosed) i then the pump was started in the reverse direction to 
snpty the contents of the Lw.t. and the duet into the reservoir tanks, 
the air vent on the du et being opened to speed the process. The purpose 
of t."1is fast draining process was to enable us to lift off the probe 
block and change the probes quickly. The way in which the rest of the 
cirwit was used in the experiments will be described in chapter g. 
It is ·worth commenting on some of the details of the flow circuit. 
! 11 bore #1 wall 9 flexible, industrial 1 PoV.C. tubing was used throughout 
the cirooitj) except in the pump, and was found quite satisfi'l,ctory .for the 
high pressure s involved9 (up to 43 p.s.i). (Elastic P.V.C. tubing is 
liable to fracture at these pressures) .. In order to ea.se the removal 
of various parts of the circuit we needed easily detachable joints in the 
tubing . These were made of stainless steel to the same design as those 
of Alty ( 196,) e Jn all the joints between the tubing and the various 
Weir ta"1ks 9 taps, e'tc , stainless steel connectors were used, which were 
made on the same principle as those on the main duct, jubilee clips being 
used to seal the connectors t o the duct 9 tank, etc. Another example of 
their use is given in figure 6/(' where the on=off tap used in the circuit 
is shcwn. These taps are made by le Bas for pressures up to 200 Po:SoL 
=109= 
in circuits with rigid plastic tubing; how~ver, when modified for use with 
fle:xible tubing by means of the stainless steel connectors shown in fig . 
6.7, we found that these t aps did not leak and 9 beeause of their positive 
action, their resistance to the flow was always the same when in the on 
position. (The importance of this characteristic will be more apparent 
in chapter 8). To provide a. variable resistance to t he flow we first 
tfied the method used by Alty P namely a micrometer screw adjustment of a 
bung in the outlet of the uow.t, but it turned out to be unsatisfactory 
owi.ng to the air being entrained in t he .flow round t he bung which in turn 
led to bubbles i n the flowmete:r immediately below the u.w.t. We there= 
fore used a le Bas t:hrottle v-alve below the .flowmeteri as shown in 
figure 6. 6, the valve bei ng modified .with stainless steel connectors fr1 
the same way a s with the on=of'.f taps. 
6.6. Instruments. 
6. 6. 1 • Manometer . 
To measure pressure differences we used a manometer similar to that 
built by Alty . ( 1966) 9 though allowing the mercury to stand vd t.h a free 
surface in the tubes connected to the t a.ppings so tha. t the existen~e of 
a pr~ssure drop (if' great eno~) or the existence of bubbles in a 
tapping lead could be observed -by t he difference in height of the free 
surfaces. Alth ough j theoretically the manometer could m,Hi,su:roe pressures 
aecura tely to ·within 0001 11 meths 9 in fact it was rea:rely possible to do so 
for the many reasons described in chapters 7 and 1:3., One aspect of the 
design which has not been considered before i s the electrical contact 
between the tappings in the manometer. In some .situations if such a 
contact exists the fact that the two tappings or the tapping and the 
probe are at dif'ferent potentials can lead to circulating currents which 
affect the r" tsur ed pl"essure difference. Therefore we examined whether 
the pincllcocka used in the manometer, when closed, electric ally isolated 
the mercury either side of the cock as well as eliminating any flow 
through it., We found that 1 if the cocks are tightened up well and placed 
centrally in . the pinchcoc:k, they do inftact isolate the mercury either 
Bide of them., In the manometer we could either use meths over mercuz•yll 
as Alty ( 196,) did or air over meths over mercury as Sherclif'f ( 19$5 ); in 
general we used the latter method o 
=110= 
Potentiometer o 
- . 
We used a Pye potentiometer to measure steady voltages, across the 
shunt in the electro=rnagnet circuit~ between the electrodes in the 
eieetrically driven flows of chapter 7 9 between the eopo probe and 
tappir.gs in the walls of t.he ducts and across the flowmetero . The 
pctentiometer used with a galvanometer can :rre asu re up to 1/ A.,,-V9 &'1d it 
is sometimes possible to estimate to !µV if the voltage is steadyo 
When the voltage was rather unsteady we tried to use the galvanometer 
direct since it is calibrated ~ but .we found that the resistance of the 
circuit was too great for accuracy a 
7• Jll_x:periments on_!!lectr
i.cally driven flo~o 
7o1o Jntroducti~no 
There were t'wo main reasons for investigating electrically driven 
flows as well as pressure driven flows; the first being that we :had 
&(Jt!.lysed these kinds of fJ_ow and found some interesting resultsy (des= 
cribed in cilapter 3) 1 which -we thought would be vrorth while investigating 
experimentally, the second being that we could use these simple flows 
induced by currffit passing between two circular electrodes to test the 
probes and probe mechanism "IAt.ile tne rest of the flow circuit was being 
construe ted o 
Be.fore c:onsidering the exrerimmts we fi!·st describe our apparatus, 
showing how we used the probe mecbanisrn to examine these flowso We then 
apply some of the result,s of our analysis on the use of probes in 
chapter ,4. to predid some of the likely errors in these experJ.ments and 
in considering the best design of probeo (Ma.ny of the theoretical con= 
cepts described in chapter 4 were developed after these experiments were 
completed, so that our probes migltwell have been better designed)., 
In §7 a3o we describe the e:xperiment-:s on the flow induced between 
f'ini te ~irrular ele;etrodes o First the flows were examined to see how 
low the ~ur.cmt needed to be for no secondary flows to o~cur .9 so t hat 
our results eould be compared vlith the theory of g3.3o 3 o Then we 
measured the va:r·iation of the resistance between the electrodes with the 
that the resista.nc e varies linearly Hartmann nur.ober 9 M~ successfully shcwing J_ 
with M 2 when M »i ~ 
predi~ted in §30303. 
and that, as M"""°° 9 the resistance tends to the value 
We us~d electric potential probes to investigate 
the curious layers joining the edges of the electrodes and were able to 
demonst,rate that the thickness of these layers in 1rmich large potential l 
gradients oc:cu.rred is O(M°"'2) o To verify the existence of the velocity 
in these layers we traversed them with a pitot tube and examined the 
Variation of the ire loci ty with the currento Finally we compare the 
results of the two kinds of probe to draw conclusions as to the errors 
of the probes and th,?,---·nature of the flows. 
~112= 
"'1 
--+~ ,; 
._:, 
\ 
!" b 
'.l- ote.-
P.V [ _ tihbe.-
,~--
'· '·.,  
' ""'- "'-, 
' ~.
' ·, ~ 
' , ' 
I I ,, 
"'.,, ' 
i f'I, 
7 2 The experiment al apparatus • 0 - -
7o2o 10 The duct and the electrodetto 
Our experiments on electrically driven flows using mermiry as the 
conduc ting fluid were performed in the rectangular du et whfoh had 
originally been constructed as a prototype i'o r the main 6611 duct 9 
described in ~60 3 9 a..11.d which was then. modi ffa d :t'o r examining electrically 
driven flows (see figure 7 o 1) o For the first set of experiments two 
Perspex blocks 9 !" thick.11 were made ·with copper discs irn in diameter · 
(=Qb) and frn thick 9 let into them,, the surfaces of the copper discs 
being flush with those of the Perspex blocks (fig o'7. 1a). The blocks were 
placed either side of the duct with a gap of 1 ,,47n (=ea) between them.11 so 
that e = b/a = 0 5120 (The copper wa,lls of the duct were isolated from its 
interior by rubber sheets t o avoid any short circuits). Each disc had 
two wires connected to i t 9 one to supply the current and the other to 
measure th e ele ct;rl c potential of the disc. A fifth wire was pla. eed· i .n 
one of the tu.bes leading out ot the ductj) its pur pose being to measure 
the potential mid=way between the dj_scs. 
We fm.nd t .ha.t with this f:irst apparatus the layers emanating from 
the disc edges were not sufficiently thick compared to the width of th,c;l 
probe and also~ sineae we were interested in examining the flow at a 
dif'ferent vaJ:ue of e ll we made a second set of experiments in ,;.fuieh one 
of the perspex blocks and the insulation OE one of the duct 1s copper walls 
were removed so that we . were effectively examining only one half of the 
space between t,wo dises o The distance from the disc to the ~opper face 
wa:s L 97 18 (~) and thus ( = b/a = o 190 as shown in fig. 7. 1bo In this 
case we e,ttached two 'W'i.res to the plate to measure potential and transrrrit 
the current. 
With the duct placed in the electro=magnet in t..he first apparatus 
the maximum value of the Ha!rltmarm nurriberj M1 based on a was about 600 
and in the second a.bout 1600 0 In order that the flows developed were 
Similar to those described in the asymptotic theory of S3o3o3, two con= 
ditions had to be satisfied by the apparatus. Knowing the rre.ximum of 
M attainable we can now state these condition.so The first was t hay any 
error in the alignment of the tw::i discs had to be very much less than the 
.1.}ri.ckn.ess of the r egions ( 1) s Leo the layers emanating from the disc 
l, 1 
edgesl' whi tllh are O(a.M-Z) = Oo( ,.03on)o We can confidently M.y this 
condition was satisfied by the two discs., with the one disc of the second 
apparatus t his condit ion did not a.pplyo The second condition was that 
the thickness of r egion ( 1) should be very much less than the r adius of 
the electrodesr which required that 
1 e 1 
b » a.M=z or M2 >'> 1 o 
l 
The naximum values of e M2 attainable in the two apparatus were 13 and 
70 70 r e spectively.11 mich Eihows t nat the first apparatus met the c:onditions 
of the t heory better than the secondo (We did not use larger electrodes 
for f ear of t he regions { 1) touching the top and bottom walls of the duet) o 
70 2020 P:robe me~hanism. 
Wher1 descibr ing in S6 .. 4 the -design of our mechanism for moving 
probes in a du ct 'tttere the flow was fully developed,. we mentioned that 
we ~oul d also use the me chanism for examining electrically driven :f'low. 
To examine t he f l ow bet ween two circular electrodes placed in non=conduct= 
ing planes oppos i te each other wi. th a magnetic field parallel to the line 
joining their centres 9 we only needed to examine the flow in one plane, 
0 ""' const ant v (to 11.se t he notatim1 of figo3o 7) 9 because.., for low enough 
veloc:ities 9 the flow is axisymnetrie o Since 11\re wanted to use t he mech-
ani.sm and the probes designed for examining duct flows 1rd.th t he rninimum 
number of alteration s 9 we iehos e to examine the flows in the plane '6 = 
Jf / 2 . As a result of t h is deci s i on we mounted the mechanism on t he 
duet as shown . i n f'i gu:r0e '1 . 1 o co 
The method of moving the probe in the plane, S = 7r/2 9 using. 
the notati on of f'ig o 7. 1 9 may be understood by referring to fl..gu.re 7 o 1a.P 
wher e the locus of the pro be stem is shown in the r=Z plane ($ = 0 ) o 
To move the pr o'be li£. in the 0 = 1\ /2 plane JI we could not keep th~ 
a..ttgle '6 between the probe and the duc t axis 9 ()= OgtJ = o) constant; 
br1.t 9 by s pe~i f ying the dist ance 9 z, we c ould calculate the angle cA a.t 
'Which the probe spindle., S ~ should be set in order for the probe tip to 
lie in t he 0 = 7' / 2 plane" To set the probe we first turned S t,o t he 
appropriate value of 0<., and then t wisted the probe on its own axis 'u."'ltil 
the di stance between the dulllll\Y pro be and the durrany duct in the Z= 
directi on ·wa s the same a s that required b etween the p:robe tip an d the 
elettroo.e 9 the dummy duct 1:eirig rm.de so that its wall was ve:~tically above 
the electrode-. {is a consequence of this method of mo,ring the probes 9 
being ,in frofa ·their tip to the centre line of their stem9 the probes 
were only able to face into the flow when z'::-ta,j'.Q. 1 and 9 in the first 
&pparatu:s at z ~ =a, f~=1 j and in the second at z = .,25a, j \sto25. 
Theref ore 9 without making new probes of differing lengths, the pi'l:,ot 
t·:1bes eould only be used near these value s of z 9 such a p:ro be being 
accurate to 1% if it faces into the flrn..,r to within 10°. However, the 
eleiGtric probes could be used ~t a ll values of z 9 since they do not 
critically depend on pointing into the flow~ or being at right angles to 
the ~u:rrent patho But 9 if ~ is the angle between the line joining the 
probe os tip to the axis of its stem and the duct's axis~ (see figo 7 o 1 a~ 
b\ it is clear from our analysis of chapter 1i. that the probe errors are 
reduced if )( is kept to a minimtlnlo When the probe tip was on the 
du~ t O s ©ent re line 9 ~ took its marlmum Yalue 9 about, 45° ,, 
We mention here a few of the considerations which led to the 
design of the e.,po probe we used in the experimen:ts9 bearing in mind 
the a:nalysi~ of chapter 4., On the or1e hand su.c:h a probe needs to be as 
srraJJ. as possible 1'lhen used in flows such as these where the gr adients 
::>!' Ye.loci ty and electric potm ti al are large 9 for the rea sons given in 
.:-riapter 49 the probe size aloo needed to be minimised to reduce the size 
of the vortices shed by the probe 111lhichi; beiing CC\..'rrted: round 9 
w,,uld tend to affeet the potentia...l at the probe tipo On. the other hand 
the probe must be Stlf'fieently rigid for its po.5i:tion to be determinate 9 
partie:;ularly since in th..1..s experiment the probe would not always be 
. f'ating into t he .f.111:nij also the conduating region insi de the probe must 
have a suffi©ient: diameter for its resistance to be reasonably low and 
this w.ire ha~ t...o be insuJ.at ed from the probe 9s exterior if made of rneta,L 
The ta.et that t he strongest small diameter tubing easily available 
is made of stainless steel determined our choice of the material !or the 
probe 0s exterioro 'I'he main alternative was PIJrex tubi.ng which Lecoaq 
us~d, this sui'fers from the disadvantage of being brittle and therefore 
ea.&1..ly broken if the pr obe was inadvertently pushed against the wall and 
also it is less easily bent to the requir'ed shape than stainless steel., 
=115= 
lj 
I 
Po.. \\ad ~UM oY-I p 1 Ol-hf\\ln'I V\JI re., 
g()W'ever the advantage o! Pyrex tubing is that it does not require in= 
sulation from the conducting material inside the probe. We could 
either choose to have mercury inside the probe a.s our conducting region 
or we could use an electric w:ir e. 'I'he advantage of the wire is that it 
can be made to protude from the end of the pro be and thus present a small 
area to the flowo We chose to use Palladium and Platinum wiresjl the 
thermoelectric potential of these wires being close to that of' mer.m.1ry, 
we did not try out the method of Lecocq (1964) who used Platinum 
wires coated with sodium, "Which 9 apparently 9 reduced the ther.mo= 
eleetrie potential difference between the wire and the mercury to zero o 
The :iiml design is shown in figo 7 o2(a)c Note the two diameters 
of stainless stell tubing used, the use of flexible plastic tubing a.s ' 
the i nsulator between the steel tubing and the wire, and the coating of 
the exterior of the probe 'With a. thin layer of non-conducting Perspex 
eement,, 
We now cansi der t he various regions of the flow between the disC!s ~ 
as described theoretically in §3.3o3, in order to predict the kind of' 
errors to be e:xpeeted~ 
,Region (&1 
In this region the current densi"t;y is uniform and the velocity 
ifj zero so that 1 if th e probe is at right an.9-ges to the current 9 Leo 
~""' o9 from our symmetry result of §403,,2 9 no error woul d be induced., 
However, as we have explained~ the probe could always be ,. at right angles 
to the plane 0 = 1\/29 so that we could expect some error due to blocking 
th.e current s~ Then the error in ~ compared to 6+; the potential of the 
dis~ is easily :seen to be Cl(d/a) g· where d is the probe diametero (There 
\ 
is another possible source of error caused by a local velocity 9 \f 11 induced 
t '"""' by the displaced currents_; then the '1 x B electric field could affect 
c/> m9 the measured value of .J. 9 but {'; th'": region (1) B is parallel to 
'P ...,£ 1 t::/~., and it may be seen that no error can result f'rom ~ x B
0
) o 
.&!gion Cu 
In this region, 1/>here severe velocity and potential gradients 
e:xist and the approximate relation 
-M T Lr Q 'to ~ 0 7 ., 2 o 1. 
O'f"' 
hOldS true we can use the res1~ 1 t. (i+o3o!5) to estimate the order of 
ins.gnitude of the error betweeri 4t a..,_d q mo Changing the n:,ta.tion of 
~4. )o 2 we ba:ve ~ Q. f i"V\ ~ ~-+ k d B0 d~/ay-, 
wtiere from §4o3o2 we @an assume k to be a positive constant of order 
unity. {In our analysis we assumed Rd»1 where R is the Rey:rrolds number 
based on the probe diameter, and that the flow was independent of Rdo 
However if Rd ia lc,w enough the flow is dependent on Rd and then kin 
(7 0 2 0 2) becomes a function of 1s6 ) o For the purpose of this approx=_ 
iDJate argument we also assume that the probe is pointing into the flow9 
i.e. z~.a.,11 otherwi.se the error in (7 02 .. 2) migj.1 t be proportional to ir'9-
as 111n,ll as dtre/ d:r' 0 
Using ~7 .2., 1), (7 o2o2o) beeomes 9 
~ N\ ~ q> + f<.- d ~-c) )/o<"2-.. 
Two :rrain results i:,tem from this. If we consider a very simplified 
expression to represent the fall of cp through the regions ( 1 ) 9 say 
f= I- e-irf (e_ ') where(' =('<-0/aM~ 9 then we can first see that 
°'aef::>rn /ar has t-wu maxima one for t < 0 and one fore I .> 0 9 and secondly that 
if k is low enough 9 the maximum value of'd~M/<>e,' is less than that of ocp .. /ci::.,' 
0 
As is obvious if (7.,2o3) is written in tenns of ~I 9 ioea 
~YY\ ~ </:> + ~- d 'L ]\I\ ~Q_[f-;i_ I 7 o2o4o 
a~ up ~-
the @ondition for the error term 't'o 1;:,e negligible is that g,~<.,<..1 o From 
(7o2.1+) we also see tr.iat9 if Rd?"co 9 (j)YYlcC ~ Ol'.' Iy the ~rent passing 
between the electrodes 9 whereasj if the flow over the probe is sensitive 
to variations in Rd 9 \ <:/) m would m:>t be proportional to <£ or I., If the 
error term is appreciable i, it . foii;ws that JI even if cp is a function off I 
at a JxJ.rticular value of j 9 (/J m is noto Si.nee the potential nteasured 
by a probe . is the :rrean of the potential across the face of its tip 9 a 
circle of .,025u~ in diamter9 to calculate the error we had to average ?:/-p_ 
or du-8 across the probe faceo OY-2.. 
~ 
Finally we note ·that we could either calculate cp purely fr<::i.'llc/>vr,., 
by int.egr1ating (7 o2.h) or we could use the results of the pi tot tube . 
readings Lr8 as well as the readings of <{!; rn to calculate (p .from (7 ,,2o2L 
=1i7= 
2 4 Fitot tubeso 7 0 0 0 
The considerations lea.ding to the design of the pitot tube 
l(eTe very similar to those lea.ding to the . design of the eopo probe, the 
only difference being tlw. t the probe should not have too small an 
internal diameter because of the need to reduce t he time for taking a 
reading of pressureo Again we used various sizes of stainless steel 
non-magnetic tubing for the probe 9 each tube fitting inside the othero 
The tube was coated with Perspex cement in order to minimise the effects 
of the probe on the electric fi elds , however this was not really necessary 
as the contact resistance of stainless steel is so JBrge as to render it 
effectively non=conductingo The final design of pitot tube is shown in 
figure 7 0"'1b) o 
From the theory of chapter 4 an.d the experimental results of East 
we expeci"J,d that the MHD probe errors could be calculated from the formula~ 
.ro :::::. p-+ lpu-...._2 (r+cx:. c:s- Bo2 c1) 
11 6 :::, 2 '-- "" ------ I e ue 
where p
0 
and p
8 
are the total and static pressure respeetively. It turned 
out that N = tr [Jc:?d was about unity so that higher order te:rms in the 
e u-e, 
expansion should have been usedo However, the , results of East gave ol = 
039 and our own results of chapter $ gaveo(:::: ,,4 i~or values of N of 0( 1) ~ 
so we assumed ol = o4 in our ca,lcula.tions of velocity. 
The other source of error to be e:x:pe ~ted was caused by measuring 
velocity in a shear flow the length scale of wiich was comparable with 
the diameter of t.he pitot tube. However we show in §So2 that using a 
pitot tube to measure the velocity of a. pl&"".le Poiseuille now in a duct, 
the width of wbdch is only f'our times that of the tube, leads to negligible 
errors over the central half 9 but appreciable errors ( ~ 50%) wqen the 
pro be touches t,h e wallo Therefar e in a free shear f'low such as this we 
do not expect errors in velocity greater than 10% due to this ef'fecto 
We decided to measure the velocity ind'\;lcea in the region (1) by 
means of a pitot tube only9 the pressure in the pitot tube being measured 
r-elative to the pressure at a tapping in the wall of the duct, Pw" The 
static pressure was not measured in the layer-, eve-fl thoµgh there wa.s a 
small static pressure gradient through the lay-er due to the rotating flowo 
Howeveir we were able to calculate this pressure gradient using the equation 
I 
I I 
I 
,I , 
.t:. \ M = 13'70 1 ,.{._ = · I°! 0 
o · M -= sii . t ;'., s 12., / 
x . IV\ = 2.04 ) t .:= ·512 /6 
• M = o , t ;. ~ s12.. 
150 
I I 
0 
25 x 
x 
loo 
So 
2.S 
I it'\.a.mps 
I · o ,) 3,0 
~1~~-k,f\h~l_b~e., d 1se,s Ap-. ~~ C 1AY(O'\,l 1I r-ai vo l(;ous _ N\. 
error in p
0 
for the moment, we have the usual relation 
and p
8
, the total and static pressures, i.e. 
ro -;::;. f>s + i eue2- 7 .. 2.?o 
E].i:mir~ting p from (7o2.6) ro1d (7o2.7) we have : ~ 
a R. ::::: -+ p \.T ~ '<-- -+ ..e d LT e '2._ 
c)Y Y 2_; a)Y 
y 
S ( r~ ~ Po ) dr . e 'f -v «) oY 7 .,2.So 
ThusSJ by onl y mea.sur~l0 9 we could calculate lrg.. Since t he static 
pressure effect is o(Oftj( <.<. 1), we could a.pproxi.ma.tely a.Jlow for t he MHD 
error SJ which was greater than the static pressure effect, by using 
successive approximations to calculate U--e , viz 
Lf,,-,. =- ( 12....,,_ l(r ~ ~) dY )V(i + · ~~-o--J \_ ,,1 
1:7 h.- er ~ er e 1..re n _
1 
I ~ 020 9o 
(We needed to use faro iterations at the most in our calculations). 
7.3. ~rimental results. 
? o3o 1 o ~le@trie Roten tial measurements. 
Since the aim of the experiments was t o reproduce conditions 
e.tarrd.ned in the theory of §3.3.3 we first had to establish that a 
laminar now regime existed in which the secondar<J flow was negligible. 
It is clear from the equations of motion that when the secondary flow 
occurs 9 Leo a flow o@curs with a radial component of velocity driven by 
the :radial pressure gra..dient~ <etrf}/r), the relationship between6q>, 
the mean potential between the discs, and I, the total current flowing 
between the discs, would become non~linear. Thus, by measurlng6(p against 
I at a given value of M, we were able to find out how low the current I 
needed to be for us to acllieve the required flow. ti..g. (7.3) shows t h e 
--6cp =I curves for the two apparatus at various values of M. (For the 
l!econd apparatus we riave doubled the potential b~tween the discs and the 
=119= 
I 
0 
f\() 
I 
:B 
r;p 
II 
~ 
• 
0 
c-
rn 
I i 
z 
x 
• :><. \ 
x 
\ 
• >< \)( 
\ 
lf) 
I 
-
Jj--
'° 
\..cl 
<) 
II) 
0 
"'" 
-0 
f'() 
() 
~ 
() 
tJ) 
t:.---
t 
Ci 
f 
µ 
-,....l 
~ 
~ 
~ 
::, 
u 
I , 
-,,a11),, Note that 9 in the range of_! considered!) .6cp 3'., I when M = 0 
and that 9 ·when M ~ 0 j the eurve of .6ep aga.i:nst I is a straight. line for 
I suffic:iently low 9 but as I increases the curve eeases to be a 
straight line imlie ating the onset of secondary. flow" There are two 
curves for the first apparatus (two discs 9 Q, = .. 512) taken at M = 5S8 
and M = 20li, ,, N0te that t he value of I at which the flow becomes 1.L'l':lstable 
111 higher when Mis higher)) thus indieating that an increased magnetic 
field tends to suppress the secondary flow" This may be 
1 
explained by the 
fact that 9 since the velocity 9 Lre, is proportional to (n.f'2 ) !)· the ratio of 
1ne:rtial forces inducing the S?eondary flow to the viscous forces restrain-
ingj being proportional t o (s-8 9 decreases as M increases for a given value 
of Io H0 wever a more detailed discussion of the onset of secondary flow 
is impossible withm1t knowing the form of the secondary flow distribution~ 
whieh at the present time is obseureo 
We oontinued this imre~iigation of the onset of secondary flow by 
examining the relation between~the difference in potential between that 
measured by the eleetric potential probe and that on the lin.e z = 0.1 , 
and the current I, the second apparatus (one-disc) being used., The probe 
W!3 placed at a radius of' 1 ,,271; and a value o:f J = =,, 95 !1 Leo in the 
region ( 1 ) 9 so that any secondary flow effects could be markedly demon-
strated,, As is shown in figo 7 o4 the curve becomes non=linear at a much 
lower value of I fo:r. the same value of M than in the Lqi - 1 curve for the 
~econd apparatus , shown in f'ig,,7,,30 There may be three explanations for 
thie, t he first is that the 6.cp- I curve is simply cu :erlng awe,y from the 
straight line curve gradually and there is no definite point at which the 
curve ceases to be linear, the second is that the probe itself induces 
a local secondary now l<hich has no ap preciable efi0ect on the 6~-I curve; 
the third is that the errors in the probe created this non=linearity as 
predicted in §702,,20 However~ since the Reynolds number of the probe, R, 
was greater than 100 for t he values of I considered, and since the velocity 
dis tribution round the probe does not vary greatly with the Reynolds 
number in thi3 range~ it :seems unlikely that this non-linearity is due to 
the probe error o Also9 1'hen the wecondary flow occurs the radial 
Velooity will tend to thicken the region ( 1) o For a given value of I 
this reduces the current density i n region (4) and thence reduces Lt 
=120= 
o N 
CS- lD 
,0 Q 
I 
I 
I 
I 
I 
I 
I I 
I I 
I I 
I I 
I I 
I I 
I I 
I I 
I I 
I I 
I I 
I I 
;/ 
I 
I 
I 
I 
I 
I 
I 
I 
I 
I 
I 
a/ 
0 
I /o 
• 
• . ,. . 
• •• 
• •• • I 
I lo 
0 
• 
• 
• 
• 
• 
• 
" . 
• • 
• 
• • 
' 
85 we see in f'igo7o3. However as the region (1) thickens, t '1e poten-
tial gradients fall and therefore if the probes position is a'r, a radius 
greater than that of the disc I <PI rises and whe:r.e lessjl falls. This 
e,cplains why, when the (p-.l.. curves in fig.7o4 become non=linear,/ cp/ 
rises. The last and most praetieal point to notice in these figso 
:,;;---" -7
0
3 and 7 o4 is that the values or (p and b.<p in the regime are always 
below 100 JA V and often below 10 fA-V in the regions ( 1). Since the 
potent iometer only measured to 1 f- V it follows that the random errors 
to be e:xpeeted are sometimes as much as 1o%o 
Having found the values of I below which j_t was necessary to operate 
at to avoi d secondary flow, we then measured the variation oi"' R = bcp 
with M~ for the two apparatus. We showed theoretically in §3.,303 that 
wmtever the distribution of current density across the electrodes, as 
R ~ R =, (=2a/ 1f o--b2) • 
We al:so demonstrated physically why we could expect R/R...o to be a linear 
· 1 
function o:f'( l M·;;q=1 when M ~) 1. Therefore, in presenting our experi= 
. l 
mental results in fig.7~5 we plotted R/Ro<:> against ( lM2')~i for the 
tm electrode configurations in whieh 1 = .512 and 0190 res:i::ieetively. 
We draw three rrain conclusions .from these two sre:its of results. The first 
J. is that R/R oc, is indeed a linear function of M 2 o The second is that 9 
to wi t.hi11 2%9 Le. well 'Within the experimental error 9 R/R c:;,0 = 1 .oo when 
the lines of R/Ro0 against ( { M})=1 are extrapolated to the point where 
M"" a() 9 and the third is that the slopes of the two lines are different 
being within 60% and 20% of the valt:1.e of the slope found theoretically 
by assuming a constant current distribution aeros5 the electrodes ( see 
§3.3.3). We take up this point in §7 .,3 .. 3 after discussing the detailed 
results of the probes. 
Having danonstrated that some of the external characteristics of 
the behaviour were approximately as prediQ,ed theoretically, we then 
examined the internal flow structure.. In our theoretical discussion of 
§3.3.3 we first postulated the existence of various separate regions and 
then made various deductions about them some of 1rfufoh we have been able 
to verify e:xperiment ally. In the central region between the e1 ectrodes, 
region (.4) 9 we con~l.uded that men M >') 1 the velocity is zero and that 
=121-
0 
+/· 
-·2.. 
-·t 
+·Z +·6 
0 t ==- • 5 12. 
18) M = 1550 I ,t = . 1'1 ° I ~ ~ 0 
® /vl ~ 1550 , J!.. .:.'. ' I <fl\ cp < 6 
0 - · 2... . J~ 
- ·1-
• 
• 
-%1,b VaflJl<Sv\ ~1.e>'\+.io,L r (=p l:" b' /'1,~j\ie,, = ··t;e.J_OC,_~ec .. 
e..kclv: Ddft_~- ~~-.M~>-'-- ~x,_d ___ M _  :-:. __ Q _______  
l'J 0 
in (r -. 
II II 
~ ~ 
~ 
'° f (J-
-
+ + 
~ II cY-) 
• 0 
--r {'{) 
./ 
I .. 
~ /. l~ 
............... • 
• • 
• 
. .. 
~ 
lr) 
( 
tl 
~ 
-
8 
lf) 
• 0 + 
I\ I "<-""""") 
0 
+ / / -\' I /. 0 
./· cJ ·/ + 
0 
~I I 0 
I J 
0 
-\-
l 
0 
I 
0 
I 
0 
+ 
0 
0 
0 
'-9 
+ 1' 
• _J) 
~ 
' 
c-J 
f-
I 
() 
I 
--------
Cs; 
I 
~ 
I 
I 
I 
I 
I 
I 
I 
I 
I 
the current density and electric fie ld i s 1.miform, provided t he ~rrent 
density is uniform on t he electrodes in t his region, as, for em:atilple J> on 
perfe~tly conducting electrodes. In fig .. ?0 6 we have plotted(-cp)11crb2;bT 
against J a.long the centre line (r = 0) v,.h en M = 0 and M ~ 1, since 
t~)fi t16~l~ 1>/Jefe 1d'I, is the value of <p on t he disc men M =oo a~ the 
s!lllf, value of L This figure shows tha.t JJ when M = O, the current density 
j z:-- is lower at J = 0 than at J = 1 because of the spreading of the current 
line~ J> and that when M >) 1 the c:m rtes become straight lines in.di ca:!:,ing 
that the current distributi on is uJlifo rm in t his region and that therefore 
the spreading is eliminated., The slopes of the two stra i ght line~ a.re 
di fferent because the R/R oe:, against 1/ (tNil)curves have different slopes, 
however.11 when M--==> oo 9 the lines shoul d merge, cutting the ordinate at 
" 
We should not e that the potential measure= (- ~) 6 b7.ii /-Cct -= I· o 
ments in this region were only likely t o be in error to order (d/a..) j ie., 
less than 4%P due to MEW effects 9 {§7 .,2.,2 ) 9 but the random errors were 
about 5%o 
We now consider the results of the r adial traverses of the ele~.tric 
probes in the region ( 1) o Fig., 7 o 7 shows t he results of a potential 
traverse wh~n M = 0 9 which a~ts as a reference with which to compare those 
when M .::>"), 1., . We then made three sets of measurements of (p in the fir~t 
apparatus 9 Leo the two disesll when J = 0291.J. and M = 175~ 330 and 610., 
(The notation we u~® is the same as that of. fig., 3 .. 8" I being positive when 
parallel to the magnetic field)o fo fig . 7 ., 8 (a) we have plotted the 
results in the form of one graph of~cp)c-11bt/I agains·t(,r- b)fa'\. 1Vd--) (-- t' f\/1-Js.) A' ' I i v1 Qin order to show firstly that t he t h ickness of the 
, l ' 
region ( 1) is unquestionably of order (a,1,1=2 ) and secondly to show that 
the distribution of cp and therefore u e is simil2.1.r for different 
values of M in these layerr-s., We have drawn t he best line through the 
experimental points j) because they show only a sma ll systematic departure 
f'rorn thi:s lineo We could perhaps say that the best line through the 
Poi nts f'or M = 175 wuld have a greater slope than the curve drawn., whieh 
WOUld be expected from (702.4)0 However 9 considering the randomness of 
many of the errors involved, the curves do not i ndic ate any large ~ea.le 
' l 
departure from similarity exc~pt where (r=b)/(a.M""'2) < - 3 , when the values of 
(- <p) are lower for the lower values of Mo This result is to be 
=122= 
II 
cJ. 
I 
r-
I 
<:J<l I 
t{) 
r-
11 
~ 
• 
0 
r,I) 
(V) 
I( 
~ 
'/__ 
.0 
'° II 
~ 
<J 
0 
~ 
I 
0 
~ 
t 
0 
- D 
J 
"1-
<S-" ....JJ \,, 
if) 
er er- 6 ,0 
+ 
. 
·-t- -r 
+ ii I II I 11 <::-_, r~ 
~~ -
-r 
...._() 
<::") 
II 
~ 
... 
-c-
!'!) 
11 
z. 
Rf" 
<Kl 
I xx 
) < 
<() (./) 
~ i:-lJ) 
11 I\ 
~ ~ 
0 ~ 
0 
0 
9 
- "'J 
I 
D 
<:'.) .._j} 
\) Cs-
. 
+- --t-
I I I I ~~ 
- -
[; ~ 
['() t() 
II .j\ 
~ ~ 
-/,.. .. 0 
I \ 
c:i;::, r- ~ 
,'() ·~ 
.() ,e. 
~'j/'X 
I 
0 
I >( 
I 
I <f 
I 
I 
I 
<]'(! 
I 
.X.>( 
J 
l 
"I 0 
j 
I 
(f.1 
I 
I 
I 
I 
x 
I 
I 
I 
() $ J Q 
0 <l 
0 <.1 l.~ 
) < 
0 <I " 
I ~ j 
"4- I"' 
,0 ~ r::' ,c 
"-'..l. 
~ ~!J_J>(J)-) 
~ 
N"i 
~ 
'-..-I. 
! ·I 
n~ i ~ ~ ~ I 
-~ 
d ~1 ~ ~ 
~ a c 
::-£:\ 6 ' '!()' - I 
\ 
I 
~ -'' ~ J.sfJ - 1 
4,.) 
,j ~ . i i . I I 
~ · 
~ gr:, I i 
I ' _, ..u! } ~- ) ~ 1 :::, ~ -\::::- 1 S3 i..:,I b ~ 1-1 I '.'.) ' - ~ 8 
I r 
~ · 
..µ 
1 
O' 
€1 
CS! ~ - 1 ~ I 
• I cl j ~ I 
81 
I 
I 
(I 
I 
0 I 
" 
0 
,o _ Q 
N) 
-
0 
. 
~..,_rl 
~ 
-.::6' 
I 
,a.:_, 
. 
-
0 
l 
() 
rl 
I 
0 
I~ 
I 
.() 
+ I 
I 
_f9 
l!) 
I 
6 
eipeeted eince in the central region 9 (t) , as ~ rave already noted, 
~~) er 6 '.l.. /a I va.rl es linearly 'With M=z 9 (--4 ) increasing as M increases., 
The results sh a~ in fig o 7 o g (b) were taken in the second apparatus 
the dis c and wall 9 and were plotted ila the same way a.1' tho!e in fig. 
7.s(a )o Tu thl~ ease we have same results taken with the probe just 
touching the wall or very close to i t ll J <(' .,01+3 9 the:re being mo 
detectable difference due to moving the probe very slightly near the wall. 
'TI,.e most interesting point about these results if3 that near the edge 
die@f·f) a.ndl' what is more; the c'l:1. stanee in which t his drop occurs is 
Toerefo re.._the potential on the electrodes ea.nnot be considered as 
of the 
1 
o(~) . 
resembling that on perfedily conducting elect:rode$ 1 the question of whether 
it resembles that on electrodes emitting current at a constant density 
is settled by reference to fig o 7 o B (e) where we compare the experimental 
points with t he potential profile produced by rueh an electrode. The 
discrepancy for r <( b elear, ly shows why the a. ssumpt ion of" a. constant current 
density i::J erroneous )) but the c omparatively good agfleement for r .> b 
indicates 9 perhaps 9 that in this reg.Lon the profile is less sensitive to 
the ~urrent distribution on the eleetrodeo Note that if the e lectrodes 
were perfectly conducting 9 the probe would have reeorded a eorrntant 
potential aeross the electrodeJat least to O(d/a) o The second point 
to notice is that t he profiles of 1 for all the Yal~is of j 9 examined 
infigs o7 o8 (a.) and 7.,S(b) ar0e functions of (:t>=b)/(a.M 2 ) 9 fmd therefore 
it follow::s from OU!' argument in. §Jo3o3 that the currAnt distribution on 
the electrodes near their edges was a 1\metion of (fr--bJ/g,...,Mt). . 
It :.w interesting that the thidrness of t he layers are approx:i.m.tely 1 . 
the same in all cases 9 being about 6 a:M""z . - This nay indicate 1-by the 
profiles are not completely si milar and why the sJonns of the two ourves 
o 1 )-1 17. rf < I - 1 
of R/R 00 against ( .tt_ MS. are different in fig , l .~h since i.fc)'::'tJaM 'S.-
the ratio of the t hi t'lkness of the layers to the radius of the electrodee 
is <Sib = 6 ( R_ M + ~ )-I ~ For the maximum value of M in t he first 
apparatus 6 /6 = oh-6 and in the second apparat us c:5/b=o 780 Therefore 
the approximation we imde in § 3 .::3 that, o(o <:> .:>) i is not really 
justifiable i .n analysing our experimental sittw:t.ion 1 and consequently 
We do not expect that the eurves of' R/R ~ against ( Q. Ml )=1 to be 
identieaJ_ for the two apparatus . Neithe r do 1rre expect the cp profiles 
=123= 
_(} 
'NV 
t,.') 
<) -c.; 
I\ 11 
)- >-
-~ L ol f 
0 ~ Cj 
'4 r- be r'1 
- ~ Ii 1\ 
~ 2 
0 • -~ 
0 
0 
-
0 
I.I) ~ ~ 
() 
0 
~ 
1\ 
~ 
0 
J I 
0 *~~ ~ 0 \() .o · E • '° 
"<] £~ ~ 
l.J) 
. 
{'() 
• 
b~ 
-c,J 
' 
0 • 
c ~ 0 
'° • 
. 
to be the same, even if they had been measured a:t the same value of f .. 
7
0
30 20 Pi.tot tube measurements. 
As · with the electric potential probes 9 when we began to use the 
pitot probe we first checked that we were :measuring a velocity low 
enough to be in the required flow regime. Wit h the pi tot tube we also 
"•d to ensure that the measured velocity was high enough for (p - p ) IJ<L O S 
to be proportional to the square of t he velcc ity U--(:) ., Since in the 
flow regime · examined in ~3.3 ., I.re is proportional to I and since it follows 
from (7,,2,,7) that p
0 is proportional to lrG'2 i> if the MHD error i5. 
negligi. blei> we had to find the value~ of I for v.ih.ie h 6 p,.::. p O - f w , was 
proportional to r2 mere Pw is the pressure at the tapping on the wan. 
In fig.7 0 9(a) ·we plot 6f agains t I 2 when r = 0 9 M = 13709 and 
11
1 
when r = 1 .33b, M = 1355, these and all subsequel'lt readings b~ing taken 
in t,be second apparatus 9 Le. f= ., 190., 1\lote that when r = o,tp < 0 I 
since Lf (7- = o and the stati~ pressure is below th9-t outside the discs due 
to the radia 1 pre1rnure gradient • When r = 1 • 3 3b i> l p ;i. 0 sh cwing 
that in region U) the rise in total pressure is greater than the fall 
in static pressure 9 mich is 9 of em!'se, to be expected., The interesting 
point a.bout these two curves is how they show that the static: presemre in 
the c: entre is less sensitive to the 
recor'lied in the f'lmi region ( 1 ) • 
"--"" 
onset,secondary flow than the pres:sure 
The other interesting fact we found 
when investigating the onset of the secondary .flow was that :1 a~ the Clurrent 
was raised, initially the flow was steady 1 but 1rhen I~ 5 amps a steady 
o~c.illa.tion. developed wi. th a period of about 2 seconds~ which was easily 
observable on the manometer by the rise and fall of the meniscus. This 
indicates that 9 as I in.Greases~ an unsteady flow occurs rather than a 
stronger secondary flow~ as in the experiments or Lehnert ( 1955L 
Realising that the most critical region for examining the onset 
of secondary flow was region (1)j we then measured6p ~t two values of M 
'When r = ( 1.267 )b. The results 1 plotted in fig o 7 o 9 (b) 9 show fi Mrtly tllat 
the velocity decreases as M increases for given I 9 as is to be e:x:peicted 
from the theory of ~ 3 J 3 o Secondly they show that the ons et of 
secondary flow is suppressed as M increases, which i s the same conclusion 
-124= 
0 l •O 
\ 
Z.•0 
..3• 0 e' = (Y--b)/C({ ~,r~) 
-% 1, 10 lbe.- VQfltAtl6Vl ~ V(dDc~ ~~ md1us trLO_\~--
_lf 9 bt<M~ JoiL.1 -rr:: Gb)/(o, M-V,J/~c 51,S,_ 1 ~, 1A 
that we reached in examining the electric potmtial measurements; 
thirdly to operate in the required regime we needed to measure p!<essures 
of the arder of .,03ou1 meths which are about as low as can be measured 
l'lith any degr ee or repeatability. This meant we had to operate at 
high values of M, with the associated disadvantage of using the pitot 
tube when the thickness of region ( 1) was least. Fourthly we have to 
presume that though the }'!RD probe error f or these values of I is ~ 6·56. \ 
the random errors preclude any conclusion as to the exact linearity of 
the 6 p~r2 relation. 
. We measured the radial distribution of £J p at J = 0 991 andJ = 
.972, at only one value of M, 1370i> since we could not lower M enough 
to obtain appreciably different yet repeatable readings and at only one 
value of I 9 • 7 amps. From t he radial distribution of /j, p we cal~u= 
lated lf ~ using the relations (7 .2. 9). Since 6 p is positive i.."1. region 
(1) and negative in (.4) it is zer o on the boundary between these two 
regions and consequently it is impossible t o calcul ate the vel/ity t here 
at all accurat el yo We have plotted [ ( lr ()) 61 '2. /Vl)/2 Jovt 1] 
against (v--bXO\ M--~2.) in fig.,7~10 so that ii' a suitable theory 
ea.n be developed it may be compared wi"L,h these results . We note that the 
velocity is greatest nearest the wall which i s predictable since the jump 
in potential across t he layer is greatest when (p is greatestj Leo near 
the disc. Also note that 9 as r decreasesll U--e, gecre™ more sharply near 
the wall 9 which is to be expected sincell if the wall :i.s highly conducting , 
the current must leave the electrode at right angles 9 thus redueting the 
shear stress and consequently the velocity at the ·wall. We may :note that 
. the Hartmann boundary layer here was so thin. 9 .001 11 t hickj as to be 
negligible. We compared the values of U-t::, against our only available 
theory ll namely derived from the assumption of a constant current distri= 
bution on the electrodes to see whether t he values obtained were of the 
right order and fau~d t hat at j = = .. 99 j r = b, the theoretical value at 
a current of o7 amps was 3.09 cm/sec, whereas t hat found experimentally 
was 2040 cm/sec~ at j = =o97, r = b 9 the theoretical and experimental 
values were 2o85 and 2 .. 65 respe©tively. Note that the experimental 
values are lower than the theoretical 9 which is to be expected sir1c9 
the maximum velocity induced by the theoretical current di stribution 
=125= 
\ 
\ 
'\ 
' ...... 
......... 
~D 
~ · 5 I I 
1·0 2 ,0 
~flLiiimparicev, 6f:0 ~~~~fl~L:md e.h:h:.- f Dbii(.11 ~_fr veJC(.;f,J--(J = ' 1C\3 1'\i = /7)&7) 
occurs at r = b j whereas the experimental maximum occurs at r > b, which 
is to be expected with highly conducting electrodes., 
7 3 3 Discussion. • o· • 
Having calculated the velocity from the pi tot tube readings, we 
can now compare these values found with t hose caluelated from the electric 
potential di stributionj using the rela tion J 
-"?i Cf> B 
- -- + U--e- 0 = 0 3.3.24., c)y 
Rew.ri ting this in a rwn=dimensional form we have~ 
~-cI~d?ir) 
I 
~y 
Jn_ fig .. 7i11 we have plotted tSd d (cf;/:r:.)/a-r and(Lre Mo.. fov{/1,gainst 
(r-b )a.JYI='°z ~ using the uncorrected readings of the eoP• and pitot p:rob~. 
(With the latter we have corrected for the static p1•essure gradient.) 9 We 
see that a discrepancy of.>100% exists bet~en the two cu!"Ves. We then 
ealeulated(u-e-MGt Jo~)/I using the formula (7.2.9)g (as we did ror fig. 
7.10) ~ and thence calculated the mean value of kid_ 2. c):~ IMo-. lT ~Jo~./ I] 
across the probe fa~e in order to use the correction f~rmu~a (? 02.2) for 
pm' (We took k = t being the value for the two.,.,dimensionr; prob'3 _ _ 
examined in §4.,3). We note that though d~ri/a2 e:.. ., 1 9 since ~~I.rt /~t:, r 
was so great the c.orrection was large enough to reduce the difference in 
the maxima to about 3o%.. We also note that the 1mximum of the uncorrected 
curve of t he potential gradient is at a lower value of r than the veloeity 
maximum and that, with the correction applied 9 the maximtun moves to a 
jggher value o.f r., We note that this displacement. is appro:ximately equal 
to the diameter of the probe.. See fig. 7 o 11 o This effect 1.~s predJ.cted 
ll1 §7e2 .. 2 for an t e1v potential profilej which we saw in figc7 .. 8(c) 
resembles our experimental cu :rve. (We may note that the criterion :for 
the e.,p ., probe error to be negligible, i.e. d~/a2<:< 1 ll was n£t_ satisfied , 
d~/a2 = ., 105). 
The main reason for the difference i n these t wo corrected ~ui:"'ifes 
is probably that the experimental situation did not suf'fic::iently satisfy 
the condition that cl/ar .:» 1/r and that therefore t11e radial currents were 
sufficiently large to make (3.3o24) a poor approximation. 
In conclusion we believe that these experimerrts have convincingly 
demonstrated the foll01,.r.ing qualitat ive flow :"phenomena predicted by 
Moffatt (1964) and furthar discussed in chapter 3o 
(i ) The I cham1elling' of current between the two circular 
electrodes o 
(ii) The existence of thin layer s joining the disc edges of 
the circular electrodes. We have ob:served that in these layers a 
velocity is induced, which, wh en M >> 1:. decreases as M increases and 
that these layers become t hinner as M increaseso 
(iii) The dependence on the magnetic field of' the potential 
distribt1,~ion across the eleetrodes 9 due to the finite conduetivity or 
the electrodes. 
The :rrain g:ganti tati ve results are g 
(i) That when M >) 1, the resistance between the two disas 
R = 6cp/I was found to va:ry with M according to the formula /_ v ~~ R/Roe = I- R I_ M I 
i-lhere k is a constant depending on the geomet.ry of the electrodeso This 
result was predicted on analytical and physi cal grounds in §3.,3 03. 
(ii) The potentialg ~ 9 in the regions ( 1) was fou..nd to vary 
with the radius Y-- at a given value of j a~co:rding to the f ormula~ 
(p f~ _b..._ f (t-,-b )/( °' M-~)) 
The function f varies with the disc geometry and the value of J it is 
independent of M when M ~~ 1 o We concJude from this that the thiir.;kness 
l 
of the region ( 1) was O(~). 
(iii) From the previous result, using t he theory of i\3.,3.,3 we 
conclude that the current distribution on the electrodes was a functio11 of 
[--r- bX'_ °' M --\) J near the edge of the cli.scso In t h e centre of the 
discs the current distribution was constanto 
(iv) The value of k (see resu__lt (i))i, for the second apparatus ( l = o 190) g was 1 .,22, whereas that calculated assuming a constant current 
distribution is 1 .,064 - a difference of 20%0 The values of lf& found 
from pi tot tube readings at Y = b were within 25% of values ealr.:ulated from 
the same theoryo The distribution of' ~ for Y< b did n,oi compare well 
=127= 
'!dth the theoretical profile.11 but did a~ee to within 15% for r > bo 
Most noticeable though was that the shape o:f the two curve-s was very 
similar .11 indi eating perhaps that the potential profiles for r > b are 
not very sensitive to the current distribution on the electrodeo There= 
fore we conclude that the constant current distribution thoory provides 
a. rough estimate fo r the va.lll es to be expected experimentally ll particularly 
for r > bo 
( v) The calculations of -velocity in tb e region ( 1) deduced I ' 
from Pitot probe measurements 1,,r"lra between 100% an.d h.O % higher than 
those made from electric probe measurementsi, men no corrections were 
applied for MHI) probe errors o After applying such corrections we 
found that the difference between the two sets of measurements -were 
reduces to between Jo% and 15o%, also the value of r at the maximum of 
the eoP• readings became closer to that of the pitot rea,dingso Thus ·we 
conc luded that our error eorrections were of some value 9 particularly 
that to the eoPo probe w,iich has never been used beforeo However, the 
large discrepancy remaining between the results demonstrates how little 
we understand the probeso (Some of the discrepancy may be due to the 
flows not satisfying the conditions of t he asymptotic theory j by which we 
compared the results of the probes)o 
S. Experiment.a in reeitangula.r duetso 
The aims of our experiments in rectangular duets were t wofold o 
Firstly we wanted to use Pitot. and eleetrie potential probes i11 
stabl~ ll laminar flows vhich are well u:aderstood in order to measure the 
errors caused by MHD effects, such as those discussed in chapter l+s 
and those eaused by using t he probes in regions wher01 the probe size is 
comparable with the distance in mieh large changes · in velocity or 
potential OCC:Ut"o 
Secondly we v.ra.nted to use our probes to investigate the flow in a 
duct whose walls perpendicular to the rrtagl'letfo field (BB) are highly 
conducting and wh ooe walls parallel to the :magm.etiie field (AA) are non= 
conduetin.gl' ioeo that flow examined in §204 and in our paper9 Hunt (1965)0 
In part,icular we w-a.nted to confirm the existence of the salient features 
of th~ flows found in our analysis and. to investigate the manner in whi@h 
these .flows b~@ome unstable o 
We mentioned the work of East ( 1961~) 9 Moreau ( 1966) and Lecocq 
(1961i,) on. Pitot tubes in ehapter 4o Other experiments with Pitot tubes 
in :magnetie fields have been performed by Sa~hs ( 1965) and Bran over & 
Lielausis (1961) on mea.mrl.ng velocity profiles of turbulent flows in 
na...ryow rectangular channelso As far as we are aware there have been no 
experiments to lleasure veloeity profiles of ;tami,_nar MRD !lo"W'\5 9 for the 
very good reason that su~h a task is d:iffitmlt an~ of' oourse 9 one 8rs 
measurements ean be compared with the theory o Therefore the work 
described i n this chapter is the first attempt info this fieldo 
more we describ e the results of electric probe measurements and j ,-mere 
possible 9 compare t hese result s with t,hose of the Pi tot tubeso (Al though 
Lecocq used both types of probe he did not @ompa.re the results of th~ 
two probes)., 
-1 29= 
Alty (1966) has already examined experimentally the dt et flow we 
a.na.lysedj Hunt (1965), by means of pressure and potential measurements 
on the walls of the duct, his results agreed very well with the theory as 
described in §20 5. 3. However, as the Reynolds number, R9 of his nows 
increased the flow ceases to be laminar and exhibited certain interesting 
effects; these effects could not be examined further by external 
measurement s 9 but only by probe measurements., Thus the work of §g.,.4 
is a continuation of Alty 1s worko 
8. 1.3. Summarr 
In t:l$o2 we describe .experiments to examine the flow in a n9-rrow, 
0
116n x 2i11 , duct with non-conducting walls using two Pi tot tubes, one 
with a circular .tip of diameter .02811 and the other with an oval tip 
(.01'2 11 x .062n). Th~ Pi tot tubes were supported in the thin duct from 
a wider circular duct, 111 in diameter. The main conclusions we drew 
were~ 
(a) At the junction between Sl,lch .a thin duct and a. wide duct 9 
the ex:i.t . effects are very much stronger, Le. affect the flow further up-
stream,11 the greater the magnetic field. applied perpendicular to the thin 
duct. As a re?Ult we were only able to use the Pitot tubes at zero or 
low values of M, the Hartmann number llsed on the 1'1a.lf width of the duct • 
. (b) When M = o, Pitot tube0s{eren.~p to { of the duet width , 
can measure the velocity to within 7%, the experimental e!'T'or in our . ca.se, 
whenFobe is more thAn one diameter or. thiclmess away from the wall. 
(c) When M = 2.36 and M ~ 5.03 we measured the velocity profile 
across the duct and found that the agreement between the theor~tical 
c:m.rve and experimental results was about . 7% when M = 2o36 for the w;ide 
probe and better than this for the narrow probe. The result s at ~ = 
5.03 were poor on account of the greater exit effects..., These are the 
first measurements lmown to us of the velocity profile in l aminar 
Hartmann flow. 
Il). §Bo3 we describe experiments on., the flow in a 06 11 x 3o01!'. duc.t, 
32 11 long 9 made of non'."'conducting material 9 and placed insi~e the 66 11 duct 
described in chapter 6. · We first investigated the MHD error on three 
kinds of Pitot tube for values of N(=O"'Bo1J/e/J.,,,) up, to about 3o The flow 
in ·this duet was less steady tha}'.l in the tf II duct 9 but despite . the random 
e1Tors in the manometer readings 1 two main conclusions e:merged. First 
the Pitot error did not increase linearly with Uo0 or as the square of B 0 
as predicted by the fornrula 
where p0 and ps a.re the dynamic and st<},tic pressures. ';l'his was t o be 
expected sincef N <,:' 1 was ,not satisfied 9 , the requirement for the theoret-
ical validity~f"tfI'o~~1); however it was interesting that the error was 
~131 .,;. 
I 
less than that predicted by (801.1) a~ N increased. Secondlyj using~ 
-a.s a crude measure of the probe error 9 we found that if N ·was based 9n 
the length DP 1 = 4. x (cross-section areaiperimeter) of the probe tip 9 d... 
-was greatest ( ~ 1 oO) ,for ,a flattened tube in w4ich the width p~,ra.llel 
to the magnetic;: field, a.pll = .,012 11 and the width perpendicular to the 
magnetic field, . b = .,062 11 ; it was smaller f or a circular t,ube (,:;;(.;l! ·4 ) 
and least (d\, ~ o2) for a f1attened tube in which .ap = .,073'\, bp = .,02.411 • 
This.effect was a qualitative confirmation of the r~sult predicted in 
§4.2. 
We also used electric p:robes in this duct. We first measured 
the electric field in the eore, finding good agreenient with the theoret= 
ical value for laminar flow ( < 1 .5%) when M = i.40" 2h-7 but poorer 
agreement (!"' 7% ) at M = ',/. f. In the boundary layer on the walls 
parallel to the magnetic field (AA) the exper::l,.rnental measurements 
differed :rmrkedly from the theoretical values ll 1:;iy about 50% for M = 247. 
However the difference decreased as M increased, indicating that as M-'>CO 
the agreement with the asJ'lnpj:.otic theory would improveo 
In §go4 we describe experime~ts :in our main 66 11 ducts whose walls 
perpendicular to the :rpagnetic fieldJI Aall are highly coJ;J.ducting and walls 
Aa are non=conductingll the dimensions being 2o1~86i1 x 3.,01011 • We only 
ecrr.a.rrd.ned. the f+ow at R < 1000 and M = 943., Our measurem(,'nts of static 
pressure 9 AP 9 showed that the flow ceased to . be J.amine,r at- R ,S 70 a.ndfor 
R "> 500 settled into a weD, defined second regime in which 6Po(' R and 
6P /R. was about 206 times . its laminar flow valueo The ±~low was very 
• 
unsteady for 500 > R > 70., Measurements of veloei ty in the boundary 
layer, on the, walls BB were below the values predicted by the laminar theory 
(H,mt 9 1 96 5 \ yet the velocity profile e:,µribi ted qualitatively the main 
features of,the theoretical laminar flow, namely a large velocity close 
to the wall, a negative velocity in the outer part of the boundary layer 
_and a cope velocity less than the maximum velocity in the coreo The 
electric potential measure:qi.entF la.ken.. relative to the wall A.A were 
also belaw- the theoretical, bo·th in tha core and the boundary layer, 
ho-wever · they, indicated clearly the, width of the boundary layer and the 
important fa.et that l, = 0 in the core in tl;i.e second flow :r~gimeo 
Comparing these results of st:;.tic pressure, Pitot J)l\'etsa.ur~ 9 and the 
potential measured by the e.po probe we were able to provide an order of 
]D!.gni tude explanation for the curious second flc,;,r regime, in ·,,rhich 
secondary flows are likely to occuro 
These experiments and those of 9hapter 7 have shawl:) that 1 despite. 
their limitations which need thorough prior investigation 9 Pitot and eoP• 
probes enable us to investigate .flows not amenable to theoretic;:al at;i.aly= 
sis and increase our understanding of those fl ows. We cannot 1 yet 9 . 
. j:kt.1.n-b+~·ttri,e. 
place great confidence in the,( deductJ..ons made from their measurerre11.tso 
Clearly further work is necessary. I 
These experiments ha·ve also sho'W!l that the main 66 11 duct ca...n be. , 
used for investigating other duct flows as well as that examined in §8.l~, 
thus ~ndicating its versatility. The other important fact proved by 
t;hese experiments is that it does not lea.Id 
I 
I
I 
( "' } =5<de.- -·ev-rc~ t:, ~J <.._6'J ~'" YICuC ~ ,.., • .C.,.~ 
bc1Ne l, :!::> , f xe.s .B t-.,5 _ 
\ 
\ ~ 
l>~ '\1!: l1t,-,,,~ t,.J · I l+:l-p UJ-=r-' N\At . 
· \ :: 11 'i& r,;:=,;1. / N. 
l~l }U •11 i' '., i ·, / rfi '"T---::r---__L__~ll I . II i I . 
I ' I I 11 : 
. I ,, 1 • I 
1 1 s-i. 11 1 I' 
I I 11 \ j 
: I J :, \ l 14 ; Srud , T. J'fL.sr<ZJ>( 
- 1rj___ I I I I '
1 
Ii j 11! --,;-;;.- _j -- ---- - - - ---.-----(1- - - _,__ ' o ·· .. , 11 a - \I ~ ~ -- . -_ + ' lj• - -- · I <a f<'=--- "~ \- ~ - -1 -1 
-1-ltl-,, "I r - r/' ,--proli-e... . 4~" - fi,. --- - . - 3 ; JE-1~~?"~ - · -r? 6-:lJ:- - ~1-- "··--r--
.2.. 2.. _L__ __ , t=le>vv --~,1~ ,. \ ; v r\~"'\' \' )? ,,_-- -:h.__-~ ' II l/ H I 
.......: l.!... \ 1\.-f y ,1·-,,.:-~ ' !--",1 4- _ ;,> ~ 
- \II 
· ~~?' ... \~ 1/ \ '\,' :0',\_), -- -1i'------f.l 't-
1
_ 1-
11 
-
--
_11 _ _ _ "l:'....'l , ;:;:::- r 
!_; ---~ ---=--f--p --_t11r-, f-,1- L1 "":j(I) 0 - -- 9 - - -- - I -- q -~ ~4) 
- .~ -- + I --=-- .:1 , ' I - -----~ .. -A t' ;e..ce., .::x::.. , .!. • h,us0s lc:Jk. i; 'YL0.11,ve, N._jloV\ \.'VCJS~;Y\c,- \J r t A II E.. , '<lJMuved w\.iev,. MSE'.rl:J tmbe,. f},v...,uJ j:,•VltS S c..vubbe..r::__; ~ - 4 - --';:,,> 
nJ e... 
h D \ 'J;:' 
CJO.Nl,k 
----
A 
1 
~~~!f----f> 
-0 
i 
0 
I''ne purpose of our first e.:xperiment on the use of probes in 
rectangular ducts was mainly to examine the errors caused by using a Pitot 
tube in a shear flowo We found that most te.:z...>t books on this subject 
merely recommended using Pitot tubes 'Whose diameters were much less than 
the distance in \'fu.ich the velocity crianged appreciably. or else gave a. very 
erude factor for measuring the displace:irent e.f.feet eogo Rosenhead ( 1963') 
p0 620 o In air or 1rrater now~ if the velocity profile is required very 
near the wall. 9 hotwire anemometers-P Preston tubes, or heat transfer 
measurements may be used; all such measurements would be very eomplieated 
a..'1d uneertain in MHD flows. 
Duet . As a result of these considerations we designed a rectangular 
duet with dimeinsions !rn(= 2a) and 2i11 (:: 2b) ~ in section, to be used with 
PU,ot tubes of diameter (d) up to 0028·11 9 Leo d < ol+Ba. In order to 
avoid blocking the channel with the stem of the Pitot ·we decided to 
increase the duct width downstream of the Pitot tipo The duet 9 as 
const'Vi.."t:lted~ is mown in figo8o 1 o It '!tffi.S made up from various pieces 
of 11 1-erspexu ooing glued a.nd bolted together. Its length was sufficient 
to enable us to measure fully dev·eloped i~lows and a settling ehamber was 
©on:str··mrt.ed upstream of the slot. To reduce ~ny tmsteadiness, t his gap 
w~s filled 'With a nylon washing-up s~r.rubber" The pressure t.appings 
were constructed by the same method as those of the 66 11 duct described in 
chapter 6. 
Erobe~. The P-ltot tubes were constructed to point sufficiently far 
upstream into the na:rr ow portion of the duet ( the slot ) as to avoid any 
effects caused by the duct widening and the blo~king of the flow by the 
probe stemo The first Pitot was constructed so that its tip would be 
protuding } 11 c~~IJI...) into the slotp sinee we believed that with 01" 1'rl. thout 
a magnetic ii eld the exit effect would be . negligible this :fa.JS' into the 
flowo (How wrong we were is shown in §g.,20~~) o This was ~onstrueted 
in a similar way to that described in §702.,3j) namely by the fitting 
together of different stainless steel tubes, the tube at its tip being 
o028U Oodo 9 00155¥1 i.,d. (See figocio2a.). 'I'he seeond Pi.tot had a 
(")]f-d. "';,\ti..D[g;,\o.r-1~-_ 
i 
I j.--tJ3e. cf ~:t,\.\, d\J(C,t' , 
I 
different tip.9 being a .,OlJ-21! Oodo tube which was drilled out until 
its i.d o was .,03611 and then flattened until its external 1,rldth wa.s 
.012n 9 (see figo8o2b )o We used the Pito~ uninsulated and insulated, 
but f oun:i no difference in their beha:viouro 
Probe mechanismo The probe mechanism was not used in quite the sam/!9 
way a.s described in §6e4o Instead of using the eranks to maintain the 
c:rank=bar B (see figo8o 1a) parallel to t.he dret ax:l.sj we removed the 
cranks and fixed the cra.nk=bar rigidly to the spindlep S 9 by means of a 
dowel)) Do Then turning S moved the dummy probe across the ductj but 
altered its orientation 9 '(f' j relative to the du ct axis, sincet< 2° for 
this thin duct the effect of varying (5 on the Pitot was negligible. The 
spind;t.e S was turned by pushing the cra:nk~bar B with a screw adjustment 
attached to the probe block, see fig.Jo1(b)e 
With this design of du et i it was very difficult fitting the probe 
mechanism and probe onto the duct. To do this the studsJI T, retaining 
the probe block had to be removed and the probe block very gingerly 
manoeuvred onto the duct to avoid knocking the . probe, (the probe was 
fitted in the probe mechanism outside the duct.). To ensure correct 
aJ..ig;unent of the dummy duct and that the probe had not been knocked 9 we 
rem.oved the end pieee~ E~ and looked into the duct a 
The details of the duct rreasurements and the various oonversi.on 
factors for calculations of the experiment,al r esults are giYen in table 
8010 
S,,2 .. 2. Sta._tic pressure measurements. 
Since this was the first experiment in which we had used the flow 
circuit described in §6 .. 5j we ought to mention the procedure inviolvedo 
BMause the static pressure reading on the manometer between any two tap-
pings or between a tapping and the Pi.tot when there was no flow 9 (the 
zero reading) 9 tended to vary it was necessary to take this zero reading 
as often a.s possible, otherwise interpolating over a long time between 
two zero readings led to large errors in ~aleulating the pressure 
differenceso (In the electrically driven flows this did not create any 
problem as t.he flow was started by turning a m.teih)~ A typical series 
,\o 
,b 
I 
$' I o 11 I, 
~'&i__f1e.~..dmf{<6-~&~:tt.1.od o0~cluc.l: ~c:rmr f oWYa.li e.' 
of readings would be taken in the following way o We assume the f1ow 
eir~uit has been filled such that there is mei:r'Cttr'y either side o:f the 
weir in the lower weir tank and that all the taps have been closed., 
(We refer to fig~6.,6)., 
(1) Take zero reading on manometer., 
(2) Open taps (4) 9 (5) and (?) o T'.o.is does not 11:!"ad to a .flow from 
the upper weir tank u.w.,t. to the L.w .. t. s i..-r1e:e the pump seals the flow 
when it is stationary. 
(3) Turn on pump at desired f:ow rate. 'l'his makes the mercury 
circulate between the lowoto and u..w.t., returning via the overflow pipe. 
(4) Open tap (1) and regulate throttle valve. Flow now starts through 
the duct 9 settling d01,m very quiekly. We ensure that the pump is 
opera.ting fast enough for there to be some flow through the overflow pipe .. 
(5) Take manometer rea.ding. 
(6) Take potentiometer readings of voltage a@ross the flowmeter and the 
shunt in the magnet supply circmit. 
(7) Re-check manometer reading. 
(8) Take one or two more readings at differing flow ra:tes 9 by 
regu:Ia ting the throttle . valve and the pump .. 
(9) Tttr.'n off tap ('1 L This stops the flow in the ducto 
(10) 'Ihrn off pump. There may be some leakage back through the pump 
so we also !jlose (?). 
(11) Take zero reading on manometer. 
The first measurements we rre.de in this duct were static pressure 
readings between t__appings (4) and (6) 9 /J.46 ., (See f'ig.8ofo)l> the ,itot 
being in position in the duct and M = 0 o Our r·esul ts ~ which are plotted 
in fig.803 9 showed that 6.Pi,6 was proportional to Q only when Q < 7 cc/ 
se~. This surprised us s::µ1ce the flow was fully developed and we ~otJ.ld 
not believe tmt the effect of the Pi.tot tube was so g!"ea:t o Hcrv1ever ~ in , 
the linear regime we caleulated the mean value of tlP46 /Qv (see . t.B.ble 802) ~ 
and found . that. it differed from the th~oretic;:a.l value for plane Poiseuille 
flow by 1 o 5% with a standard deviation,, s"do ~ of 8%0 At these yecy low 
flow rates our pressure differenees were difficult to measure and this 
was the cause of the ra...ndomness of the readings. (The mare readings we 
took the more consistent the mean value)" 
\ 0 () ~ II 0 
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-t}e,s_lari~'PI~ re.. Jrof-lt,ri~ Y'ltL~6_1'/_r~(Q) 
_J\Lllr:: -#1e.. lVld_of -ft7e.- dvd: _ lAJhl-v\ J!l ~) l_. __ 
i 
That the cause of . the non-linearity was an erl t or Pitot effect 
is shown better by figoBol~ where the pressure difference~ be·tween various 
tappings along the duct are plottedo When, Q = 6 e.e/sec 9 the pressure 
gradient is the same as that of plane Poiseuille flow near the exit but 
not at the entrance, it.is interesting to note that the theoretical (ie €.) 
entry length = Ra/20 = 10 411 when Q = 6 ec/sec 9 whereas we find that the. 
10 411 from the end of, the duct dp/dz is 00% greater than its final value., 
When Q = Bo 55 cc/sf}c 9 we see how the pressure gradient is least in the 
cent re of the duet, thus c onfirlliing our hypothesis. 
-·-
.We then measured AP, 11 , as a function 9f Q (figo8.5) 1'ben 
M = 23.8 and M = '4,6 .. o and fo1.md; to oor s'!ll"prlse 9 that the ~ P,i.i - ~ 
relation was not linear for thf; same valuE;s of Q as the . ~p4l,- 6< relation 
was linearo The entry length 9 of oourse:p iri. this sit uation was mu.eh 
less, the .obvious conclusion was that the exit length was greater o 
Again the A P-2: eurve conf:irms ,this Yiewo We see in fig.8 .. J that when 
M = .46oO, dp/dz agr~es with the theoretical value .for fully developed 
flow .when (=z) > 4n~ but diverges from this value ' -ve:ry sharply as (.=z) 
~Oo We believe that the explanation of this effect is quite simple 
and should have been forseen. 
If 9 when M '>)1 j the flow had changed B'Uddenly from that of comren= 
tional Hartanu;m. flow in. the narrow part of the duct to uniform now in 
the wide pa,rt 9 ~t would have meant that the .current in the slot would have 
. returned from x = b to x = =b via the high relsistanee Hartmann layers on 
the walls o Inst'ead of doing this the current near the end of the slot 
could return via the low resistance , pat}) of the wide part of the duct 
where, v z and Ex a.re much . lower o Thus 1 the wide part,_. of the duet 
effectively short circuited the Hartmann flow at the exi:t, of the slot 
(!)a.using (=j.x) . in the core to be greater. and producing a component of 
~u.:rrent =jz for x > o and +jz for x <. ~ o Clearly the , ef'fect of t,he 
short, eirc-q.iting would be ,grea.te~t when M was greatest, .and would diminish 
towards the entrance of the slot 9 Le. (=z) increasing. (We re.fer to this 
hypothesis later)o 
.t 
,~ 
0 
(i) 
-· --------y---------,------~--Lo loo (c. c/ sILc )._ 
' 'fuL_fu~~-F2~re,~nincl,i_~-*~l_-0_p,P )a~ 
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0 - circ»l<A, ?',bt · ()l-t '' o.d 
x - t ~-tte&t\e..d ,p;tbt, fulc.k ne$ · 012 11 
. 2. 
-1·0 
& - ci rc.,ul~, ?', bl:. · ()).t 11 o.d 
,· 
x - +=l<xtt-ue.d 'P,W:i, ~ck~es; I 012 ft 
-1·0 
-.l 
+hrore..b'CfA{ CJJfUf) 
plara<. P6i.::,e.uille.. ~c.lN, 
- ,'L 
. I . 
I 
I 
. 2. 
0 
EL2.3. Pi.tot tube measurements 
We first me"-sured the pressure dff :ference b~tween tapping ( 1) 
and the Pi tot tube 9 llH p 1 when M = 0 at y = x = 0 9 :r. = =tn. We 
noticed that the lag in the, PJ.tot reading was much greater than that of 
the :st9-tic pressure so that j since the flow over the weirs '!,\,,as not very 
steady 9 these readings were more prone to r andom error than those between 
two stati~ pressure tapp1ngs. · 
We measured I\P,p as a ~.metion of ,Q2 , since if the !low 1,\Pcl.S .. __ , 
fully developed liP.p ae.. Q2 • Our results 1 which are plotted in fig.806 9 2 . 
show that b.,P,r o(_ Q for Q (,,., 7 cc/sel/2. and for Q greater than this 
AP.p falls , below the theoretical :line. Wh.eth~r this was due to the 
veloo i ty profile changing or due , to some stati® -pressure gradient is. :un= 
certain. (We did not have time to investigate further)., In table 8.3 , 
we have ealeulated the mean value of l\Rp/c/ and {fplP /Q 9 for Q < 7 ce/sM 9 
compar.i.ng them to the values for plane Poiseuille flowo , We a.gain found 
that the agreement was well within · the standard deviation l' so that we 
ca-ald assume that the nean of a . few measurements of A.P1,. would be 
accurate to within these limits • 
. We f,hen measured A P,p aga.:i.nst 'Q at .various values of y, ( l. G:·-el) ) 
for Q < 7err,/sec a.r.tj th~nce eal~ulated vz/Qo F-rom our resu.:J_ts ~ .;which are 
plotted in figoBo 7 9 we draw two te~.ti ve conclu!3ions; fir st 9 th~ bloc:king 
effects of the P:i.tot a.re negligible 9 and seconcl. 9 that to within the 
experimental error the Pi tot ma.y be relied upon to measure velocity near 
a wall prov1,.ded it is further than 01'.le diameter f'rom the -walL 'fo test, 
these ideas, p~rti.wla.rly .t}J.e secondl' we then cons-trueted a s~coriJi"P.i..t_;-tj 
already·, described in §B.2o 1 j o After measuring jj ~P against Q,, (table 8.,4) 
and f'indin1s a similar order of agreement as with the first Pi tot w we 
mE3a~n.U"ed the Telocity at points ac:ross the dm;to (FigoS.0 7)., The measured. 
velocities a gain follow the theor~tical eurve 9 differing by less than 
t he experimental error for @-=y)< dj whieh covft.rm J our conclusion mentioned 
above . The reason why the points all fall ~low the theoretical curve in 
the second c,ase as eomr,ared to fa.l+ing ~ in the first is obscure. 
Despite the possible random errors,9 it certainly seems a significant 
differenceo 
-·<"( 
-·7 
-·2 
- , I 
0 
b. ?,p ~V\ 
. l~(.,~l.,A ('t\~1;;, 
hi 
II 
, I 
I 
I 
0 (V'\ ==-'Z<3b 1)'()(.,()(l'(l(Ye.ol 
t iip rA~ eonedd. "'°'ke.. 
8 M == S-·D~ uncoou:1-e.d 
i lip 0tt O:irreded uo.lue... 
'4 
'""lhe6ret,~( curves : 
ij,.v--·b~\C:Hvt flow o,± M; e.,:Sh 
r\-ttr-bMo.VI~ f ow oJf M =- ~·tS!. , 2.. 
trob"' '('It cl&. 
ro Sco..le 
~· '1 _ Mwsured ,1/(,JQ_a§._rQf b w~~ M ~ 2.·.% ' S,t,_"';;,' .... 
Urc.(J0r ~~t:oi h1be.. 
- --- -- ------· 
I 'b 
v~/ a 
\V\U',fl/te.G; e,. c. 
4 
'%$,IQ _ MrosuJtlLveJ9(A~ff>.k,_ll,lbm MO z.,._:,_(. :- .. 
'}lolit_V\ tcl_ ~ttoL_ilLc_ 
11 
Since the exit e.ffeets were WO}~S~ for the statie . pressri.re 
measurements i'fuen M >> j than when M = Oi1 w~ expected a. similar effeet 
whe~ using Pitot tubeso This was~ indeed 9 the sase as is shown by 
figoEL8 where AP,p is plotted against Q., . We see first that b.. P,p 
is rie.w:,~_ive and second that./:!. P,f' varies approxilr..ately as B0 
2Q at the 
same value of x and z o This !'el?'1lt would eon!irm our hypothesis that 
near ~t @f the slot ;-men M .>> 1 ~. large cmrrents eir~ulate in the x=z 
plane 9 ~in~e the presence of' jz leads to a vertical pressure gradient 
(=j
2
A,) 9 whi~h would p:i;oduee ,fi negative pressure .in the Pitot 9 To 
further test this idea 3 we measured_bP,p at x = .,511 and found 9 . as 
would be expe~ted on our hypothesis j that ( - bP. p) was redu(lledo Toa:t 
the effect diminishes as (=z) increases is verified by the measurements 
-of' [.) Pip ta.ken by the thin Pit.ot at z = =!11 ., we, see that (- AP,p ) i.s 
redu~edo After first using an mitnsula.ted Pitot 9 we coated the Pitot 
with non°-,conducting Perspex ciement 9 but found no difference o We note 
t.hat the MHD eirt"Ol"~ of' chapter. 4 would lead to a. positive b. P, p and so 
diminish the negative pressureo 
, We, then deeided to measure two velooity profiles at M = 2.,:36 and 
• N!ffi ,:,o...e.cf,or1 ~flfi"' , 2: . 
M = 5o0'.3 9 t1s1.ng the/...~• ~ .~A Fjp~ i70B0 . Q9 where B0 is rrea.sured :in 
wo/m2 ~ Q in litres/ se~ l' M.d 6. P,pin L"lehes o.t rneths o . We found that 9 at 
these low values df M9 b'ir was approximate'.cy proportional to Q2 so that 
i 
V1Fa @ould~o~p;i.re our r~sults with the theoretical Hartmann flow profileo 
Our results 9 (figo 1:t 9) 9 show how even a.t these low values of M the 
correction factors a.re a.ppreoo.able and how unreliable the l'."esults ar~~ 
inconsequene!:leo However)) the results for M = 2o36 when , i!- = y> d a.gr>ee 
wi.th the theoretial to within the experimental error and eertainly 
demonstrated the flatteinirig of the, velocity profileo 
Our results with the thin probe are better 1 f'igoSo 10 and three out 
of the .four ·points. (eall'.!;h takm from a graph of h. P,pa.gainst Q2 ) for,a = Y> d 
agree very closely,, Poo:i;· as the :results of f'igsoSo9 and ·8., 10 are, they 
are the ,to~ measurermnts Y we know 1 of velocity profiles in Hartmann 
flowo 
} I 
~fr-.l·· · . ~ 
G 
I 
I 
~ 
, I 
The :rmin purpose of. the experiments in this duc:t was to 
investigate the Pitot and ~ .. p 9 probe et".l;'ors when the interaction para.meter 
base on t..h e probe dia!ll.eter, N.~ was O ( 1 ) 9 sine e the .. t'!lJiPeriments Tu.lb.ieh had . 
been designed f'or this . purpose in the 1In duet f ailed when M >> 1 " We, 
needed a duct w.t th non=eondueting wall s 9 (so that E ":\ 0 in the core) 11 ..... 
sufficiently narrow for ;reasonabi :, a.remurate readings of Pi.tot pressure 
a.t at,tainable flo-wrates 9 and yet wide enough for the probe g_ot to block 
the .flow. 
Dutto ,We de~ided, to make a duet .,q-a x 3o01 11 in section and place it 
inside the ma.i.n 69 11 du~t so .that we eould use i;.he probe :mechanism.easily" 
The , i.'!'lternal duetj which was made of VPerspexn 9 is shown in figo8o 11 ., 
Its length was 32J18 i> with a rrunded en try~ the probe rneehanism was' 911 
f~m the end to avoid any e.xi t -effects whi@h would be smaller with this 
duet anyway a The internal duet f'itted into the main duet suf.fi~iently 
snugly f.or the leakage between its walls and those of tqe main du.tit to 
; 
be negligible.a Even with the little leakage there was~ we felt that 
short ~ir~uits @ould only .be eliminated by cw,rering the eoppe1~ walls 
of the du~t w,ith semta.p~ o ' The only questionable f'eatur~ of its 
design were the ispaeet50 ~ shown in . section in figoBo 11 (b) 0 -whi@h were 
needed to keep the two walls q.parto Aeeording to the entry length e~.= 
· flte5,e. 
c:ulations of Shercliff (1956) 51 .the ef'f'eets orJ 0spaeer.s0 nea:t" the probes 
would be negligible at high field strengthso 
-We might mention that the su~cessf'ul eons:l,ruetion. and use of this 
inner du@t demonstrated the versa.til,ity of the,- 66 11 du(:to 
P.t-obe.s o One of the aims of these experiments '\II.Ta~ to see if .t.he ·shape of, 
~ probe affected its MHD error in .order to ver1.fy 9 at least qualitatively9 
our theoretical coneilusions of, S!i-o2o We had already constructed a 
cirru:t.ar and a flattened Pitotj v.rhose dimension parallel to tht;i nagneti@ 
field, apfl (=0 012 11 ) was very miieh less than that perpendi&alal." 9 .bp 9 (=.,06218 ), 
so now we construeted one with a.p = 0 073 18 and bp = .,0241~ {fig.,8o12a);the 
reason we increased its size was to reduce the lag of the Pltot ·reading 
=/40 = 

relative to the s:tatie pressure rea.ding9 whieh was even more 1.mportant 
in this duct -where the flow was considerably less steady than in the ta 
duct., 
As an el~etrl tt potential pro b~ we used one already eonstrn~ted 
1 . . 
for the g'8 duet 9 We used it in preference to tha;t construeted f or the 
ele etric ally driven flows be oa.use the now. at its . tip would be less 
a..ffected by the stem in this design., (Fi gogo12b)., 
We first made .static pressure mea.surements 9 -when M ~ 135 9 between 
tappings (4) and (5) G We. found that rma.surements of ti.~ were not the 
same as those of (.:.. t. P4r ) ., ( The notation refers t9 the way in whi@h 
the pressure tappings are eonnected to the manometerj either tapping (4) 
or (5) could be oonnee.ted to the .. micromete:r side of the JMnometer 1 if 
(4) 9 we measure APn; and if (5) 9 bfl.,_r )., Whether t his was ea.used 
by the :rmnometer or some spurious connection we are not sureo Ho~ver 
the results were repeatable and the mean of llP.n.1 ,and (-AP'f.r) mtlS 
found to agree elosely with the theoretical value, as shown in tabl~ :g.,60 
·. . ·. . 1 We .found the flow in this du~t more unstable than in the 81 du~t ~ though 
the flow was lamina?' the pressure was all the time osdllatingo This 
may partly have been caused by.the f.a.l@t that there was litt le eonstr'i@tion 
betwee!J. the du@t and the lqw.,tj wheJ;"e the flow over the weir ~s not 
. . 
steady~ so that though the !lowr_1t..,t~9 as rre asured the flowmete1•, 11~s very 
steady 9 the st~.,i~ pressure level in the duct .was fluetuatingo On 
a.eeount of the different lag in the leads ne~essarily led to random errors o 
go3o3o Pi tot tube· ,measurements O 
The first readings were made ·with the e:ir eula.r Pi tot used in the !10 du~t (still :tnsulated)o The pi.tot tip W9-S in upstream of the pressure 
tapping so that» in ~alrula.ting the veloe ity 9 we had to allow for the 
stati~ pressu:re drop 9 6 '1 o . We first measured t1R., against V 9 the . flow-
meter rea.ding,,when,M = 1350 The results a.re tabulated i:rJ. Table 8o7(a), 
the difference 9 .4 9 between the theoretical value o:f t>P 9 = b Pc,..;.t:,fls 9 ' 
·IDO-
II Me.-1'5 
b. .c:ffo 
•0'&0 
•?Y]o 
,o{.o 
·~o 
,oto 
•03o · 
•6'2- (). / 
3 , c 4'0 5 · o 
\I L~ v) ,f-f,,. ~ t.Cl\d :5 
~ _$,__I 3 __ Var:tat,t\cWLff YLLoi_ -bbe,, __ &Jlor __ j~~l~--fb~_mlL_ 
wb M = Z!i3 ... __ (J\~.ct 13 7 
I I 
i. I ) U~Pf>"" !4~ ~ (?r+!J e,"'-1.(f;:,, 11;_.,~ being calculated from.Q an<j)l, and t,prf 
Since A~-p was subject to random eir1°ors of at least ± .,01011 9 h ·was 
5t1bjeGt. to the same errors and since/ Li/< lAPs-p\ the percentage e:rr0 01"'s 
are la.rge o This explatns why the values ol o<..t''"" -4-- , 9 lJ,ave APo:JC.N 
so mueh scatter., It is interesting tbat the rrean value of' ex.= ,,.,38,, 
albeit with a standal"d d~Yiation of' ± .,h 8 whert.::1s Rast ( 196h.) .f'ou.w1d ol.,;,, 
0 39 for a similar Pi.tot tube o Also note that the mean valu~ ,:,f N := og9 
so that, the basic assumption that N , . 1 9 was not s~.tisft ed o Owing to 
the seatter of the results we ©.s1,..not tell if b""' Q,; Ci /i( l1 or f>.,.., (),1 <-/q -t cL/ t.t\ 
as would be the case higher order terms in N were appreeiableo 
We then used t.1-ie Pitot !) already desieribed in fig,, 7 .2 9 to see if 
the Pitot Jag was. reduoied by such a design of tubeo 'I'h'!',re was only ill. 
small :r..mp:ro,remento We used the tube to m!Htsur0e ,b.11.-, against V wh~n 
M '"" 11-7 and found that the ire an value l'.rt OC. = ,,49 r; but since o<.. noti~e0 ~ 
ably dt:i~r.eased a~ V increased the result has little :meaning (See table. 
So7(b))o Clear~r in this ease the higher order terms were appreci&.ble o 
We used the t,:rJ.n ntot tube w..i©h md been used already in. the k1u 
di.1et, 9 (fig o8o2o(b)) 9 t.o l:ie,e i.f ::1..ts MHD eiY-.".r'Ol'." would be ~e.~ tha.i"1 that 
to random error on a,~eotmt of bhe lo:nger lag of tbJ.:s tube" The, :r0esults 9 
tabul2:t,cd in table SoS and plotted in fig.,80 '13 9 show,the ~ti!'iation of h, 
~nd o<.. ,d,th V' for M.:::: 137 and f'cr.r0 M ~ 2530 We seej firstly that the 
gene1"'al increase of b, wi. th V is n;ore ,::lis~ernib1e 11m~n M 0:: i 37 than 
wht~n M ~ ~53 o J:n;th.~ laU,er @a.se,, the poin:ts do not really ind1.e:ate 
&1y pattern at.all9 on account of the random errorso We have hesits;ntly 
drawn on 1J.go8o f3 a line 1-mi~h might represent the irar:i.a.tion of' /::. ea.used 
by the second ang higher order terms in the expansion in N of the pitot 
erroro However 9 if we merely regard ex: as a. crud(;? measure of' the pi tot 
error these results demonstrate 
error is greater with this kind 
(area) /(perimeter) of the probe 
one important fact 1 namely that the M'rID 
of probeo If' N is oosed on D = 4 x p 
tip (:Leo 4 x hydraulfo mean diamete'.r.') 
then o( :i.s somewiere-between 3 and 4 times greater for this prohe than 
for a tl'lircular prob~ o It is puzzl.i.ng that the .error we find is 5 times 
~~~ than that. predicted by the t.heory of 311-02 0 We ha:1Te no expl~.nation 
for thiso 
=142. = 
I 
· ~II ~ 
J 
,61o 
,o~ 
~ t'i,'' .1 /nV 
\JN\~ 
)( 
· x>< 0 
,/I 
/ 
I 
0 .... .. .. _, __ , _ __ ' - ---·- ,-- .... - ... 
1;_.,_) Z.. t) 
I 
0' c, 
I 
+·o ' s-·o l~·o 
. Eneouraged by this confirmation of our theoretical prediettion 
of S4o2 we then examined the MHD error in a Pitot tube f'or whi~h a >> b p p shown in i'igo8o12(a)o The results are tabulated in table 809 '1hence 
we see that d.. _ is not appreeia bly bel?w that of a circular tu be, w.. en N 
is based on D __ The tube used had a ra. ti o of a /b of 3 eompared to p p p 
. the ratio or b /a in the former Pitot tube of .. 5. It would be P P how interesting to investigate in more detailj this ratio affe~ts the MHD 
error. 
s.,3.4., Electric potential masurements. 
Before describing_ our measurements we first note.that we eould 
not form an adequate loop of wire in the magnet gap to compensate :tor 
the f'lux linked by the oircuit between the lead to the tapping (5) 
and the electric probe so that our reading of potential was subject to 
oseillati9ns caused by_ small variations in the magnetic fieldo , (We 
could notD using the 66 11 duet mich just fitted into the magnet 9 use the 
method of Sher~li.ff (1955) mieh would have entailed having the tapping . 
lead leave the magnet gap i n the same plane perpendieular to the pole 
fa.ees as the probe)., 
As a reference for the . probe poten ti<J.l we IJl.easur~d the differ,erice 
between the eleetric potential o:f the probe, cj>p 1 and that of tapping (5), 
'$}!;, 4'br =,pp-</).r,, We have to, confess that we were not measuring th e 
potential at (5) but at (51), a fact onlydisc;iovered a±'ter the a.ppa.!"atus 
had been dismantledo Therefore we refer to lifi!t'r,,, In the boundary layer 
traverses we measured (/>p relative to 'Ptt 9 the potential tapping of 
tapping (lr,)., Then we refe,r to 111~ Ip o Since we could . not find any 
difference between !:::.c/1;P and f:4rip we only refer to A cp o 
We first, examined the variation of Arp with Q and found that 
over the .tlihwra.tes and magnetie field strengths available D1~6l o In the core we measured the difference in A(/;J / V measured at x = 0003 
and x = 0753 for values of M = 41oS 9 140 and 247 and found sati~fa~ory 
a.g:f!eement with the theory. of lamimr flow in a re ~ta.ngular duet 9 using Shereliff 9 s (1953) theory o , Clea.rl7 no , systematic MHD error was disee,m= 
,, ible from t here results mfoh, of course, was to be expected from the 
= 143 = 
2· 
f•O 
ol___~~~~-,-,~~~~~-,--~~M_~, ~ 
•C>l 'oZ.- _..,... 
4q)* 
VMY:i.. 
In m\J(rnY 
·, 
'oo<;;-
I 
_J 11-ote- ,._;.d_(ii ~ I (M~~SJ J 
~,!-....~~---'-~~~~,~~~~~~~-,~~
~~~ 
· 5 I ' p X ~ (_b- x.. J IV\ i-2-
symnetry arguments of §4.30 (Table Bo 10) o 
It wa.s when we ~omi:ared the aetua.l value or brp/ V. with the 
theory that curious discrepancies oecurredo In fig.,8.14. we have 
plotted D.1/Vr,tlfvagainst X = (b = x)/a~ 9 (It may be deduc ed from the 
theory of Shercliff (1953) that to .O(M=~), in the boundary layer on.the 
walls parallel to tl).e magnetic field Al}., Arp J,"fi' /flt, M-Vi .. 6( :::: J!,{?<J ., 
Whence ll in our flow 9 llf /M 1/1.. V ~ffl..) 9 since llf'1p is measured relative 
to the mid point of the ~l AA ,, 2. It also follows that if LJ,;Jc.. is t hei 
· v:alue of /jf on th.e wa.lll' x = b, if the.electric field in the boundary 
layer .were the . s~e as that in the core, then h..<ic/Mt/i=. .,003 mV/mV in our 
ductL Figo.Bo H. shows how .the slopes o.f all the lines for the core flow 
a.re the same and equal. to that of the theoretical lineo (We have used 
the data taken at x = 0003 and ,x = 753 .. to.calcula.t~ some of the lines) 9 
However the inter\C'epts on the ordinate, .. I 9 of the various :J_ines differ,ll 
all falling below the theoretical value. To see "Whether1 .as .M~ 9 
the value of . I tends to the theoretical value we drew figoBo 15. It shows, 
very roughly j that we c.an expect I to be between 2o5 and 3 "2 as M --::>oo,, 
Bowever with the the·oretica.l error only of 0(~1) it is hard to see why 
such a discrepancy exists when M = 247 o 
Little further .. insight into this -curious result .wa~ aff orded 'by 
the traverse of thee.po probe in the bou:t;tdary layer, fig oS'o'l6. cw~ 
measured li<ft.,f' here to avoid blocking effects) o These readings were 
1rery difficult to take since llq)4p was very low and the random error9 
proportionately la.rgeo We note that in our plot the ·pointw ~hQ,_uld 9 
if they were to agree with the results of the asymptotic theory )l a.11 fall 
on the same curve. Clearly they do noto 'lbe errors in this e.xperiment 
cannot be ascribed "to the probe itself as in the electrically driven flows 
on a.©count 9f the relatively en.all size of the probe... They may be due to 
the flow not being as fully developed as it should bej e og,, the 1/e_ entry 
length criterion vhen M >> 1 of Shercliff (1956) -was only just satisfied 
when M = .42o Of one thing we feel surej that these discrepancies cannot 
be explained by. the fact that we were measuring. relative to (5 u) or (4) 
rather than_ (5)o We regret there was not more time to spend on these 
I 
e:xperimerrt so They ought to be repeatedo 
t 
J/g '' 
' I 
I 
·7 
·I 
·:i· 
I 
•).. 
$DO 
----------
~b,,.;nar fi,.i t.to'._'.J 
1 
looo 
,02. 
I 
I 
I 
11 
130 4. 2! 11 :x: 3" ducti conducting and non=conducting wallso 
In fois experiment we hoped to use our probes to examine the 
curious flows predicted by our theory of 82o4 and Hunt (1965) and t;;' 
continue the investigation of Alty (1966) into the instability_of the 
flowso As was intended in the desi gp. we cou ld use the main 66 11 duct 
for .this.purpose with the copper walls exposed to the mercuryo (See 
fig o 6 o 1 ) ., To reduce the copper ~,, ercury contact resistance we le.ft 
the duct full of mercury ±~or 3 da.ys ;o 
The only new probe used in these experiments was an electrie 
potential probe ~ic}J. was designed to minimise the effect on the now 
near the tapp:i.ng ( 5), the Probe is shown in figo 8., 17 o 
The basic data of the apparatus are presented in table 8of1. 
8040 2o Static p!'~ssure tne"'asurements • 
. We first measured the static pressure between. tappings (.4) and (5\ 
t:>.Pc,~ j as a function of Q w1en M = 943. We assumed that the flow was 
fully developed since the entry length, given by(a R/M2) was mue:h less than 
the duct width. Also Alty ( 196,) had shown that for higher values of R 
and lower values o±• M t!J,e flow was fully developed in a. yery shor t 
distance. 
figoSo 1$., 
Our results 9 a.long with the theoretical l:ine 9 are plot t f'Jd in 
In taking the readings ·Jt was noticea.bl~ that the flow ,,,as very 
mu.eh less steady when R <. 500 than for R > 500j when the readings were 
very reproducible., This would seem to relate to the curious kink in 
the graph wien R < 500., It is pertinent to note that Alty found that 
his A P, against R curve .touched the theoretical laminar flow curve at 
R ::::! 1000,j) gradually moving away from it «s R increased. The,differenee 
may be caused by the fact that our curve was taken at M = . 943j as 
compared to Altyqs value of 228, however we showed in ~2o4o3 that th~ 
critical Reynolds number is theoretically independent of M as M__. oo 9 
so this is an unlikely expla.rta.t.±on., A more plausible one is that his 
flow circuit produced a more smooth er fl-0w than ours thus causing a 
smoother and later transition to the second flow regime we seem to f'ind 9 
I 
I I 
I 
\ 
i 
0 
~ 
l 
8 x Q' 
-CJ,) c 
.... J + 
(' (, 
('• (' 
01' '(Ii-
~ h 
(--- --, (' 
j\i ?O 
11 (J) 
... !. ....0 
()' -~ 
(] -..._.....-
--
:~ 
'['.< 
fl 
er 
3;' 
JI 
arguing by analogy with non-MHD flows. 
Thus our readings indicate that the minimwn critica::J:_ Reynolds 
number for M = 943 is < 70 and that . fo :r R > 500 9 a. stable 9 but , not 
laminar second flow regime is set up.. (This second flow regime !I if , 
t or.:-pl .de.~ 
1·ully de1,reloped.11 ean:not bepaminar because there is only one laminar1 
fully developed flow as shown by our uniqueness theorem of §2.3) .. 
. We also note that this second flc,i,,r regime is -still basically la;rr,.inar 
because . 6P is proportional to Q and not Q2 ., We .refer to this a gain 
in §s.,.4.5. 
8 .. .4.3. Pitot ;taj.be measurements. 
We used the f'la.t Pitot tube (a.p = o073n 9 bp .= ,,02.411 ) of fig .. 
S&12(a) to investigate t~ boundary layer on .the lower non=conducting 
wal.l p arallel to the rragnetic field at y = o. Realising that the flow 
would· be r ather unstable we attempted to measure h. q...,, within a small 
range of Q so as to plot u.z/c;t at one or two parlicular values of Q. It 
was exce~dingly difficult taking the readings on account of the 
unsteadiness of the flow and the lag . between the Pit.at and . static read4.ngo 
Qxr results a.re tabulated in table 8. 1'\a.and plotted it:1 figoBo 190 Note~ 
firstly t,hat we computed v z by using the result of §g;3 in taking o<.., 
to bei ,,4 9 when N was based on b o . ~ Second we assumed the m .. essure1 to b~ p , / " . 
C;011sta:nt across .the ducto Thirdly 9 since the Pi tot was placed i 11 down= 
... . 
1 • ' 
stream of the ta.pp:i,ng (5) we used the' static pressure drop measurements 
to calcul.a.te l.l~ 9 the static pressure drop between (5)' and the probeo 
Fourthly 9 where the, velocity was clearly negative and the readings 
even ,more unsteady$) we have plotted the apparent values of =Vz on f'i.go 
B., 199 merely to indicate the presence of such backflow. 
The results of these readings indicate; first, that the existence of 
the general features of the flow predicted by our theory was confirmed; 
a large velocity close t o the waJl in the boundary, a negative velocity 
and a 9ore velocity less than the rraximum boundary layer velocity~ 
second, that the form of the velocity profile_ varied as Q varied, the 
lower value of Q leading to values of v /Q closer to t he theoretical near z 
! 
I 
1 
0 -· 007 • .x...::. l 
64 J x .):'.._ !: 'qg3 I / 5P My· ! gJ Jc..= I, ?>4.3 
.,It, . t~ 
I 
1 
:,c..=- I· .Y7~ O,'M1V\O...v · '2..J ! w 
I I 
I 
I 
,/ 
I 
I. w -::c.. :c ),4 I 3 / 
·IS~ f:) 0 x_ ; j . l/4.3 
./ I 8. .:x:. .c.- i·4- t'6 
£ x.., = I· 4 '17 
© .),:_ ,,_ I· 4ei.3 
. >( 
I I 
I 
R 
lbOO 
+- ' YoJe.. Wt 
' ! 
•4- / 
3-o 
. ~ Ii ,21 Pohlu.L t+tc, ;)'\ bcul\d':'.'.J le'.:'!'-' W:o .x_ = 6-~: 
M .= q 43 
the wa.llz third 9 that the velocity shea:r at the wall was apra.rently 
gre~ when the flow was unstab:J_e; this e.ffect would be e.x:pect~d from 
the static pressure measurementsj but the readings here are uncertain 
because the pro be was so_ clos~- to the wall; fourth I that the apparent 
vru..ues of velocity in the eore were between 50 and 70 times as great as 
the theoretical value for laminar flow .. (More about this later). 
8e4o4,o .Potential 32robe mea.suremer~. 
We only had tine to ~asure the potmtial on the line y = o~ and 
we only measured the potmtial relative to the tapping (5) after assuming 
that the potential qetween the probe and the copper wall was negligible 
when x = y = o . N0te that the probe was vertically above t,he tapping (5) ,, 
We measured A<P!"'P as a function of.Q at various vaJnes of x 9 fo r M = 943 9 
the results being plotted in fig .. 8 . 20. These measurerner;ts were all prone 
t o the same diff i culty in taking -them _as those of SSo3o4v namely t he probe-
tapping circtJ.it linked flux and consequently lllf}·p was subject to rMdom 
fluctuations • Over and above these fluctuations irre could discern ~ 
noticeable difference between the steadiness of the rea.dingsj being nm.eh 
steadier in the core (x < 1.,35) than in the boundary layer. 
. . 
The ma.in results of .fig.,8"20 are these 9 (i) The relation between 
~TSi>and Q is linear in the core for R < 1000.11 the highest value measured" 
(ii) The electric field in the core is zero since all the points fall on 
the same Acp!.P = Q curve. (iii) In the bou:r.J.dary layer 6cps-P is only 
proportional to Q if Q is sufficiently lowo It .1iv'a.S diffi cult to determine 
how low owing to the randomness of the readings. 
Taking the best straight lines through t;.hose points which might 
reasonably be ~f;w,J4-to be in the linear refirne 9 at the various va,lues of 
x we plotted e<1n, /Q against X = (b=x)/(a.Ii.f-Z) in order . to com~re the 
l ee -Pi9 .8,ZJ theoretical la.mina,r profile with the expnimenta.l one. First1 we notice 
that the e:xperimental curve falls below the t heoretical for all values of 
X and second we note that thE;l e:xperimental curve does not have a maximum)) 
unlike the theoretical curve, where it occurs because v ~ 0 at this point 
in the laminar theory. 
We may observe from the fact that E = 0 in the corej that the x 
copper walls ha.d a negligible resistance and therefore the ccndi tions of 
the flow were as intendedo 
Our aim here is to provide a plausible explanation for the 
differences between the results for ~'5--~, (hereafter /J Pi v!?l/Q1 and 
6,ps-P /Q and those predicted by the lamina.r theOFJ' o We start from the 
incontrovertible fact that E = E = 0 in the core in both th~ si tua.tions j x y . 
hereinafter denoted by the suffices 2 and ! respercrl:.iV"elyi furthermore, the 
velocity is uniform in the core in both cases and therefore~ clp/clz = -a Vi-Bo" 
and 
A p"l.. - Uf; o,.d,2.. 
6P,. . ~p,-.e) 1 . . 
Now vve see from figo8o18 that l>ft':::!.-2.,6 AP, .1 t,.)fi€Yile.. 
( vz() t-e.)'" ':2,1(,r;;t)~), 
But in fig 8019 we see that 
( VZ0 f€h. ~ ~O (VZote) 1 
Thus some di.screp&.Tlcy needs to be exphinedo 
We see f:rom the Pitot and eop .. probe trav~rses in the bounda.ey 
. ( I . 
layers that . the turbulent: boundary ley-er thickness is po.t,, greater than 
t,he lami.naro Therefore we can still nw,ke the approxi.mation in the 
boundary layer g-
= dm /'ltx = v B = 0, Bo4o.4o T z o 
with an eJ::Tor of order (.i.r / cr1r .t. .. . )., the suffices referring to the 
core and bouncta:z layer re speci;.ivelyo From (8,,lt.o.4) it follows that 
6.</>c -:. '80 .[ Uz d (6--;;.t},80 fi,1't_wher~ Qf,.{ is tl).e VQlUJJle flow rate in 
the boundary la.yer o Now in laminar flow (Hurtt~ 1965) 9 Q=,Q&-( + 6(G 
where QC..~ '? ;it f:?tr .{ Now in our flow we found that 
(-'$\ : , l-41~ , whence fI'Qlll the consemtion of flow it follows that: 61 ( r ~ , l l¥cJ ~ ( ~' J + · 5 ( ~6- f) 1 ~ ~ G 1 ( l-t • 3 ~ -: I) 
for our duct men M = 943,, Thus! 
~ (51 
(UZh 
=l4i = 
which is in better agreement with (8.4 .. 2) than :is (8 .. 4 .. 3) .. 
We believe that the explanation .for (8 .. 4 .. 3) is that a }:'.".'essure 
gradient ~/dx (< 0) exists in the centre of the ductne~rthe wall x::::: b., 
This pressure gradient may be caused by turbulent seconda.r ;1· .f l ow whfoh 
impells fJuid ~ places where the shearing stress is r.i;T •,at est .u1.d . 
towards places where it is least, to .paraphrase Pr~ncttf ( Vi'.32,., p.,1.49; 
such a flow,would induce a velocity vx.C> 0) and thus a pressure gr adient 
(dp/tbc) < 0 9" Q .. E.D. 
This apparently comrincing . argument has two ]Jlain flaws g (1) Why 
is ·APs,., proportional to Q in the second flow regime~ if this r egime is 
dominated by secondary flows? (2) Is the reason for the discrepancy 
between Alty 0s results and our vs simply due to the greater unsteadiness 
of our flow"? or does the value of M matter? We have to leave these 
questions un.answered .. 
We note that it follows from (8oli.o4) that we can calc:.mlate v z 
;in the boundary from the potential profileo In figo8o 19 we have plotted 
3 points by measuring the slope of the l4>.!P/Q against X curve of figo8o21o 
The agreement between these values and .those .:rreasured by the Pi tot tube 
is qu.ite good ( r.l 5%) near the wall but is poor as X increaseso 
We conclude by observing that these measurements of stat ic pressure !I 
dynarrdc head and electric potential, taken.together can be correlated 
and they show how the use of Pi tot, and e.,po probes enables us to increase 
our understanding of very complex,, umrl,:,eady flows .. 
14q 
9. Conclusions.o 
Su.m:mari~s of the work descr:1.hed in t.his thesis are to be found 
at the begirming of each chapter. In this chapter we mention some 
po3Sible Jines for ft1J'.'th(n." :!"esearch wh:i~h a:r~ir:ie out of our work and some 
improvemsnt s 1ohich might be rr.ade on our e.Jr.p-:: :ri.menta.l apparatus. 
Ttte most important development of the theoret.i,:;~l 1rJOrk en MHD 
du~t flows desc:cibed in cha,pter :2 should be the e.xt.e,nsion of the analysis 
to comp~ssible flows. We believe that such ~, developmer;.t is not only 
possible, hut of practical us e . 'Ihe reasons f or our belief were set 
out in Hunt (1966,b); however the out]i.n(': calculati ons o.f that paper 
·w:i.JJ. obviously have to be worked ont in gren.ter detcd}. before these . 
calculations are taken seriousl y by the designi:T·s ,::, f MHD generators. 
Th~re arA still some interesting aspe cts of J..a:minar dn ct flows to be 
analysed, e.g .. t.h1:, boundar-J layers rrn the walL"l ,pa-ra.ll-81 t o the magnetic 
field in the di verging ducts considered in §ze 7. 9 or +..h~ flow in a duct 
with a sudden change in eras s secti ona.l area .11 (not the sudden change in 
the r a,te of d:ange we exa.min('}d). 
Clearly the thoory of . Pitat. and electric potential probes o:f 
The theory will only a•hr.mce 1rihe.n we 
ur,derstand the flow over bodies placed in various kin,i:": o f' flow 1A".i.th 
the magnetic fields .9.t various orienta,tions. 'l'his a.s:pect of IvlJ-ID theory ' ,, 
ha.s been pooh-poohed in the past (by Professot• .SlH?rcliff' in particular) 
-~Js being entirely acadernic; tl1e stud:,t ... of Pitot tttbes is definitely not 
acad~nic and therefore this practical appl .. "tce.tion may stimulate further 
theoretical work on the M,lID f l~w over bodies. Also .. with th t, advent , 
i.:>f the use of MHD probe9, we . can now i:nvestig1!l.te the ,flow over bodies 
in ducts experim.entall y , ,:: . g. Tsi.n0be:::- et al, (1963)~ i.fnich ~hould 
provide 1m.othe r stimulus to this t .. heory and 9 most importruit 1 a check 
on it. 
As r-ega.rds the possible uses of' MHD probes 1 the number of flows 
which w19 cannot analyse and which need irnrestigatin~ are Jj_m.itless, 
However, it is more important to concent,rat e on flows ;;ib6ut which we have 
some theoretical understanding. One such is t he onBet or turbulence 
in narrow channels; this type of flow has IM-inly been exarrtlned by . 
en~mal pressure measurements~ and upon these a huge body of semi-
empiric al theory is built. This theory badly needs verification .by 
interral probe measu.rerp.ents.fl1s9 this particular flow is technologically 
the most important and, we feel, the one ·where most effortshould be 
concentrated. 
Our flow circuit is subject to rrany s erious defects v.hich mst be 
elliinated if more accurate im, asu.rem:mts are to be made. First, our 
choice of pump was disastrous, the f'low it produced was not smooth and 
the trouble caused by the wearing down of the pipes in it produced more 
spilling of .mercury than ever a small leak from a centrifugal pump 
would cause. Having seen the flow circuits o:f Moreau arrl lecocq 1 we 
feel sure that, if a cir~~it with two free surfaces is to be used, a 
centrifugal pump is quite sa.tisfa<;to:ry. HoweverJ from the published 
accounts of the flow rigs at Riga, it is not clear whether the Russians, 
'Who use closed circuits with electromagnetic pumps, achieve more or 
less steady flows tha,n those using centrifugal pumps. This point must 
be resolved before any satisfactory measurements are to be made on the 
onset of instability in }1HD flows. Another point to be resolved con-
cerns the best design of weirs for use ,,r.i t h mercury. The one we 
designed seemed to induce an tmsteady flow over them, perhaps because 
they were circular in shape. Some e:x:pari r...ent s should be done to find 
that design of weir whic.;t/t*e least amount of mercury and yet produces 
a steady flow over it. 
-151-
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f r')!' sma.11 1\, .• 
?:.A.r-"I.M., J±l, 9. 
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flO'v-J. 
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The st.a1:d.lit2r ,)f plt.n ~ pa.r e,l l el :f1c.ws 
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,::,~,nc c~,..,i·J f-h .Ll. c,Jor; J, Oc 4 '>t. 1 l. \..., o •.:14H S> - :..,._ ,:; I 
_.. Q ~j • C t., ~ 
'I'heoretica1 H;;rd:J'.'odynamics .~(}.(.nu /lo."') 
Electrically- driven r:rt.eady flows in 
n~gr,.etoh~rodynam.'ics. Proc. 11 tn Int.Ccng: .Appl.Hech. Munich . Septe;:,her 1 1961+, 
:F:.:xperim /ttal rig of th e Fluid Jv:echanics 18,boratory of the University of Grenoble. 
" t R d n ·d S. 2c • roe" 0omp 0 . . 8n • ""ea o , CJ. • :f.2£, J _},;:,. 
The effe ct of a t.r~.nsv-erse m;e .... g:n.eti.c fii'ild on s eparation. 
Compt.Rend.Acad .Sci _, ~, 1732. 
Sachs, s. 
eh l 0 ff J ' (·1ot:: •) '> ,, crci, ·-'i · ,/-;i,,;)o 
She:r-c 1ifi', J . A. ( 1955). 
Sher cliff J .A. t. 1956) 
Shercliff, J .A. ( 1962) . 
E'.hercliff, J .A. ( 1965). 
Squire, H.B. (1933). 
~t • t ~ T (1n~1 'J , . .> U.:,,X' , "" • . • , 7 :, .. i • 
:'3tewA.rt son, K. ( 1960 ) . 
5 utlon> l-r.vJ ''-(Arl 561'..1,4,lL/-(_ I Jl t) 
Tcl!li, I, ( 1962) . 
-155-
/::,-'.,, 0.~·~rr:-: lv.intc~rf~11·_J_ ~~~~ .. 1_ .. d>"' (',f t11r~ s:\Jnila1'1J:'~ty 
cl'.' +.h.":i p.'of1L,s or" ~r-:cr-.~~ v;e;locity in 
c :::i:nfi:o ;~r~ ·1et,"l in th,e presence nf c1 
t:i:'t,n:SV,;- .·.1.~s·::: ,--, :il'.i.ll(:ti c fieJ o. • 
.. ~ .•'•~ )'(' "i. I· ~ r •d' r;;: ~ ~" :i j .... .; 2 / 2 7' c::q l; t../i .. ,,r.'J l,.; ,"1 ;! ,. <.e1 1-.. e -·-· .. vct.'..lc ,J •...-.J..;:. (. O ~./; ~ 
Ph1.id P '.".7°! '.l.l' :1 C'3 . (r:1 :11:.:kie), 
-!· 1e ro/£, pcue5'ci, . e-J 2. 3 s-
li..~ t:rt:n/· ·:"I, , 1-·~1:.~n.ac::x·~r l..!C·\Sf6j:C-~ o 
( ('\., .. ,, .. · ".:: Fr ·1 ··, D ) 
. ·· .. ·,., .· .. ~ .. u .' ... !. ...... \·Ill..:. (l J e 
11''!1ID entry- J.2°ngths c r: d~;t.~:rrni11l",d by 
Fit.ot t uht~ V''3locit-:..r p:r·of:i.ie meas1.1.rement.s; 
(') 'R T"- "'- .-,.,{ "' n,,_.,,...i. i ,,,,.,.,.)., -:Cy·-o- 11,IT'T' 19?.i, 
· d .,_, 4 . {!li::;;..._"!.1- .... >..-. . .-- ... 1:-,;·v o ... . :_,..__,1. lo : i 1-=,J .t' . ...J....._ ~ :J',;Jr, 
St,:1:viy ,·1:::t:;x,n of condtu ting flu..i.d:S in 
pipBS u:wfar tra.:nsve1°s.=, iagnetic f'ieldso 
l~"1)C (I CtJ'fL~} $ ~'!'1tlG Sac,-: li.2.9 136 11 
P:tc-bh,rns in l'~ID. Ph .1J, \'1-v:1:sis " 
C~wib o lir.i:br o 
Entry of ccnducting c.\l'td nor1=eondu,':;ti..11g 
fltd.ds 5.n pipes» P.r'Oc J'.amb. Ph.iLS.:H;,. 
£., 573 . 
Th.e th E:ory nf elect.rom;:.vmetic flow 
v 1e a .. s11.J~er:w-; nt c, 
CrJn11'J1,'0:idge TJni v ~ P., 
A t.extJ.>ook o:t magnistoh;p::!:t"ody;_-1~.mics . 
Oxfor d~ Pergtnn.on PrGs,3. 
Ori t h e st.E,:biJJ.t;r :tor three=dimensiont!i.l 
ctt ;:;-t;ur-1Janc·~::: n.f" vis C·S~ts flrti.d~ fJ_o\~ 
bet.w1.":len :pr.:tr :s>.11.,:e; 1 walls o 
P-£-o 0. R):r. :3cc. s -~ . 1k2,, 9 62i • 
On th e fs t :,b:D.it:r of v:l.scous flow 
betwee:1 pax·8.11 el :,:,J.a1¥.1 s in the presi:;nce 
of a · c,)pMn~r ru.agn "-1tic f ield o 
Pro c . R.o;y- . Soc,l\. 9 2.?l~ 189 o 
On t h6 motion cf a non-0ondu.cting body 
t hrough ~ perf('.lttly condu.ct:lng flu.id o 
,1,, Pl uid }fec.:11. §:, 82. 
J. Flu,·.::1 Me,A- 11 t L- I. 
-) 
St,<'::a.riy i'l.0w of conducting flrd.ds in 
ch<"innels tmde:r tra:ns-vt::irsG me.1:1J1eti~ 
i'i elds wi.th c ont'>id er, tion of I~,11 
effect. 
J.Aero.Sci, ~ 9 287. 
To..v-c,..S, oVJ V. A . (_ t0t,o) 
Tatsunrl., T. (1962) 
Smr ~ Phy s .JETP 1 10, 1209. 
Eag;:-1etof':-i_uid~d;yn~mtic sta btli t y and 
turbulence . 
Progr .. 'I'heor.Phys .. Suppl. 2l\;1 156. 
Taylor, J.B. & We.f3on, <-T.A. (1965 ) End loss~s .crom 'i theta pinch. 
Tsinober, A, ( 1963). 
Tsinober, A• , Sri tern·. A,, ,, &. 
Shcherbinin, E. (1963)~ 
Ufl,y:.-md, Y.S. (1961)0 
Uflya.nd, Y .. S. ( 1962). 
Velikhov, E •. P. (1959) 
Waechte, T. ( 1966). 
Wooler, P. T. ( 1961 ) 
Yakubenko, A~Ye. 
Nu.cl. P,rn 9 ;ill 1 59. 
Questions o:f' the effect of a ma.gpetic 
i'J..eld on the fiet\7 past bodies. 
Vop:r eNag.n.Grid:ro« JJ k9=8S: (Lka.d.Nauk ~Latv SSR, Riga) o 
On t.he separ,ation of a magnetohydro= 
dyna,imc 1:iou:nrlary b tyer-. 
Izv. Akad~iifauk 1 Latvia SSR no.12~ 49,, 
Flow of a c:onductJ..ng f'l uid in a 
""'&ctar,gu~.r channel in a transverse 
:mt,.gnetid · f'ielrie 
Sov. Fhys . = Tech.Phys. 2_9 1191., 
Certa.in questions in th5 unsteady flow 
of a conducting fluid through a tube of' 
constant cross.:.section in a -t rc1.nsverse 
magr,. etic field. 
Sov. Phys. - Tech. Phys~ 2_9 1031 o 
'The stab:i.li ty of plare Poiseuille f'1ow· 
of an id13ally Mnch-1cting flt1id in a 
longitudin.:-J..l magnetic f:ield. 
Sov. Ph;ys. JETP 2J 1:5.h8 . 
Ph.D. 'l'hesisr Cr,._rnbridg1' 'Jniversity. 
Ins tabilit y of flow between parallel 
planes with s coplanar magnetic field. 
P'Dys .• Fluids j lt,,, 21+. 
Certain problems concerning the move= 
ment of·~- conduct.i.nrc: fluid in a plane 
channel. 
Zh. Prikl.Mech. Tech. F'iz. No ~ 6, 1963. 
JASA. Tra.nsL F=2~.1 , Sept .. 1964. 
I 
11 
I 
Table 8.1. B • .. t .,,, 1' ' U 2·.l.1n d1.,_ .. ~t f,.J',L .. 1 a si ,c:, , ;_a · a .1 or rr x - ,. . 
v t, 
~-~~-------·~= 
Cross-section dimensi ons~ 2a = ¥ 11611 9 2b = 2.37511 o 
Properties of mE,rmn'.''Y (NK3 U:.c'1its) ~ 
Density of meths, e M 
103 
rn=J 
/ 1. 0M3 x rn·) 
~120 x iC3 
25° 
13€5 x 103 
i t!'? , =--3 
I r, .J•- .JC 10 
'I a03L. x rn6 
, 8:12 x w3 
(This data is also us .d i n tabL':le E\5 a!1d 8~ ·1 ·;) 
Rs;ynolds number , R == (-'3:-J f?°'-/-P( u = 73 ci at 2c., c " 
Pressure gradient s i.n fuUy de,ve1op1:;d la1ni11ar i'lowg 
H = O, 
1·rhence Jk/J~ .·1' .r1 li .i.'} . ,,,"'/.,"'.,,,,, J 1 / J.,,,._ 1· - · -in~, 1-, "',-.. ,~ ·r 
... . L~.., ""'o ...__. t .. ._. .. J) V' Yl..f (f i:;.: .. l.1. J.. t.; .1 .:.. .... ·.,_; V ...... 
meths per inch of duct 9 
calculr3. ti ons of d t\ / d j!. ~ 
h_ being ma.nome ',I?!' reading ,, 1-'I~ -,mit :subseqU.•\:lnt 
foll,::n,, f::•on this cne) . 
d p/12: =-
~et 36-(. +~ ... hM- 1/M) 
Velocity profiles in fully developed J..amine,r :flm,;g 
M = 0~ ? ·.:. _} Q ( /- 1:iL/Al) 
~l!). 6-
whence JAPif / I) ~ tJ 4J>Lt { 1~ 'JY a9 
(Qin cc/se~,. tlP,f' in inches of rneths) . 
Q M ( I - WJh (11 /()_) 
Z-,,-(r · M -- -t"""' t.. M 
=1 57·-
Table $. 2. Static pressure drop in duc t I when M = O. 
_Q (c.c./sec) /J.~6/a (inches of meths/cc/sec)o 
3,62. 3Q 86 
4.B2 3.13 
5.10 
5~38 
5.1+7 
5.57 
3.56. 
J.7h. 
Mean ""'3o59 
Theoretical Value = 3~6k 
standard dev, ation (s . d)=.28 
Table 8 9 3. P:ttot tube r eadings in du.et I wh~n M = O; . circuJAr fu.. 
. Q 
(e.c/sec) 
3.32 
4.00 
4.1 l~ 
4.41 
5.30 
5. 51 
6 .. 15 
6.25 
6050 
Cl P,p/fr1 / 111>,,. / I) 11/'r\elt:i/ ~c/~)l.. 
=2 3.1Sx 10=3 5464 x 10 
2.00 4,.J/? 
2.06 li-,S3 
2o51 5~01 
2.64 5. 1 a 
2.66 5.16 
2.12 h."60 
2.28 1,. "78 
2.37 k .. 87 
Mean = 2 .42 x 10-3 =2 4 .. 91 x 10 
s.d. .= .35 x 10=3 =2 c30 .x 1 O 
(i oe• 14.5% of' mean ) (i.e., 6.1% of mean) 
=3 =2 Theoretical Value = 2 .. 34 x 10 1.,,.81~ x 10 
Note: ( 1) Q c::::.. 7 c .c./ sec so that flow is in linear regime . 
(2) the square root of the mean value o:f N,,/£;7 A.fp/il..is 4_._92 __ 
and so very close t o th e me~n value of .fi;Jfr, / &t. ~.fo.P,p/61" 
Thus to calculate Mip/ [} l) we can cal culate t;.P,p/~ '>[ and 
then take its square root= a much simpler process . 
Table S.4. Pitot tube readiwrn ::Ln duct I ,,!ten M. = o~ flat·2endeid ti:Q,. 
Q 
(c.c/sec) 
3.00 
%:;,!!.n value :::: 2. 37 
s~d.= ~1h.(6% of 
.mean) 
4.10 
1~~-41 
h.62 
:;.10 
5.51 
B.35 
2 .. 31 
2.32 
2.53 
2.62 Theioretica.l Value = 2.31J.. 
2.23 
2.23 
Table B.5. Bas:!_c _  data .f9...r .611 x.3" duct (II}. 
Cross='2e~tion dimen..<:iions; 2a ""' "600 18 9 2b :::::: 3. 01011 • 
Distance between probe and du ct entry = 22 11 
Calibrat.i..on of .flmvmsiter (Mark II)~ Q = J3V; 
(Qin c.c/sec, V in mV). 
Maxi.mum flowrate ' .06 litres/sec. 
Rey-nolds number, R, = 5? .2Q 
( () ) Ha.rtman.11 nu.mb er, M = BO x 213. 3 , at 20 c t 
Theoretical pressure g1"a.dient for la.mi.nar, 1u.Uy developed flow ~ 
M=O dl - ,. 6'-, I -t Q T-i : )(, 0 
M >>1 'o ~ i M 61_ (1- ~s-2.. 11 J_ f' ) ~'df -+ 4tt3 6- fr M'l1- M db.. 
'; 
'2.! g'B !-., x I o -r . V t:j di {I - • l~t, M-'h. - f-1-,) ) 
using the result of Shercliff (1953) . 
Entl"IJ length (1) whe. .... M >>-1 ~ 
l = ~R/M. (Shercliff (1956))0 
Hence, since l = 2011 , R/M ~ 62 for full;r developed flow in our ductc 
P1.tot tube measurerrent,s in the core when M » 1 ~ 
-159-
where A • AP. + t> P, , e, Po i ,}_ t "i.' ~ f. e 6/'-I (/i,:, {,--( I - · I flt 11-.... H ~I) )._'2..., 
N = (TB r;'" {d, op,(;., )le "c, is used ft:r P.i:tr,t t ubes with circuh,r . tips 9 
and the s:m.alJ.est 1nlua of "\" or bp :ts 1J;:H:id t•o:r· flcttte:ned t5-ps~). 
ol.... is an ,13.rbitrary constar1t t l'> he measured., a:nd 
where A~ 
tip measured in the +z d..i.recti.on. 
Electric potentia l probe measnrement.s ~ 
If A.tip i s th ~ potential . dif f siremc e between the p:t•o be tip an cj 
t.he tayping (5) at y={)~ x = b = 1. 505n 9 then in the cor$ when M :>.> 1 9 
the larninar flow theo1"'y is that. 
~ =- lFc.. Bo {t- H- 1) =- ~ /- M -, ) Bo 
l::,,.x_. 4A6-(/- 1 r'1f1- 11'--/1-J 
,,lhence ~ ~ 1 -6 2. ,10 -4 Ii. t;-"- V ( I - · 18411 -,,,._ -,. D(tl-31,)) 
In the boundary layer on the wru.1 at x = b 9 
&<b4>r) - a (AcJ>J - /. ( Ii.A )! \ ~ ~ - Botlrc._-:-~ /+-OY:J -1 ) 
where 6..</>c_ is,the core value o:f ~ o Using the result of 
Shercliff (1953) 9 
b ·4 xl t)-S 8(> l1-yL l/ 
U- · ,<?,, 11 -'1~ • 
Table 8.6. St.a ti 12...J?I'e s sure d:r_:012 in duct II 1,,rh en 11____.115-. 
v :tv:r (P~/110 v M (-AP1tr-/Mi/J 
r,i II 
'' rn~ft; 1'(11\ v 'hV 
1.99 131+ 2,. 99x10=.4 1. l-i,2 134 3~66x10=4 
2.2s 131. 2,,92 L.89 139 3.69 
2.33 135 2.83 2,39 135 3,.,40 
2.97 133 3.00 2 ... 46 135 3~ 51 
3.70 131+ 3.00 2. 92 131~ 3.62 
=160= 
I 
11 
I 
Table 8.6 (cont). 
v 
Table 
M 
134 
135 
135 
133 
135 
135 
135 
135 
131~ 
M 
256 
258 
257 
257 
v 
2,96 
4~02 
135 
135 
Mea.n 2~95 x 10~4 ejl_,t, 16°0 ~1.,, 0 Mean 3 .. 55 x 10 . at 17 c 
with te1:1P• 3 . 12 x 10- 4 at 20°c wH,h tte~ip ~ 3.57 at 20°c 
correction o correc lOnc 
Average of "b cth J .35 x rn=h- 1t, 20° c 
Theor etical v-alue 3. 32 2. 1 o=4 • 
8,.70 Pi t ot tube err-or s in duct, II. (circular t i ns). ~ . 
(a) ti:£2 d:imens:l. ons~ .,028ll d~d o~ .,, "'.)· . .. :t~ (fig.,8.,2) 
v APs-r !J,Po 6 fs 6. N tL 
""' v , , rr,.1;:r:t;""" 
10 77 001 2 .ono a0050 =,0061 1.115 =~39 
2.78 .033 .019~ .0066 .OO? .91 ,,39 
291+2 . 050 o . ..,., .... 0 ,<_.._. ..) . 0070 ,,021 .. BS 1,,07 
2. J+J.., "6li.1 ~022'7 e0069 .011 .,S5 c57 
2.,e2 .026 ~0304 D00(-5'.2 ,=.Oi 3 rr75 =~55 
2s84 ,,065 ~0:308 c,0083 .,0:24 ,,'74 L 05 
2~86 .OL .. 9 .030h, 00082: ,,010 ~75 ~4k 
3. i 1+ ~077 (>037'7 ,0{)9~i ~030 a68 1 ~ 1 B 
3.ld+ ,,046 .ot,-50 Q009B ~-.009. ~6i =,32 
Hean value o:f ex, Ql l = .389 Sodo =.4 
Hean value c.f 1'J" -,- .,82 
{b) i,i.tl¥J.ensions ; .Qj!i:!! o.d."" .020!l Ldo 
>T 
\J 
1. 95 ~Ok9 .. 011~5 .0105 .045 l+o, 51~ c6S 
2,.60 .058 co0258 .,Ot4.1 .,Ol~6 3,.40 052 
3.35 -~0'77 00l~2g ~OH!i 0052 2.65 ~L-6 
3., 35 .,058 00428 .. 001 £51 ~033 2.65 .,29 
cZ" - .1+9 
=161= 
Table 8.8 7 .fi_tot, tube erl"f)J.".§ in du.et tL.. {flattened_ P<Q.,. . ..,.~<s;: b~L 
M 
139 
139 
135 
135 
139 
13h 
135 
135 
139 
(a) external tip dimensions ~012'1 x ,.062u (fig.7.2b); M ,~ 137'" 
! v 1f. ~l7,. . A PO 6 fs ~ N ol 
,.,..., (,M 11"'1!- ft: 
1.67 .027 .,0111 ~0031, .013 
1.65 
·j • <)6 
2 ~77 
3.16 
3 .. 26 
3.5'7 
.o8S 
.105 
,,074 
110 
~055 
~0302 
"0293 
,,,039g 
,.0390 
e0(.05 
~Oli,08 
r,0_508 
Ol. = 2,0 
.. 0057 \0028 
o008l, :,011+ 
•. 0080 '~0011 
,009? ~Ol.2 
.0093 ,,057 
~0095 .02,~-
~0095 .060 
·J 5{1 
5 . ..., O I 
,.35 
r.32 
"30 
.,28 
c:28 
c.2ti 
.2'7 
2,, 1 
,.., 
,, ? 
4.2 
·j c,3 
5.,2 
If N is based on DP(= !+ x 9'rea/perimeter o:f tip cross section), the 
255 
251+ 
253 
251 
251 
255 
253 
254 
2~.a 
mea.."11 value of Cl( ~ 0(1> = 1.2. 
F 
(b) saine Pi tot tube: M ,,,,. 251 • 
1. 7$ 
LB1 
2.25 
2o43 
2.81~ 
3.5? 
3.92 
.0?5 
,.079 
;,092 
, 062 
"129 
• 1 '11 
., 157 
.. 0125 
~0129 
.0:200 
.. 0233 
00318 
.. 0318 
,03913 
?050'i 
.. 0607 
s.d. - .,25 
.0098 
a010 
~012 
.-013h 
.,0157 
,,0157 
a018 
~0197 
.• 0217 
A 
c067 
~052 
~Ol/1 
.065 
,,012 
1;061 
~01~1 
-~075 
N 
i.79 
1 ~ 76 
Lli.2 
1.31 
1 .. 12 
L12 
1.00 
IX. 
3.,0 
L7 
~92 
Table 8 .. 9 .. Pitot tub-e .error in du1::t II . (flatt ened tip, ap>>bp) 
Tip dimension. .,073 11 x .. 0.21~ 11 ; M tv253 (Fig.,8o12a) • 
M v APrT' -~Po -~ fs . D. N (>(_ 
252 2.71 0055 o02SO ,,0145 ,,042 2()34 ,.6!+ 
255 2o85 c:, 047 0031 .. 0153 ~031 2.-66 r, 38 
253 3.,38 .. 055 .,01/35 ,,01a1 .~030 1.,88 ir3? 
253 3,,55 . 060 ~048 .0190 ~031 1.,79 .,36 
253 3.92 .065 ., 0585 .0210 0027 1.62 .. 29 
With N based on bp, o<.. = ~l+O 
-With N based on Dr,, (>(.~ = ,.24. 
Table 8~ 10 . Electric_12oterrtfal nrnasul"t-'Jffients in core flow of d·qg_t, IL 
x 
inches 
.. 003 
.,753 
. 003 
.753 
.. 003 
r.753 
B 
02 
wb/.m 
. 196 
.; 655 
1. 160 
M 
140 
"°lf>O .. 0 t)b' ~ ( 0 f :!, ii . 'L· 
, l,03 rt 
.271 ± "01 
.,1~.15 rn 
.4?13 :t $001 
.,250 t ,,002 
( f:4;/VJk::: 0 Theoretical 
- ~q,/ V )> t.:: - :,_1- value o 
1Y1v'/""'tf 
<>3?'7 ± ~012 -.396 
~130 = ~003 ~1305 
,.227 ± .. 003 .,2305 
Note (1) the values of Ap/V ar·e mean.vt'i11H:H, ·deri~Jing from ~~veral 
readings of &p at various values of V,,, 
(2) The pr ob e used is thc1.t shown :!.n fig~S.,12(a, )~ 
Table 8. 11. Ba.sic data of 2! 11 x 31~ d~i;,.t. (JI]l. 
Cross-section dimension s: 2a == 2 .• lil:i6 11 9 2b = 3,.010 11 • 
Positions of tappings .rela. tive to (5) in +z direction: (1) =46 11 ; 
(2) -44.5"; (3) -23 '. '; (l~ ) ~1L2511 ; (5).0; (6) 71'6 
Reynolds number 1 R = 1930 x V = '!Y.i.i5 .. 5 Q; where Q is in c.c/sec. 
and V is in mV. 
Hartmann number" M = 820 B • 11it1ere B is in wb/m2 • 
' o - 0 
Flowmeter calibmtion 9 Q =~ i3.0 V., 
Pr"~ssu:;'."~ gradient in fully devel oped l am:i.na.r flow (Hunt 
1 
1965): 
1£._ - az (··3~ . .1. )-, 7>2 - + -f - -4a3(r · l;-M3'12.- f1l.-
'11ome dk/h = ,s;. x/D-~ v 11Y•-/(1+ "M~Zj 
The theoretical velocity &id pot~"F·;~.01 profiles in the boundary lci,ye:r 
at x = h may be found from ( i,:,.;1:-;.t 9 1965) . In the c ore the potential 
relat ive to tapping (5 ) (x = b 1 y = o) is theoretically: 
' I '-t C/ x to-~ go V 
(j_-1- 4· 02. / M '10 
To calmile;te t.r~ fr•om A Psi:, we use the experimental value of dh/dz 
to :1:'ini::" 6'P_s and the value of' !)(._ for the .MHD correction factor of the 
f l-9.t t en e;d Pitot tube f r om the table 80 9 • ., i..e. _,4,; so that 
,6 Ps-P ~ l e ~ 2- (I+- ,y... o- 841..frp \ + b. f.s 
2-. '(: ,.,-i- J 
Thenct?. 1re cal ir::nla:t-e ~. Note th8,t {j & < 0 since the probe tip in 
th::1.:::; q_iase is .. ?5010 downstream. of tapping (5) * 
'.l'able 8 0·120 !'itot ~Mun duct IIIg M "" 9430 
Ti :p dimensions e073" x e024". 
v I .,,~, .. b,~p ,:s +~ 0 0 lT' lT tr:j 
"'d.. ... ~I/ 
~~·or v~'lf r,,,.l,,, , W't\ ll.U I it r-
. I . I ·Ou~ •fO,o ·otR ·D.)A"' ·0'13 · i,43 1-~ l. ,S3 . .;2L,.3 . t,2.q •l)U' 
·04tJ ·0£1 •05,'T 
. 4·1Jb . j' -o~ i: 222 l ·1t r061 
'"~ff •/l,/ 
' . 
. {) l.4 
·?J . 6'.l.F •till 
·1H1? 
'ft! ' !Jb? ·OU, •03/ 
·ot,.r 3·67- 2~63 •2.rJ 1-04 •02.0 
-on 
·Ol,J> 
'fXt8 ·, 3 ·K'o 2-?-6 •2,.o/ 
·· 01.,q I ·06 
·01..z. '013 ·on--
-on· l-1·06 (. '14 !·04 ,222 
·018' ,()1.tr 
·D7_3 
·064 ·690 -OW ' ()],..I ·OJ6 
-c.n, 03rJ 
· 0 26 ·())'l 
. 'lf'tJ ·()?.S . 02.,. 
1
. {)r-2 
·Or'V g-g;i 2·8'3 1 272 ,.,~ 
-t.:l 10 
-o{r- ()t,r 
· 0'18 3-ro 2·]6 
-1.-01. 1-'f~ 
- ·r,os- 0'11 ., ·Ol.ft, 
-x v f1' 5 + 1/= f·o, i/,v ·Ji? 
'C) 'SCJ 
·?l -·uo', ·02.1, 1-.o.i.o ; OU) 2 ·L1_r /, j-1) ',,, ff I 
·8t 
- ·OtJS '027- •(2L 
·r - ,013 
·Ol? ·O/S-
!·06 - ·o,s- ·033 • Of '6' . o, I? 2-32 ., ·Jr- · IU, 
I '50 -· 034 • (> l(Q -tio6 
H1q -· Of I) . ov6 • 036 
,114 . ts- - ·on • 01t • •035' 
'C/) 
- ·oi.i .. . 1>2.q . 007· · 01'-, (-2-05i ~,~) 
/·(.d1 
- ·065 ·O~q , 01.t 
,1 14 ,q3 
- ·()t,J '{.l2 q 
-·01, 
-2· 1'-, (: ,,r) 
- ' 
·3/11 -q 3 - ·03J' ,ou, - ·Obb 
~I.&>) t •/3 ) 
. 614 .q3 
- "l>ID O)l.1 ·0/Cf z ·'58' 1•44 . I 1. (s i;t-e_ 
'2 ·t11 
-·o7'S" 
'IIO ·04S' J·61 "2·63 '{) 8 
16.S-
, Mech. (1965), vol. 21, part 4, pp. 577-590 577 
ijagnetohydrodynamic flow in rectangular ducts 
By J. C. R. HUNT 
Central Electricity Research Laboratories, Leatherhead* 
(Received 27 July 1964) 
paper presents an analysis of laminar motion of a conducting liquid in a 
gular duct under a uniform transverse magnetic field. The effects of the 
having conducting walls are investigated. Exact solutions are obtained 
o cases, (i) perfectly conducting walls perpendicular to the field and thin 
of arbitrary conductivity parallel to the field, and (ii) non-conducting walls 
el to the field and thin walls of arbitrary conductivity perpendicular to 
eld. 
re boundary layers on the walls parallel to the field are studied in case (i) 
it is found that at high Hartmann number (M), large positive and negative 
rities of order Ml{, are induced, where l{, is the velocity of the core. It is 
ted that contrary to previous assumptions the magnetic field may in 
,cases have a destabilizing effect on flow in ducts. 
Introduction 
le design of magnetohydrodynamic generators, pumps and accelerators 
ares an understanding of the flows of conducting fluids in rectangular
 
with finitely conducting walls under transverse magnetic fields. At the 
1nt time even the case of uniformly conducting incompressible laminar
 
with no variation in the flow direction has not been fully analyzed. In this 
~ we confine ourselves to problems of this type alone. The main character-
s of such flows that need to be known are: 
.) the volumetric flow rate q through the duct for given pressure gradient 
magnetic field; 
.) the potential difference between electrodes placed in the walls; 
I the stability of the flow. 
nree exact solutions have been found for incompressible laminar flows
 
nets with transverse magnetic fields : 
I) rectangular ducts with non-conducting walls and the field perpendicular 
ne side (Shercliff 1953); 
!) rectangular ducts with perfectly conducting walls (Chang & Lundgren 
l;Uflyand 1961); 
I) circular pipes with non-conducting walls (e.g. Gold 1962 and Fabri & 
lrunck 1960). 
Pproximate methods have been developed for the physically interesting
 
I Seconded to the Department of Engineering Science, University of Wa
rwick. 
578 J.C. R. Hunt 
case of flows at high Hartmann number, M. For a rectangular duct with 
conducting walls Shercliff ( 1953) developed an approximate method for analy:~ 
the boundary layers on the walls parallel to the field and thence deduced 
and the potential distributions round the walls. By ignoring the reduction 
flow rate due to the boundary layers, he then found a first approximation fo 
in a duct of any cross-section, which was later extended to the case of du; 
with thin walls of any conductivity by Chang & Lundgren (1961). Sakao (196 
used a variational method to find a second-order approximation for q in circul 
pipes. 
In finding the overall features of the flow at high M, approximate metho 
are often best since the exact solutions are in the form of infinite series who 
rate of convergence decreases for higher values of M. It is possible, however 
compare the expressions for q at high M obtained by the two methods. The 0 ' 
case hitherto of the approximate expression for q at high M agreeing with th 
derived from the exact solution is that of flow in a rectangular duct with no 
conducting walls (Williams 1963). In the same paper, using some lengthy math 
matics, Williams deduced an expression for q at high M from Chang & Lun 
gren's result for flow in a rectangular channel with perfectly conducting wal 
The asymptotic form of the exact solution for circular pipe flow at high 
provides an expression for q which differs from Shercliff's (1962a) and Sakao 
( 1962) approximate expressions by a term due to the velocity defect in the bound 
ary layers. 
No satisfactory approximate or exact solutions exist for the most importai:r 
practical case of a rectangular duct with conducting walls parallel to the fie 
and non-conducting walls perpendicular to the field. Some observations on thi 
problem have been made by Shercliff (1962b, page 16) and by Braginskii (1960) 
Grinberg (1961, 1962) has attempted an exact analysis using a Green's functio.1 
M agnetohydrodynamic flow in rectangular ducts 579 
a,tive to the axes defined in figure 1, the equations describing such mag-
ydrodynamic duct flows are: 
jx = cr(-orp/ox - v,,B0 }, jv = cr(-orp/oy ), 
ojxf ox + ojy/oy = 0, 
(1) 
(2) 
(3) 
(4) 
iv are the current components; <p is the electric potential; Hz is the induced 
which may also be considered as a current stream function; B0 is the flux 
'ty of the imposed magnetic field; vz is the velocity; er, 17, op/oz are con-
. "ty, viscosity and pressure gradient respectively. Let 2a, and 2b be the 
s of the sides of the channel (see figure 1 ). 
t . 
Y, 1/ 
B 
2a A x,i 
A 
B 
2b method but his result is incomplete. We have not been able to solve this problem 
1 l r!GURE • Cross-section of a rectangular duct with magnetic field in y-direction. but we have solved exactly two other classes of problem of flow in a rectangu • The walls AA lie at x = ± b and BB at y = ± a. duct, in which the duct has (i) perfectly conducting walls at right angles to tb 
field and thin walls of arbitrary conductivity parallel to the field, and (ii) ~~l\e equations are usually re-written to give two coupled second-order partial 
conducting walls parallel to the field and thin walls of arbitrary cond~ctrv~ ntial equations in Hz and vz, 
perpendicular to the field. These exact solutions are in the form of_infinit 
O oH 02 02 series whose rate of convergence increases for higher values of M. In this pape O = - :£ + B 0 -" + 17 (-2 + - 2) vz we examine the asymptotic form of these solutions at high M and draw som oz 0Y ox 0Y 
interesting physical conclusions, the main one being that a magnetic field ml, ov,, 1 ( 82 82 ) 
0 = Bo "x +:;:;:: "x2+ "y2 Hz. have a destabilizing influence on the flow in a duct. u v u u 
,nd H,, are normalized in terms of (op/oz), a, er and 17, (5) and (6) become 
2. Formulation of the problem 
We consider the steady flow of an incompressible conducting fluid driven by. 
pressure gradient along a rectangular duct under an imposed transverse magn~fi 
field. We assume that the walls of the duct are thin and finitely conductUll 
We postulate no secondary flow and no variation in duct cross-section \ 
magnetic field. Consequently all conditions except pressure are constant alon 
the duct. 
a2v a2 v aH 
852 + 8172 +M """§ii = -1, 
82H 82H oV 
- + - +M- =0 852 O'fJ 2 817 
V = Vz1J / (-!:) a2; H = Hz(?J)i / (-!:) a2 ui; 
M = aB0(cr/rj)i; 5 = x/a; 17 = y/a. 
(5) 
(6) 
(7) 
(8) 
580 J.C. R. Hunt 
Also let <D = <p (<rrJftj ( - !;) a2 and l = b/a. 
We follow Shercliff (1956) in the specification of boundary conditions O 
at a thin bounding wall. If oH/on is the normalized inward normal gradient~ 
at the wall, the condition satisfied by H at the wall is 
oH/on = <raH/<rww, 
where <rw is the conductivity of the wall and w its thickness (w 1 If d = <T"ww/<ra, the boundary condition for His 
M agnetohydrodyna1nic flow in rectangular ducts 
r11, r21 = (aJ ± iMa1)! 
K; = 2 cosh r11 l cosh r2;Z + dA[r21 sinhr21 l cosh r11 l + t\;sinh ri;l coshr21 l] . 
r1; and r2; may be split into their real and imaginary parts, namely, 
r11, r21 = /1; ± iy1, 
/11, Y; = (ta1)t [ ± a1+ (a; + M2)f]( 
K 1 = cosh2f11l + cos2y1l + dA(f11sinh2/J1l - y1sin2y1l). 
.final result for V and His 
581 
oH/on = H/d. ( 2( ) . [ 
- oo - l 1cOSCl;'I'/ C;(;) - M/a;D;(;) dA[/J;E;(s) - y.F.(;)]] 
The boundary condition .on V, of course.' is that it should vanish ~t the walls. V - i~ a;(aJ + M2) l - K; - K; 1 1 , (l 3) 
In §§3 and 4 we consider the followmg two cases, defined with reference oo 2( - l)i sin a- [ (M) M/a .Q.. (t:) D .(t:) d [ .E.(t: p.F. c figurel: B= ~ 2 /'/'/ _ _ + 1 ,s + 1s + AY1 1s) + 1 ,(s)JJ , i =O Cl;(Cl; + M ) Cl; K; K . 
case I rectangular duct with walls BB perfectly conducting and walls A.I 1 (14) 
ofarbitrary conductivity; re 01(; ) = cos y1(l - ;) coshf11(l + ;) + cos Y; (l + ;) cosh/J;(l - ;), 
case II rectangular duct with walls AA non-conducting and walls BB o D;(s) = sin Y;(~ - s ) sinh/J;(l + ;) + sin Y;(l + ;) sinh/J;(l - ;), 
arbitrary conductivity. 
E;(s) = cos Y;(l - ; ) sinh/J;(l + s) + COSY;(l + s) sinh/J;(l - ;), 
3. Case I ~(;) = sin Y;(l - ;) coshf11(l + ;) + sin y1(l + ;) cosh/J;(l-;). 
In this case walls BB are perfectly conducting, dB = oo, and walls AA haJthe non-dimensional volumetric flow rate, Q, is defined by 
arbitrary conductivity dA. The boundary conditions on Vand Hare: Q f +15 +z 
= Vd17d;, 
at 17 = ± 1, V = 0 and oH/017 = 0, } -1 -z (1 , from (17), 
and at s= ±l, V = O and oH/o; =+ HfdA. 
We c~n sat~sf)'." the b?undary con~itions on~ =± 1 by expressing Vand J Q = _f 2 28 2 [z- (/1;-MY;/a-;)sinh2f;l~(Y; + M/J;/a;)sin2y;l 
as Fourier series m 17, with the coefficients funct10ns of g, 1=0 a1 (a; + M ) (/1; + Y;) K; 
(15) oo oo oo a1dA(cosh2(]1l - cos2(]1l)] v = .~ V;(s)cOSCl;1J ; H = .~ h;(s ) sina;17; 1 = .~ a;COSCX;'f/ , - (a~ + M2)fK . . 1=0 J=O 1=0 J J 
where a1 = (j + t) 1T and a1 = 2( - l)i/a1. Substituting these expansions for fo.te that th~ terms independent of sin t~e expressions for V and Hare the 
and H into (7) and (8) leads to two ordinary differential equations for v1 and ~er expansw~s of. the Hartmann solut10n and also that, as d A -+ oo, the 1t1ons become identical to those obtained for rectangular ducts with perfectly 
" 
2 M h V; -Cl;V; + 1 a; ; = - a;, ducting walls by Uflyand (1961) and Chang & Lundgren (1961). 
and h; - aJh1-Ma1v1 = 0. \thigh Hartmann numbers the fluid tends to move at a constant velocity, the 
The solutions of these equations which satisfy the boundary conditions cl velocity, in the centre of the duct with the velocity gradients confined to 
g = ± z are ~ow Hartmann layers on the walls BB. If the non-dimensional core velocity 
_ a1 [ ([1 - iM/a;] coslir21 l + d A r21 sinh r21 l) cosh rli; '' then T;; "' l/M2 as M-+ oo. The current density is also constant in the core. 
V; - a~+ M2 l - K. iRartmann layers are well understood, but the boundary layers on the walls 
1 J I ([l +iM/a·] coshr .l + d r -sinhr .l) coshr .g] 1 are less well understood and need examination. 
-
1 11 K ! 11 11 21 ' ( fe consider the boundary layer on the wall g = - l at high Hartmann number 
• _
1 
• lmake the following approximations: 
z _ a1 [ (M) ([i +M/a1]coshr21 l+idAr21 smhr21l)coshr1i£ · 
i; - a;+ M 2 - a1 + K; M-J> oo, /J;,Y;"' (t a1M)i(l ±0(1/M) ... ) "'\, where .:\1 = (t a1M )i, 
_ ([i - M /a1] cosh r11 l + id A r11 sinh r11 l) cosh r21i], (llhence K; "' t[exp{2(ta1 M)i l}J (1 + dA(ta1M )i ) (1 + 0(1/M)) 
K i "' @r)LJ [(f,Pt1 l) ( 1 "i" d A J..1)]. 
.... . N { ( I+,,.." j) o,cp(2,. Aj'() . 
582 J.C. R. Hunt M agnetohydrodynamic flow in rectangular diwts 
For clarity we take the two cases of d A. = 0 and d A. = oo and consider the V, H 
and <I> profiles in the boundary layer. If f =;+land <I>= 0 at 17 = ± l , the L------------------------r--
for d.,i = 0, as M---;,. oo, 
00 2(- l)j cosa-17 
V - ~ M 2 1 [1- exp( -\f){cos(\f)-M/aisin (.:\.f)}] j = O aj 1 , 
00 2( - 1 )i sin a· 17 
H - i;O M 2ai 1 [-(M/ai)+exp(-\f){(M/ai) cos(\ f)+sin(;\s')}], 
(17 
In contrast with the exact solutions for rectangular and circular pipes with 
non-conducting walls, the higher terms in these series decrease exponentially 
and therefore it is a good approximation to consider the first few terms only. 
Hence we see that the V, H, and <I> boundary layer profiles approximately have 
the form of exponentially damped sine waves, the thickness of the layers being 
O(M-i). In figures 2 and 3 velocity profiles when dA. = 0 are plotted for various 
values of 'Y/ at M = 100 and for various values of M at 17 = 0. In figure 4, the 
velocity profiles when da = oo are plotted for various values of 'Y/ at arbitrary 
M, provided M }> 1. Here the abscissa Mif provides a universal plot, when 
M }> 1. Note that in all cases V/~---;,. 1 as f-;,. oo. 
The dramatic effect on the flow of varying the conductivity of the walls 
AA is seen by comparing figures 3 and 4. When d A = oo (figure 4) the maximum 
velocity in the boundary layer A is greater than the core velocity though of the 
same order; but when dA. = 0 (figure 3), the maximum velocity is O(M)f,; and 
the minimum velocity is negative provided Mis high enough. We can deduce 
from (16) that the maximum velocity tends to 0·25~ as M -;,.oo, whilethe 
minimum velocity becomes locally negative for M > 89 and tends to - O·Oll ll!Ya 
as M ---;,. oo. The physical reason for the effect on the flow of varying d A w~en 
M }> 1 may be seen from equation (13) which shows that the form of the velocity 
profile depends on d A. Mi, the ratio of the conductance of the wall to that of the 
boundary layer on the wall A. Thus when d A. Mi }> 1 the currents return to the 
walls BB through the walls AA and when d A. Mt ~ 1 the currents return to t~e 
walls BB through the boundary layers on walls AA. These effects are shown ID 
figures 5 a and b. In the first case the j x B drag force remains almost as high in the 
0AI A 
0AIA 
0 
"' 
t--
6 
'° 6 
"' 6 
'<:!' 
6 
"' 6 
0l 
0 
-6 
4J) 
583 
-+:> ~ 
"' 
.st.. 
al O 
Oil 
d :rl 
~ ::1 ;::,..-
--- d ;::,.. > 
00 
4s ::I 
0 0 
...:: ·c 
0.. al 
d > 
... 
... Cl 
'2 
o-0 I 
-II II 
~; 
o· 0 
... 
II (l) >, 
"l~ 
~ ~ 
H~ 
(l) -0 
_g~ § 
u O 
. ,.:, 
<N (l) 
~ '5 
p .:: 
0 ·~ 
~i:..J> 
584 J.C. R. Hunt 
boundary layer as it is in the core and in the second case the j x B dra £ 
decreases to zero at the walls, which explains why the velocities in the bo!i;: 
layer are much less when d.,,_ = oo than when d.,,_ = 0. The reason for the la 
~ositive and negativ~ velo~ities whe~ d.,,_= 0 is diffi~ult to explain simply,~ 
it appears that relative to its value m the core, the J x B force increases at t 
outer edge of the boundary layer, where the negative velocity occurs, before 
decreases near the wall, where the large positive velocities occur. 
::,..~ 
--::,.. 
1·4 .--------------------
1·2 
l ·O 
0·8 
0·6 
0·4 
0·2 
l·O 2·0 
Mif 
3·0 4·0 
FIGURE 4. Case I: dA = oo. Graph of V/V., against ,./111.f in the boundary layer at 
g = - l for various values of r, at any value of M, provided M ~ 1. 
Figures 2 and 4 show how little variation in velocity there is in the 71-directioD 
as compared with the f direction which is to be expected since the magnetic 
field tends to damp only the vorticity perpendicular to it. 
M agnetohydrodynamic flow in rectangular ducts 585 
ence, 
00 8 { Mf(2a-)-l - d.,,_a2 } 
Q "' ~ 2( 2 M2 l + 3 3 t . i=Oai ai + ) Mai{l + d.,,_(iaiM)} (22) 
st term in this expression represents the velocity flux due to Hartmann 
tween the planes 17 = ± 1, while the second term is the change due to the 
ary layers on the walls AA. Note that the form of the second term depends 
Boundary 
layer 
Boundary 
layer 
Current 
lines 
---f 
Current 
lines 
----------- f 
Williams has worked out an asymptotic expansion for Q when dA = 00 al (b} 
M-+ oo in terms of 1/M. It is possible to use a simpler method than he used li\E S(a). Cross-section of the duct when dA = 00 and dB = 00 (M ~ 1). (Not to 
derive the same result and this same method may also beusedforanyvalueofd~) (b) Cross-section of the duct when dA = 0 and dB = oo (M ~ 1). (Not to scale.) 
We consider the expression for Qin equation (15) as M-+ oo and make thE 
same approximations as in equations (16) to (21). As M-+ oo, Myi/ai ~ Mil; /rJ:.he value of d.,,_JMi, the ratio of the conductance of the wall to that of the 
and /Ji c::: \, where i\.i c::: (iaiM)i. Hence for low values of j, such that ai P .Mbdary layer. If d.,,_ =OO, 
Myi/ai'?>/Ji. Also, as M-+oo, cosh2/Jilc:::sinh2/Jilc:::}exp(2l\) and henOI 00 l 39 00 1 
cosh 2/Ji l '?;> cos 2/Ji l and sinh 2/Ji l '?;> sin 2yi l . Therefore from equation (IS) Q ,..., _±!,___ (1 - __!:_-) - ~ 8 f 23 c::: ~ (1 - 2-) - ---i-a ~ 3 , 
as M-+ 00 M
2 M i=O MzaJ M 2 M M21r• i=O (2i + l )z 
Q i; 8 {l+ (Mi\.i /ai)i{exp(2li\.i)} summingtheseries leadsto 
"'i=oaJ(aJ + M 2 ) 2i\.Ji[{exp(2li\.i)} (l+d.,,_\)J 
aid.,,_ i{exp (2l\)} } Q ,..., ;;2 (1-! -2l~~ + 0 (~2)) • 
- (aJ + M 2)lf[{exp(2l\)}(l + d.,,_il;)J ' 
586 J. 0. R. Hunt M agnetohydrodynamic flow in rectangular ducts 587 
Hence the mean velocity 
v ,..., ( - op/oz) a2 (1 _ ___!:_ _ 2·40a) 
z 17M2 M bMt . 
satisfy the boundary conditions on s = ± l by expressing V and H as 
series ins, with coefficients functions of 17, 
In the exact expression derived by Williams the coefficient of the third term 
2·43. 
If d = 0, 4l ( 1) 00 64 Qrv - 1- - +~ 
M 2 M j = O Mi!- (2j + l)t 
00 
1 = ~ ak cos cxk;, 
k=O 
1r 2(-l)k 
cxk = (k+f)z and ak = cxkl 
1·20 4l ( 1 ) 
,..., Mt+ M 2+ 0 Mi . 
Hence the mean velocity 
·uting these values for V and H into (7) and (8) again leads to two 
differential equations for vk and hk 
-; ,..., ( - op/oz) a2 {~ 0·30 .2_ <!.o (-1 )} 
Vz 17 b Mt + M 2 + b Mi . 
These results further demonstrate the interesting physical effects due to 
conducting walls. From (23) we see that the velocity deficiency in the bound 
v%-cxivk+Mhi = -ak, 
b~-cxibk+Mv~ = 0. 
Jutions of these equations which satisfy the boundary conditions (25) 
e some formidable algebra. The results are 
layersonAA, i.e. J+lf +l 
(T;;- V) d17ds, 2(- l)k cos CX1c£ [1- (1 + tanh r2kfdBr2k) cosh r11c ?'/ 
-l -! . . . . lcx3 (coshr )(M2+4cx2)1/r +sinh(r11c+r2k)/dnr21c coshr21c is O(M-t) , when all the walls are perfectly conductmg, 1.e. d..4. = oo. This 18 1( k Ik 1c 21c 
than that in the case of the rectangular duct with non-conducting walls wh (1 +tanh r11c/dnr11c) coshr21c?'/ ] 
itis O(M-t) (Shercliff 1953). The reduced velocity deficiency is due to the veloc (cosh r
2
k) (M2 + 4ai)t/r11c + sinh (r11c +r21c) /dnr1kcoshr11c ' 
in the boundary layers showing an overshoot; relative to its value in the c 
the velocity first decreases, then increases above its value in the core, and fin 2(- l)k coscxks [ (1 +tanh r21c/dnr21c) sinh r1k ?'/ 
decreases to zero at the wall. In fact there are an infinite number of fluctuati O lcxf (cosh rlk) (M2 + 4cxi)t/r2k + sinh (r1k +r21c)fdBr21c coshr11c 
in the velocity profile between the overshoot, just referred to, and the core, bi 
they are sufficiently small for us to ignore them (see figure 4) . Note that in bot 
these cases the thickness of the boundary layers is O(M-t). 
(26) 
(27) 
When d A = O the velocity 'deficiency', as defined above, is negative and = 1 ( + M + {M2 + 4cx2}t). 
bl d lb M t) t fth fl ( t" ( r11c, r21c z - k d find that for a reasona y square uct (a ~ - mos o e ow equa ion end = 0 then all the walls are non-conducting which is the case analyze 
is in the boun~ary_ layers on AA. If the ?oundary layer h~s thickness O(M ercliff (1953). Putting dB= o in the above formulae and adding V to H 
and the velomty m the boundary layer is O(MT;;) = O(~- ) then the veloc Shercliff's result (equation (15) in his paper). 
flux through the boundary layer is O(M-t) as compared with O(M-2) for the to end =OO the duct is the same as that analyzed in §3 when dA = 0. It 
velocity flux in the core. A practical consequence of this would be that, whate be d . able to check that the two solutions were the same but this proves 
the value of dAi for a given pressure gradient a system of thin insulating ha It si:: to examine the above solutions near g = ± l the higher harmonics 
placed parallel to AA at. a distance O(aM-t) apart would pro~ote. a grea e~pansions of V and H have to be considered. It would be easier to ch~ck 
volume flow rate by creatmg more boundary layers. The e~planat10n is th~t the expressions for Q derived from the two solutions are the same. Usmg 
dominant retarding force on the core flow is electro-magnetic rather than vrsc wh d _ . es 
and the baffles will reduce the currents and hence also the electromagnetic retart en B - oo, grv 
ingforce. Provided the baffles are at least O(aM~t) a?art, the decreaseinelectr1 Q = 5+1 5+ 1 V d?Jds = L --; [l + r2,c tanhrvcfr11c(M2+ 4cxi)t 
magnetic drag will be greater than the increase m viscous drag. -1 -z k=O lcxk 
2 2 
! 
- r11ctanhr21c/r21c(M + 4cxk)-]. (28) 
4. Case II . nay expand this expression in terms of l/M as M --o> oo, using the methods of 
In this case walls AA are non-conducting, d A = 0, and walls BB have arbrtraliains (1963) and it is easily seen that the leading term is O(M-!) in agreement 
conductivity, dn. The boundary conditions on V and Hare the previous result (24). The significance of this term has already been 
at 17 = ± 1, V = 0, oH/017 = +HfdB,} (21Ssedin §3. ·~ 
and at s = ± l, V = O, H = O. ·oin (26) we may see that V and Q depend on dB M, the ratio of the conduc-
"I' 
I 
I 
, I 
'I 
588 J.C. R. Hunt 
tance of the walls BB to that of the boundary layers on BB. Decreasing d 
makes the current induced in the core return through the Hartmann 1 B . ayers BB and hence reduces the electro-magnetic drag on the flow. This in turn d 
out the sinusoidal form of the boundary layers on AA and for some fini~ 
no negative velocities will be induced in these layers. 
In examining the case when the walls are non -conducting, Shercliff 
derived the exact solution for ( V + B), but found that this solution gave lit 
information about the boundary layers at s = ± lowing to the slow converge 
of the series. Then by assuming that, in the boundary layers on s = ± z, 
he found a self-similar solution for ( V + B). Thence he was able to work out 
velocity deficiency in the boundary layer. Shercliff's method is not applica 
to cases other than dB= 0 and d A = 0; no other type of self-similar soluti 
has yet been found. 
M agnetohydrodynamic flow in rectangular ducts 589 
yers on the walls AA is O(M) vz, and since the thickness of these boundary 
is O(a.11£-t), the Reynolds number of the boundary layer 
Rb.1. = O(aMt) v2 /v, 
vis the kinematic viscosity. Hence 
(29) 
R is the overall Reynolds number of the flow in the duct (R = v2 a/v). 
, for given R, Rb.I. increases with M; hence the critical overall Reynolds 
ber at which the boundary layer becomes unstable is reduced by increasing 
Note, however, that away from the remote walls AA the flow would be very 
e. 
w consider an approximately square duct with a/b = 0(1). We see from 
tion (24) that in this case most of the flow is in the boundary layers on AA. 
mean velocity in the boundary layers on AA is O(M) ve, where ve is the core 
ity, and since the thickness of these boundary layers is O(aM-t), the overall 
fn velocity is given by 
5. Conclusion v2 ::: O[(Mvexa2M-t+vexab)/ab]::: O[Miveafb]. 
Though this study is far from complete, it does indicate the need for fort 'f Jb O[l] R O[MJ. / ] d · R - O[Miav /"] ce 1 a = , ::: 2 av v an since b 1 - e .- , theoretical and experimental study of M H D flows in ducts with conducting wa e · · 
First, it is not difficult in a mercury experiment to raise M to values greatt R::: Rb.I.· (30) 
than 100, and the effects predicted by the theory should be observable. r h" fd t .c • R R d t · "th M Co par· 
. . . 1or t 1s type o uc J.Or given •, b 1 oes no mcrease w1 . m mg Secondly, the stab1hty of the boundary layers on the walls AA m the prese d ) . d" t th t th th· · t. h d t th th t• fi ld an (30 m 1ca es a e mner e uc e more e magne 1c e 
of excess velocity and reversed flow needs theoretical examination. The anal d b"I" th fl · th b d I AA to esta 11ze e ows m e oun ary ayers on . 
of the steady-state duct flow problem does not depend on the value of . 1. th t th .c f th 1 ·t fil f t• . . . tis important to rea 1ze a e J.Orms o e ve oc1 y pro es are unc 10ns Reynolds number R or the magnetIC Reynolds number Rm, but the stab1hty d R Th 1 .t h t d dfl · th b d an not us ve om y overs oo an reverse ow can occur in e oun -
such a flow depends on R, Rm, and M. In most practical situations Rm 1 ·AA t b"t .1 11 R ld b w ot ayers on a ar 1 ran y sma eyno s num er. e cann assume, 
and we can ignore the Alfven wave motions associated with Rm~ 1. c h h b d 1 1 t bl M ,..,... th ·11 . . . 1 1ore t at t ese oun ary avers are a ways uns a e as --,,.=: ey w1 Rm~ 1 the stability analysis depends onlyonR andM. Lock (1955) hasana Y bl 'b bl t ffi . tl u II R Id b h t th 1 
the stability of Hartmann flow and found that in realistic cases the magne a y e sta e a su men y sma eyno s num ers, w a ever e va ue 
field stabilizes the flow by its effect on the equilibrium velocity profile and h II f th d t II .c tl d t· (d d ,..,...) th 
• • • • • • • • • • ...J en t e wa s o e uc are a perJ.ec y con uc mg A = B = = e by mh1b1tmg the growth of small disturbances, smce this 1s dommated byviscU1 .t fil f th b d 1 1 t · · t f · fl · (fi 4) effects. We can then make some qualitative predictions about the stability chi Y pro _e .0 Me oudn arythayRers a soldcon ainbs pomt 8 hio. Ihn tehxwnb gudre d ence ra1smg re uces e eyno s num er a w c ese oun ary 
the flows studied above, based on our knowledge of the stability of boun b bl B t. thi .c M::,.... 1 th I ·t · th b d I h th · . fi ld 1rs ecome unsta e. u m s case J.Or ? eve om y m e oun ary ayers w en ere 1s no magnetic e . . · d · h b d 
L t · h b·1· f h b d 1 h 11 AA · a dwlrs on AA 1s of the same order as the core velomty an smce t e oun ary e us examme t e sta 1 1ty o t e oun ary ayers on t e wa s m thi k . 0 M-1 . . . . . aJlr c ness 1s (a 1!) 
with perfectly conductmg walls perpendicular to the field and msulatmg w ' R = O(M-t) R 
parallel to the field (dA = 0, dB = oo). As M-,,.oo, there is an increasing number c b.l. •· 
points of inflexion in the velocity profile which indicates that the higher M thelo"'ovided a/b < Mt the shape of the duct does not matter.) Therefore in contrast 
the Reynolds number at which the flow in the boundary layers becomes un~~a~the former case (d A = O; dB = oo ), raising M at given R may first tend to de-
(figure 3). The degree to which the magnet~field is likely t? be dest~biliz~ilize the flow in the boundary layers on AA and then ~o stabi_Zize_it. . 
depends also on the shape of the duct. If a/b ~ M-t, a very thin duct with wa.Jriie only tentative conclusion we can draw from this qualitative analysis 
AA much shorter than walls BB, the mean velocity in the duct closely approach\hat, for flow in a rectangular duct with conducting walls, the value of the 
the core velocity and most of the flow is in the core. (For a/b i M-t most of ~ran Reynolds number at which the boundary layers on walls AA become 
flow is in the boundary layers on AA (§ 3)) . Then the mean velocity in the b0Ull'ltable decreases as the Hartmann number increases. This may be contrasted 
590 J. G. R. Hunt 563 
to the case of flow in a plane channel where it has been shown both th . 
. • eoret1ca 
r Mech. (19~5),. vol. 23, part 3, pp. 563-581 
iii Great Britain 
and experimentally, that the magnetic field stabilizes the flow. 
Thirdly, we have only discussed flows in ducts whose walls are either 
conductors of insulators and it would be of interest to study the cases w/er£ 
walls have finite conductivity. When all the walls are non-conduct' ere t 
M h 1 . 
mga 
~ 1, t eve oc1ty profile in the boundary layers on the walls AA has no . 
agnetohydrodynamic flow in rectangular ducts. II 
By J. C. R. HUNT 
of inflexion and the flow in such a duct is probably stabilized by the m/
0 
field. Hence it is likely that uniformly lowering the conductivity of th/;e 
will tend to stabilize the flows in the boundary layers. And lastly we have a 
considered contact resistance, though it would not be difficult to include it in; 
analysis. 
Central Electricity Research Laboratories, Leatherhead* 
AND K. STEWARTSON 
Department of Mathematics, University College London 
(Received 26 April 1965) 
I should like to express by thanks to Dr J. A. Shercliff for the help he h 
given me and for the interest he has shown in this work. 
paper is an extension of an earlier paper by Hunt (1965) on laminar motion 
conducting liquid in a rectangular duct under a uniform transverse magnetic 
The effects of the duct having conducting walls are further explored; The work has been carried out under the sponsorship of the Central Electrici 
Research Laboratories, and is published by permission of the Central Electrici 
Generating Board. 
's case the duct considered has perfectly conducting walls parallel to the 
and non-conducting walls perpendicular to the field. A solution is obtained 
REFERENCES 
BRAGINSKII, S. I. 1960 Sov. Phys. J.E.T.P. 10, 1005. 
CHANG, C. C. & LUNDGREN, T. S. 1961 Z. angew. Math. Phys. 12, 100. 
FABRI, J. & SIESTRUNCK, R. 1960 Bull. Assoc. Tech. Marit. Aero. no. 60, 333. 
GoLD, R. A. 1962 J. Fluid Mech. 13, 505. 
GRINBERG, G. A. 1961 Appl. Math. Mech. (Prik. Mat. Mek.), 25, 1536. 
GRINl3ERG, G. A. 1962 Appl. Math. Mech. (Prik. Mat. Mek.), 26, 106. 
LocK, R. C. 1955 Proc. Roy. Soc. A, 233, 105. 
SAKAO, F. 1962 J. Aero Space Sci. 29, 246. 
SHERCLIFF, J. A. 1953 Proc. Oamb. Phil. Soc. 49, 136. 
SHERCLIFF, J. A. 1956 J. Fluid Mech. 1, 644. 
SHERCLIFF, J. A. 1962a J. Fluid Mech. 13, 513. 
SHERCLIFF, J. A. 1962b The Theory of Electromagnetic Flow Measurement. 
University Press. 
UFLYAND, Y. S. 1961 Sov. Phys. -Tech. Phys. 5, 1194. 
WILLIAMS, W. E. 1963 J. Fluid Mech. 16, 262. 
,iigh Hartmann numbers by analysing the boundary layers on the walls. 
solution involves an integral equation of a standard form. 
is found that in this case, unlike the cases studied in the earlier paper, 
velocity profiles in the boundary layers are monotonically decreasing. 
effect of an external electrical circuit is examined, although it is found that 
ies not influence the form of the velocity profiles. 
Introduction 
nefully-developed laminar flow of uniformly conducting and incompressible 
~ through ducts under the action of a transverse magnetic field is attracting 
iderable interest at the present time, mainly for two reasons. 
Cambrid~st, magnetohydrodynamic generators, pumps and accelerators are devices 
iactical importance in which conducting fluids are passed through transverse 
netic fields. The analysis of the flow in these devices is formidable for one 
·have to take into account the variable conductivity and density of the fluid, 
plicated potential drops between the electrodes and the fluid and the fact 
·the flow is usually turbulent. In order to make progress in the understanding 
1e phenomena therefore, considerable simplification is necessary which may 
·various forms, e.g. (a) an assumption of slug flow (Neuringer & Migotsky 
I), (b) a reduction of the problem to one-dimensional gas dynamics (Resler 
ears 1958), (c) a two-dimensional analysis of the development of the laminar 
ndary layer on the walls in the direction of the flow (Kerrebrock 1961; 
e & Kerrebrock 1964), (d) a two-dimensional analysis of the flow down the 
!assuming that it is inviscid (Sutton & Carlson 1961), or (e) the form used in 
present paper, two-dimensional analysis of the flow variation across the 
tassuming that it is laminar, fully developed and that there is no variation 
"Uid properties throughout the duct. These various idealizations are comple-
I Seconded to: School of Engineering Science, University of Warwick, Coventry. 
36-2 
I 
I 
564 J. 0. R. Hunt and K. Stewartson 
mentary in that they each extract some of the basic physical ideas, and colle 
tively it is hoped that they will prove useful in interpreting more complica 
physical situations. 
Secondly, this theory of duct flows can be tested in laboratory experimen 
with liquid metals. The uncertainty in the experimental results can be reduc 
to below 1 %, and consequently these experiments provide critical tests fort 
theory, in marked contrast with the majority of magneto-fluid dynamic e 
periments. 
Since most magnetohydrodynamic generators and pumps have a rectangul 
cross-section, we shall confine ourselves to examining rectangular ducts. Exa 
solutions have been obtained for laminar flows of uniformly conducting inco 
pressible fluids through rectangular ducts with thin conducting walls und 
transverse magnetic fields by Chang & Lundgren (1961), Uflyand (1961) a 
Magnetohydrodynamic flow in rectangiilar ducts. 11 565 
ually the walls AA are electrically connected and either the duct supplies 
nt to a load or a potential difference is placed across the walls AA to drive 
0w. We show that if the walls AA are sufficiently highly conducting, the 
al electric circuit has no effect on the mathematical problem and that it is 
'al calculation to work out its effect on the flow parameters. Some examples 
rnal circuits are given. In comparing the cases where the walls AA are 
ucting and the walls BB are non-conducting and where all the walls are 
ucting we find as before, that the conductivity of the walls has a marked 
on the flow in the boundary layers on the walls AA; we also find that in 
cases the conductivity of the walls in the corners is important since the 
nt distribution in the corners affects the rest of the flow in the boundary 
Hunt (1965). Chang & Lundgren and Uflyand analysed the case in which all t he formulation of the problem and the basic solution 
walls were perfectly conducting. Hunt analysed (i) the case in which the wa e consider the steady flow of an incompressible conducting fluid driven by a 
perpendicular to the field (walls BB, in figure 1) were perfectly conducting a ure gradient along a rectangular duct under an imposed transverse magnetic 
those parallel to the magnetic field (walls AA) were thin and of arbitrary co We assume that no secondary flow is generated and that there is no varia-
ductivity, and (ii) the case in which the walls BB were thin and of arbitra either in the duct cross-section or in the imposed magnetic field, with dis-
conductivity and the walls AA were non-conducting. Thus he included the P z along the duct. It is also postulated that any external circuit connected 
vious author's analysis as a special case of (i) · Hunt also examined the form oft e conducting walls of the duct is continuous and unvarying in the stream wise 
solutions for large M where Mis the Hartmann number, and found that var · t' (Th' d 't' b 1 d ·f tl d t' 't · ffi · tl 
' 1011. IS con 1 1011 may e re axe 1 1e con uc 1v1 y 1s su men y 
the conductivity of the walls AA dramatically altered the form of the veloci 1) Th 11 h . 1 t't' t · d d 
t f R 1 t· 
. . us a p ys1ca quan 1 ies excep pressure are m epen en o z. e a 1ve 
profile in the boundary layers on walls AA and also theveloc1tyflux through them d fi d . fi 1 th t· d ·b· h fl . 'le axes e ne 111 gure , e equa 10ns escn mg sue ows are: 
,vhen the walls AA are non-conducting and the walls BB perfectly conducting, hi 
found that large positive and negative velocities of order M ve are induced, whel'I Jx = er (- orp - vzBo ') , Jy = er(- !if>) , (2.1) 
ve is the velocity of the core. This fact indicates that the magnetic field may de . . ox Y 
stabilize the flow in certain types of duct. It is this effect of the conductivit: OJx + OJv = O, jx = oHZ, · oHz (2_2) 
of the walls on the flow which gives the problem its physical interest and suggest ox oy oy Jv = -8x, 
the need for solving the outstanding problems. . 0 = _ 0 /oz +. B + -(0
2vz + 02vz). (2 3) 
In ducts of most practical value the walls AA are conductmg and the wall 'P Jx O µ ox2 oy2 · 
BB are non-conducting; this case is not included in any of those exami~ed 1\jx, jY are the components of the current, if> is the electric potential, ~ 
Hunt and at present no complete analytic solution is available. Grmbe~e induced field and may also be regarded as a current stream function B 
f . t ' 0 (1961 , 1962) has, however, reduced the problem to the solution o a_n m eg~eflux density of the imposed magnetic field, vz is the velocity, er the conduc-
equation, whose kernel is the Green's function for the problem and mvolves ,'J, µ, the viscosity and op/oz the pressure gradient which is a constant. The 
double infinite series of Bessel funtions. When the Hartmann number M 1itions can be re-written to give two coupled second-order partial differential 
large, only the leading terms of this series need be retained and he was thl\tions in Hz and vz, viz. 
able to solve the simpler equation. In order to determine the current an: ( 02v 02v ) oH op 
velocity distribution and the mass flux down the tubes, however, furthernuroel'l ji ox; + oi +Bo oyz _ oz = 0, (2.4) 
cal work needs to be done. In this paper we approach the ~roblem_ of the f!o, 1 02H 02H ovz
 
at high Hartmann numbers using a boundary-layer techmque whrnh has _th -(--t + 0 t) +B08 = 0. (2.5) 
advantage that the analysis is more transparent and it is easier to form a physief, er 
0
~ Y Y 
picture of the properties of the magnetic and velocity fields. Expressions ~etake the lengths ~fthe sides of the channel to be 2a ~nd 2b (_see figur~ 1) 
obtained for the leading terms in the expansion of the flux through the due~ I suppose that the sides y = ± ~ (BB) are non-conductmg, while the. s~des 
descending powers of Ml. Also diagrams and a graph are displayed show:±b (AA) are perfectly conductmg. It follows that the boundary condit10ns 
representativevelocityandmagneticfieldsin theneighbourhoodofthewalls · vz = 0, oHzfox = 0 when y = ±a, (2.6) 
where their structure is complicated. Vz = 0, oHzf ox = 0 when x = ± b. ./ (2. 7) 
566 J. C. R. Hunt and, K. Stewartson M agnetohydrodynamic flow in rectangular ducts. II 567 
Thus on the walls y = ± a, H,, is independent of x and consequently we can mo . e consistency of the results. For large M the interior of the duct may be 
ed into five parts, as indicated in figure 2. These are: (2.6)to 
Hz = H1 when y = a; v,, = 0, H,,=~ when y =-a, 
where Hi, H2 are constants. The net current I leaving and entering the walls 
per unit length of the duct is simply related to Hi, H2 : 
) The core region consisting of the majority of the interior but excluding 
eighbourhoods of the walls. 
) The primary or Hartmann layers, of thickness O(M-1), near the walls 
+l but excluding the regions distant O(M-t) from the side walls g = ± c. We 
-e word primary for these boundary layers to emphasize their control of the 
t 
Y1 'Y/ 
f 
B 
·~ 
2a A 
A 
r 
'H; B 
2b -~ 
FIGURE 1. Cross-section of a rectangular duct with the magnetic field in 
the y-direction. The walls AA lie at x = ± b and BB at y = ± a. 
(2 in the core (a) and to distinguish them from (c). 
O(M-1) · 
O(M-11•) 
A 
(c) 
I 
I 
B 
(b) 
----\------
(a) 
Primary or Hartmann 
boundary layer 
Core 
-~---~---------------(e) I (d) I (b) 
B 
Secondary boundary layer 
F IGURE 2. Cross-section of the duct showing the various regions of flow 
when M ~ 1 (not to scale) . 
The. gove~ning equations ~n_d boundary conditions may now be reduced to I The secondary boundary layers, of thickness O(M- 1), near the walls 
non-d1menswnal form by wntmg ± c, so called because they are determined from the core flow and the primary 
g = x/a, 1J = y/a, M = aB0(cr/µ)t, (2.1\dary layers but do not exert a decisive control on them in return. 
a2P 
Vz = -=-V(£,?J), µ 
The equations satisfied by v, hare 
I) Those parts of the primary boundary layers at a distance O(M- i) but 
(2.l"[-1 from the side walls g = ± c. 
)Those parts of the interior of the duct within a distance O(M-1) of the four 
(2.l~ers. 
Hhese regions (e) is of the least importance and the most difficult to treat 
n M is large; we shall discuss it only by order of magnitude arguments. 
other regions can, however, be discussed in detail as follows. 
~t . 
2.1. Coreflow (a) 
subject to v = 0, h = 0 when 1J = ± 1, ,e this region extends over almost the whole duct it follows that 8/8£, 8/81] 
d v = O, 8h/8i: = 0 when i: = ± b/a = ± c. (2.1,0(1) . Further, from the boundary conditions and differential
 equations h 
an s s \tbe odd and v even in 7/· Anticipating that v and hare of the same order of 
We are particularly interested in the properties of the solution when M ~Jnitude we then have from (2.13) that 
and to find them we proceed by a heuristic argument, relying for justificatic h =ho= - ?J/M, v = v0 = g(g)JM, (2.15) 
568 J. C.R. Hunt and K. Stewartson Magnetohydrodynamic flow in rectangiilar ducts. I
I 
where g(g) is a function off, to be determined, and terms of order M-2 have b ·tue of (2.23), together with the boundary conditions 
neglected. oh8/of, = o, v5 = - 1/M at f, = c, 1111 ~ 1, 
2.2. The primary boundary layer (b) near 17 = 1 
569 
(2.26a) 
(2.26b) 
The core solution (2.15) fails to satisfy the bundary conditions at the walls a 
in particular at 17 = ± 1. Consequently there must be boundary layers near th 
walls to make the necessary adjustments in v and h and, since they are ex hy 
aving the immediate neighbourhood off, = 1. These obvious boundary con-
ns are, however, insufficient to solve the differential equations (2.25) 
Jetely . In addition we must know something about v5 , h8 at a station or 
thesis thin, in them o/o?J }> of of, . (2. 
· us of 17. In the same way that region (b) provides the additional information 
etermine region (a ), the regions (d) provide the extra boundary conditions 
Writing (2. ed here. We shall anticipate the discussion ofregions 
(d) here and state the 
and concentrating attention on the boundary layer near 1J = 1, vP and hP sati 
'tions 
a2vP + MohP = 0 8
2hv + M ovP = 0 
017 2 017 ' 017 2 817 
in virtue of (2.16) together with the boundary conditions 
ing the reader t o (2.52) below for their justification. 
(2.1 order to solve (2.25) it is convenient to write 
h8 = a (17) /M at f, = c, 
(2.27) 
(2.28) 
hp = 1/M, VP = - g(f, )/M at 1J = 1, lsl < c, (2.19 hmeans that, in effect, we are specify
ing the current distribution on t he walls 
Further write (2.19 X = V8 + h5 , (2.29) 
and 
on leaving the immediate neighbourhood of 17 = 1. A consistent solution of (2.1 then, since V8 is even and h8 is odd in 17 , we have 
satisfying (2.19) is only possible if v8 = t[x(?J ) + X ( - ?J)] , h8 = t[X(?J)-X(- ?J)]. 
g(f,) = 1 (2.2( 
and then vP = _ ! e-M(l- 1J), hv = ! e - M(l- 1Jl . (2.2 X satisfies 
. . . 
. . h . f the boundary conditions (2.26) and (2.28) become 
Thus the core velocity 1s determmed by the cond1t1011 for t e existence o tl:i 
primary boundary layer and, as anticipat ed in (2.15), is of the same order c X = l - a (17) i [X( )- X (- )] = 0 h 
magnitude as the induced magnetic field. Further the t hickness of the primar; M ' of, 17 17 w en 
f, = c, 
(2.30) 
(2.31 ) 
(2.32) 
boundary layer is O(M-1), and the associated defect in velocity flux is 
J+c Joo dY - df, vP M = 2c/M2, 
· O on leaving the vicinity off, = c, and X = 0 at 17 = 1. Leaving the con-
on on oX/of, on one side for the moment the general solution for X is (2.22 
- c O X _ 1 r (c- f,) Mi c - f, f 1 a (171) d171 { M (c - f,) 2} - - - eriC + exp - . (2 33) 
whereY =( l - 17)M. Theprimaryboundarylayernear17 =- lmaybetreate M 2(1 -17)! 2(7TM)i ,1 (171- 17 )! 4(171-17)
 · 
by a parallel argument but we do not need to deal with it explicitly here since~tisfy the co d"t " c,X/c,i: ·t r 11 
d·.cc t · t· (2 33) "th 
. 
. 
n 1 10n on u us , 1 iO ows on 1ueren 1a 1ng . w1 respect 
1s known to be even and h to be odd m 17. ·and setting r, = c that 
2.3. The secondary boundary layer (c) near f, = c f 1 a' (1J1) d171 _ a(l )- 1 = f 11 a'(111) d171 + a ( -1) + 1 (2.34) 
The core solution is now fully known and does not satisfy the boundary conditiot 11 (1/1 - 17)! (l - 17 )-! - 1 (1J - 1J1)t (l - 17)l . 
at f, = ± c. Consequently there must be boundary layers near these walls, the problem has been reduced to finding the value of a (17 ) which lead
s to 
make the necessary adjustments in v and h and, since they are ex hypotheatnstant electric potential on the walls AA. 
thin, in them o/of, }> o/o?J. (2.2llie equation (2.34) may be cast into a recognizable form by writing 
Writing v = vc + v8 , h = hc + h8 , (
2
.2! a(17 ) = l - A (17) (1 +17)!. (2.35) 
and concentrating attention on the boundary layer near f, = c, V8 and hs satisfjUplying it by (?; - 17)-! and integrating from - 1 to ?;with respect to 17: 
o2vs Mohs = 0 o2hs M ovs = o (2.21 
ot;,2 + a17 ' ar,2 + 011 
A (?;)- ~J+1A (17 )d17=+ 2 . 
7T -1 17 - ?; (?; + l )i 
(2.36) 
570 J.C. R. Hunt and K. Stewartson 
Magnetohydrodynamic flow in rectangular ducts. II 571 
This equation has a known solution (Rott & Cheng 1954) for a general ri 
bstituting from (2.41) into (2.42) we obtain, after formal manipulation, 
hand side, which reduces in our case to g ( - ! )! 2t (+!)!Mi 
(2.43) 
8( 1 - 1]2)"! J 00 s2 ds 
a(?J) = l- 1r(l+?J) 1 (s4+ifr)(si-l)t' (2.3 he flux deficit due to this boundary layer. 
where iJr = (l-17)/(l +?J) . Thus a may be expressed in terms of a hypergeomet 
function. In particular when ?J ::::::: 1, iJr is small and 
a(17) = 1- (l ~r)l ( ;t?' + O(l -172)! . (2.38 
a---,,. Oas ?J---,,. O and a is of course an odd fµnction of ?J· Knowing a(17) we cai 
calculate X, V8 and h8 from (2.33) and (2.32). 
In an earlier paper by one of us (Hunt 1965) it was shown that, if the wall 
g = ±care non-conducting and the walls ?J = ± 1 perfect conductors, then tbt 
velocity in the secondary boundary layers is an order of magnitude greater thai 
the core velocity and contains an infinite number of reversals of sign. If all fo 
walls are perfect conductors, then the velocity also oscillates an infinite numb, 
of times in the secondary boundary layer, although in this case there are IM 
reversals of sign and the velocity is of the same order of magnitude as in the core 
It is of interest therefore to examine the nature of the boundary-layer flow in tht 
present problem. At large distances (in terms of ( c - ;) Mi) from the wall 5 = c 
the structure of the boundary layer in X is given by the behaviour of a neli 
?J = 1. From (2.38) 1 - a,..,, (l -17)! as ?J---,,. 1- and hence, from the sim.ilaritllURE 3. 
solution of (2.31) satisfying 
X =O at ?J=l, X---,,.0 as g---,,.-00, X- (l-17)! at s =c, 
we find that when (c - s) Mi is large ring 
l·O 
0·8 
0·6 
;:,u 
--
;:, 
0·4 
0·2 
l·O 2·0 3·0 4·0 
Mi(c -g) 
Graph of v/vc against Ml(c - £) at rJ = 0 in the boundary layer at g = c. The 
value of Mis arbitrary, provided M ~ 1. 
2.4. Primary boundary layer (d) near the corners = c, ?J = 1 
V = vc+vp+V8 , h = hc+hp+h8 , (2.44) 
(l-17)! { 
X,..,, M2(c-s)aexp 
M/(c -s)2} 
4(1-17) ' 
(2.39satisfy the governing differential equations provided we can neglect terms 
[ - 2) . The boundary conditions on the walls s = c and ?J = 1 are also satisfied 
so that the number of oscillations in vis at most finite. A graph of v as a functi<lvided we can neglect exponentially small terms and provided
 we exclude th; 
of Mt(c-s) for ?J = O is given in figure 3 and shows that in fact v never chang~bourhood of the corner s = c, ?J = 1. Specifically the conditions are not 
sign. Lines of constant hare shown schematically in figure (5) . fied when s = c and ?J = l - O(M-
1) and when ?J = 1 and s = c-O(M-l). 
In order to calculate the overall vel~city fl~x, we have to ~ork out the fllltl the second o~ these wit~ which we are concerned here and w
e shall briefly 
deficits due to the boundary layers. Smee h 1s an odd funct10n of ?J, the fl~rto the other m §2.5. If mstead of (2.44) we write 
deficit due to the secondary boundary layer is given by 
-J+i d17f 00 d; v = -f +1 d17f 00 d; X, 
-1 o Ml s - 1 o Mt 
where "; = (c-s)Mt. 
From (2.31) Jct) d; x - 1 Jl (l-1X(?J1))d?J1 - o Mi - M(M1r)i 7i (?J1 -?J)~ ' 
so that the flux deficit is 
- -
2
- 1
J+1 [l -a(17)] [l +17]ld17. 
M(M1r)"2: -1 
v = vc+vp+v8 +vPi' h = hc+hp+h8 +hPi' (2.45) 
(2.48ire vPi' hPi satisfy the same differential equations as v, h, then, on the wall 
=!, we must have VPi +v8 = 0, h8 +hPi = 0. (2.46) 
(2 ~ ough the values of V8 , h8 on the wall ?J = 1 var
y rapidly with s, the scale is 
· hlarger than the scale of the primary boundary layer, O(M-i) compared with 
-
1) , and hence we are justified in assuming that v h satisfy the condition Pi' Pi 
1 ~ o/os, whence 
(2.4l v = - v (i:: 1) e-M<I-11) h = -h (i:: 1) e - M<I-11) P1 s =,, ' P1 s :,, • (2.47) 
I 
1 1 
I 
I 
I 
I 
572 J.C. R. Hunt and K. Stewartson M agnetohydrodynamic flow in rectangular diwts. I I 573 
It follows that a consistent solution is only possible if 
(2. 
as assumed earlier in (2.27). The other condition in (2.27) follows from a p 
e type of duct whose walls BB are non-conducting and walls AA are 
tly conducting has many applications. Shercliff (1965) has recorded how 
,duct for which b }> a acts as a pump, flowmeter, generator or brake depend-
ara 
argument for the wall 1J = - 1. It is noted that, if the wall 1J = 1 is a per£ 
conductor, v = 0, oh/011 = 0 there, and a parallel argument shows that t 
condition satisfied by vs, h8 at 1J = 1 is then 
the value of -E:i)B0Vm, where Ex is the electric field in the core and Vm 
rnean velocity; Vm = ~(1-1/M). If b - a and the effects of the walls AA 
nsidered, a new parameter has to be defined. In interpreting experiments 
signing equipment the following five parameters are of most interest: 
(2. /8z, B0 , I, and l::!..<p, the electric potential difference between the walls AA 
by 
2.5. The corner~= 0, 17 = 1 
(3.1 ) 
The assumptions leading to the primary boundary layer (d) and these d ly we are given three of these parameters and we wish to find the other 
boundary layer (c) fail when both 1-~ and 1-?J are O(M-1), i.e. in re;~:~· terms of these, e.~. in designing an electroma~netic pu~p we would want 
No simplification in the governing equations is possible therefore in this · culate I and l::!..ef>, given Q, op/oz, Bo, and the fluid properties. 
However its effect on thefluxissmall beingoftheorderofthemaxi'mum re]g
1
· find!::!..</; integrate equation (2 .1) from x = -b to x = band from y = -a 
-
, Ve OC! • 
multiplied by the area, i.e. O(M-3 ), and consequently we have not attempted "a, which leads to b BoQ 
elucidateitsproperties. - 1::!..<p = aaI +2a. (3.2) 
The leading terms in the asymptotic expansion for the flux of fluid through t . 
tube when M is large may now be written down equat10n shows that to an external electric circuit the duct is equivalent to 
fJil.f. U; = tB0 QJa in series with a resistance Ri = b/aa. The replacement of a 
vd~d?J = - - - - ·-. - - +O(_ll!J--,l-) . (2. ma .c. e ec nccrrcm yane.m .. an ares1s ance1s am11artoe ectnca rf 4c (- ! )1 2i 4c · · d 1 t · · "tb f d · t · £ ·1· 1 · 1 • duct M ( + ! )! M·2 M 2 eers as Thevenin's theorem. By the same theorem any linear external 
The term O(M-i·) arises partly from the neglect of o2v/0112, o2h/0112 in theseconda 'tmay be regarded as a resistance Re and an e.m.f. Uc Hence, in the general 
boundary layer and partly from the deficit due to the primary bom1dary lay ~1, op/oz, Q, Mand I may be calculated from the following relations: 
(d). In principle it can be calculated using the methods of this paper but wi f::!..<p = -'- u. _ R .J = u + R I 
have not done so. ' ' e e ' l::!..<p = -tB0Q/a-(b/aa)I = Ue+ReI, (3.3) 
3. Practical implications Q =f(tB0 I/a - op/oz,M). (3.4) 
In§ 2 it was shown that, whatever the value of I, the net current leaving adl2 we calculated the function (3.4). See (2.50), which may be re-written as 
entering the duct, the problem of calculating the velocity and current distribu (B I op) 4aab ( 0·956a 1 ) 
tions could be reduced to the solution of two differential equations with a sing} Q = 2: - oz µM 1 - bMi - M- · · · · (3 .5) 
set of boundary conditions. It follows that the velocity distribution is alway 
the same, though the magnitude of the velocity depends on the values of I, ie now examine three special cases. 
and op/oz. It follows from (2.12) that the distribution of current density doe . . 
vary with the external circuits, but in a simple manner: jv varies in magnitud 3.1. Open-circuit case 
but its distribution does not change: jx has two constituents one ofwhich,jx,, n the duct is on open circuit, I= 0. In this case the duct is a flowmeter. It 
constant throughout the duct, but varies with I, and is given by ws from (3.5) that 
jx, = t(H1 - H2)/a; Q = ( - op!_oz) 4a3 b (l -0·956a _ __!:_ _ ~ O(M-t)) (3_6) 
the other, jx,, whose distribution is always the same but whose magnitiule varie1 µM bMi M b ' 
is given by . a-iaP oHz &om (3.3) that l::!..<p = -tB0 QJa. (3.7) 
Jx, = (µ) i By' lfact has been noted many times before (see Shercliff 1962, p. 16). 
where P = B 0 I/2a-op/oz. 
In §2 we also found the velocity and current distributions and a relatio 3.2. Short-circuit case 
between Q, op/oz, I and M. In this section we use this information to exaroine_t•hls case the walls AA are joined by a circuit of zero resistance (Re= Ue = O) 
effect of electrically connecting the walls AA for the various practical appli~consequently l::!..<p = 0. This device is an electromagnetic brake or generator 
· tions of the duct. · hort-circuit. 
11 
I 
I 
1111 
Ill 
574 J. 0 . R . Hunt and K. Stewartson Magnetohydrodynamic flow in rectangular ducts. II 575
 
Equat ion (3.3) becomes, is interesting to note that in the core jx = 0 since op/oz= 0. All the current 
2bI O = - + B
0
Q, the primary and secondary boundary layers (see figure 5 ( b)). 
and hence (3.5) becomes o-
iorn an examination of these three special cases it follows that the para-
f which describes the particular application of a duct for which b ,.., a is 
Q _ (- B5Qo- _ op) 4a
3b (l- 0·956a _ 1 ) Mft/B0 Q as opposed to -Ex/B0Vm in a duct for which b ~ a. 
4ab oz µM bMt M... . (3 
Q = ( _ op/oz) 4aab [ l -~ -1 .. · ] Discussion 
µM2 0 956 ,
 1 §2, we analysed the flow through ducts whose walls AA were perfectly 
. 
1- _· _ _ a - O(M - !) Juctingand walls BE were non-conducting. The flow was assumed to be lami-
bMt incompressible, uniformly conducting, and fully developed; also the Hart-
It follows that 
and re-arranging Q = ( - op/oz) 4a
3b [ 1 0 M - i ] 
number (M) was assumed to be large. In §3, we showed how the results of 
µ- M 2 l - M - ( ) . (3 l . ld b d £ &na ys1s cou e use or ducts with various electrical connexions between 
Though the form of the Q- op/oz relation is different in this case from theo alls A A. In this section we compare the previo
us results with those obtained 
circuit ~ase, the velocity flux deficit due to the secondary boundary layers p ther types of duct and we 
s~ow how some o~ the effects of the conductivity 
proport10n of the flux through the core is the same, since the velocity distribut' e walls on the flow may be
 mterpreted physically. 
is unaffected by external connexions. There is no term of order (M- t) in 1!or a_duct whose walls AA are perfectly conducting an
d walls BB are non-
bracket, as one might expect, because the core velocity Ye is given by uctmg, we have been abl~ t.o analyse the flow only for ve
ry large Hartmann 
her. For our purposes this is no great disadvantage since a solution at high 
Ye = ( -8_!;:) a 2 ( 1 + 0·95:a + O(M- l )) . (3. sufficient to illustrate the essential physical features of the flow and also it 
µ bM uite usual in liquid metal experiments to have M grea
ter than 100. In this 
Thus the flow in the core is not the same as that for flow in a duct whose wa.fssion we shall concentrate o
n flows with M ~ 1. 
are all perfectly conducting. In that case thigh M, provided the condu
ctivity of each of the walls BB is constant along 
tngth, the value of the conductivity of each of the walls BB does not affect 
V.: = ( -op/oz) a2 form of the velocity profile away from the walls AA. The velocity is constant 
c µ-M2 . ipt in the narrow primary or Hartmann boundary layers on y =±a. How-
This difference is explained by the fact that in the latter case Ex = o in the cod, the magnitude of the velocity in the core depends
 on the conductivity of 
whereas in the former case, even though !::..<p = o, Ex ,to in the core because wallsBB as well as on the pressure gradient an
d any external electrical circuit 
the defect of v x Bin the secondary boundary layers. nected between the walls A
A. 
hough boundary layers are also found on the walls AA as M-+ oo their 
3·3. Purely electrically driven case n changes considerably with the conductivity of t
he duct walls. It i; these 
When the pressure gradient is zero and the flow is electricall driven by ttdar~ boundary ~a~ers w
hich have been little understood hitherto. The 
t t' l d':ffi th 11 AA
 h d . . 1. . .
 Y £ f MH:° mam characteristics of these boundary layers are the velocity profile 
po en ias· 1 e:e
1
i:,ce ac
0
ross te· wa(
3
s 
5
) b' t e evice is a imitmg orm O current distribution and the velocity flux deficit, as defined in (2 40) I~ 
pump. mce up uz = , equa 10n . ecomes , . 
. . . 
· · 
inot seem possible to provide convmcmg explanations for the shape of the 
Q = !B~Ia2b (l-~0·956 _ _..!.._ + ... ) • (3.1:city profiles, .a P_rior~ , but it is possible to provide a physical explanation 
µM b Mt M the current d1stnbution and velocity flux deficit and th
ence to explain the 
Combining this result with (3.2) leads to pe of the velocity profile . 
.etus now compare the flows in two types of rectangular duct: 
Q = _ 2at::..<p [1-_!_-0 (~ )] . (3.1:'ase (i) Perfectly conducting walls all around. 
Bo M bMt, (ii) Walls AA perfectly conducting and walls BB non-conducting. 
In the two previous cases either I or !::..<p is zero, whereas in this case we canobtaiCase (i) the velocity profile in the secondary boun
dary layers has the form 
a useful relation between I and !::..<p by dividing (3.12) by (3.11), damped sine wave and the velocity flux is 
given by 
!::..<p = _ Mb [l -0·956a _ O (~)] . 
I ao- bMt bMt 
(3.1 Q = (-op/oz)4a3b[ _ _!__ _ 2·43a ] 
µM2 1 M bMt +... ' 
111, 
I I 
I. 11 
I 
I, Ii 
576 J. 0. R. Hunt and K. Stewartson Magnetohydrodynamic flow in rectangular ducts. II 
so that the flux deficit is O(M-i) times the flux in the core (see Hunt 1965t king the usual assumption that in the boundary layer on walls AA 
in case (ii) the velocity in the secondary boundary layers monotonically decrea 
to zero at the wall, and t~e flux defici~ is O(M-!) times t~e core velocity (see§ o/ox ~ o/oy, for (a - y ) ~ aM- 1 , 
These values of flux deficit were obtamed by mathematical rather than physi tion (2.3) leads to 
arguments. To show that the difference in flux deficit b_etween cases (i) and ( 0 _ _ op ( . . ) B _ a2 
is explicable physically, we now give arguments by which the orders of mag!ll - oz + Jc + Js o + µ ax2 (vc + vs) · 
tude of the flux deficits are estimated. 
.ejcBo = op/oz and ovc/ox = o, (4.3) becomes 
4.1. All the walls are perfectly conducting-case (i) 
We consider the secondary boundary layer on the wall A at x = - b (see figure 4 
Let . . + . + . (4. the thickness of the boundary layer be 8, then Jx = Jc Jp Js, 
Vz = vc + vp + vs, (4.S ~~; = - 0 (~) and j 8 = 0 (::J2). 
where, as before, t he suffices c, p and s refer to the core, primary and seconda,1 • • boundary layers, respectively. low smce the current m the core enters the wall A, Ex\x=-b = O(jc/a"). 
=O at y = ± a, oExf oy = 0 ( - jc/a<F) for y > 0. But 
B 
- - - ~- ---- - - -----
' 
I 
--1-----------
A 1 _ Current 
stream lines 
(a) 
Bo 
B 
- -C-~]R __ _______ ---~~~:~i 
I ' 
: I 
I I 
oEy/ox = O(jy/<F8) and oExf oy = oEy/ox, 
hence jcfa = -O(jy/8). But ojy/oy = - oj8 /ox since ojc/ox = 0. Hence 
js/8 = O(jy/a) = 0 ( - jc8/a2 ) 
j 8 = - 0(82jc/a2 ). 
1rom equations (4.5) and (4.6) we have, 
µvc/82 = 0 ( - 82B 0 jc/a2 ) . 
e Ex = 0 in the core, B 0 Ve = - jc/<F. 
bining ( 4. 7) and ( 4.8) we deduce that 
8 = O(aM-!). 
;sing (4.9) we now have an expression for j 8 : 
577 
(4.3) 
(4.4) 
(4.5) 
Since 
(4.6) 
(4.7) 
(4.8) 
(4.9) 
(4.10) 
- I _ _J Q 
I -
I - --Core current 
w consider ! E. dl taken round the path PQRS on figure 4 (b). s· E o :r 1nce Y = 
Secondary 
current 
(b) 
I 
I 
I 
I 
I 
I t---------- - -- ----i O(M-1) 
B 
FIGURE 4. Cross-section of the duct when all the walls are perfectly conducting. (M ~ 
not to scale). (a) Shows the actual current streamlines. (b) Shows the core, prunsrY 
secondary current streamlines. 
he core and along PS, and Ex = 0 along RS, 
. IQ i1l steady flow f E. dl = 0. Therefore P Exdx = O, and hence 
Fluid Mech. 23 
I I 
I I 
578 J.C. R. Hunt and I{. Stewartson 
But from (4.8),jc/cr + B 0 v0 = 0. Then, using (4.10) andassumingthatj8 ismain] 
positive so that J: j8 dx = O(oj8 ), 
f0 (µMvco)' o vsdx= - 0 cra2B5 ' 
and (4.11) 
Hence the ratio of the flux deficit due to the secondary boundary layer to the 
flux in the core is O[M-f], in agreement with the exact analysis. 
4.2. Walls AA perfectly condilcting and walls BB non-conducting-case (ii) 
The notation is the same as for case (i). We can then proceed to equation (4.5) 
as before. In this case the walls BB are non-conducting and consequently 
f+a - ajxdy = I 
is a constant for all values of x (see figure 5). 
Therefore all the secondary current j 8 leaving the wall A will have to return 
at the corners via the region (d) on wall B. If j 8 = j; in region (d), then, for con· 
dition ( 4.12) to be satisfied, 
Magnetohydrodynamic flow in rectangular ducts. II 
A I Streamlines -~-------
' I 
I 
I 
_..,
O(M-t) 
(a) 
Secondary 
current 
(c) 
-external 
circuit 
579 
j;a/M = O(-aj5 ), 
and hence 
URE 5. Cross-section of the duct when the walls AA are perfectly conducting and the 
I;, BB are non-conducting (M ;,> 1, not to scale). (a) Shows the current streamlines 
(4.12;n the duct is on open circuit. (b) Shows the current streamlines when the flow is 
·en by a potential difference between the walls AA. (The current in the walls AA is 
We can deduce the value of j; from the equation of motion in region (d). Ifll'Il sche~ati?ally.) (c) Shows the core, primary and secondary currents when the duct 
, · h' · h n open circmt. 
V8 = V8 111 t 1s reg1011, t en 
. . ., _ 02 , fow consider f E. dl round P' Q' R' S', where P' S' are on the wall A, R' is 
0 = - op/oz + (Jc+Jp +Js) Bo+µ oy2 (vc+vv + vs)· the wall B outside the secondary boundary layer and Q' is in the core (figure 
,)) . Since Ey = 0 in the core and along the wall A, 
The boundary conditions on V8 are (i) v; = V8 = -O(v 0 ) in the main partoftlu JR' IQ' . J 
secondary boundary, layer, i.e. region (c) , and (ii) v8 = 0 on y = ±a. Sinceth1 8 , Exdx = P' Exdx, smce jE.dl = O. 
thickness of this layer is O(M-1 ), I R' _ J 8 j; 
_ 
W Exdx - - dx, 
., o( µ Ve ) (4.13 S' 0 (T 
Js = - Bo(a/M)2 . I IQ' E d = f0b.dx + B f\ d t 
P' x x O (T O O s x· 
Hence, from (4.1 2), (4.14:1king the same assumption as for case (i), that f: j;dx = O(oj;) we have 
I/If a v8 dxdy = - O(vca2/Mi). 0 -a 
, J8 f8 Thus 
O 
Exdx =!= 0 in this case, whereas 
O 
Exdx = 0 in case (i). The consequences 
lhis were discussed in § 3. 
But, from (4.5), 
and therefore o = O(aM-k). 
37-2 
580 J. C. R. Hunt and ](. Stewartson M agnetoh
ydrodynamic flow in rectangular ducts. II 581 
Hence the flux deficit due to the secondary boundary layer is O(M-t) times t1Jhis may be illustrated further b
y comparing case (i) and case (ii) when the 
flux in the core. ns AA are sho
rt-circuited. The Q- op/oz relation in these two cases are both 
We see from theseorder-of-mag~itud~argumentsthattheformofthe bounda eform 
. ( - op/oz) 4a3b ( 1 a " ) 
layeronthewallsAAisbestexplamedmtermsofthesecondarycurrentsinduc 
Q = _M2 1-M - bO(M-,-) ... 
in these layers, much in the same way as the Hartmann layer may be explainac 
µ 
by the decrease in current in the boundary layer relative to that in the core. n though the
 velocity profiles are very different (see §3). If we were to alter 
both these types of boundary layer the current is less than its value in the co duct of case
 (ii) and make the walls BE conducting for a distance O(aM-!) 
because of the reduced v x B induced electric field and consequently t the corner
s, then we would not alter the Q- op/oz relation but we would 
electromagnetic j x B force can decrease in the layer relative to its value in ti.nge the velocity profile, the core velo
city from 
core to the same extent that the visco~s _st_ress increases . . If this were ~ot so 
(_op/oz) a2 [ 0·956a] ( _ op/oz) a2 
the boundary layers would grow or d1mm1sh. A comparison of cases (1) aw: - M 2 1 + b
M! to - M 2 
(ii) shows that the value of the secondary currents relative to the core curreu µ 
µ 
can differ by an order of magnitude. l the ratio of
 the flux deficit due to the secondary boundary layers to the 
1in the core from O(M-l)toO(M-1). The reason is that the extra pieces of 
In case (i) ,j8 = - O(j0 M-1). 1ductor would increase the secondary currents and hence reduce the flux 
In case (ii) , open circuit,j8 = - O(jc). 
I ( .. ) 1 d . "t . O(. M
-l) icit and potential difference across the secondary boundary layer. The valu
e 
n case 11 c ose circu1 , J = - Jc · · 
d h c · 
' s 
he core velocity would be reduce , butt e .1orm of the core and primary layer 
Yet expressed as a fraction of ve, the values ofj8 are of the same order inboivwould of course be una
ltered. 
cases. This is necessary for the balancing of viscous and electromagnetic for~e have state
d already that we can give no good reasons, a priori, for the 
in the boundary layer. We make this observation because the approximati pe of the vel
ocity profile in the secondary boundary layers; all we can do is 
made by Kerrebrock (1961 ) and others that the current is constant through t deduce the shape from the flux d
eficit. In cases (ii) and (iii) the flux deficit 
secondary boundary layer, i.e. j 8 = 0, even though js is a very small fracti<tJ(M-!). Since the thickness of the second
ary boundary layers is O(aM-1), 
ofje, will lead to results which may over-estimate the rate of growth ofbounda re is no reason to expe
ct that the velocity does not uniformly decrease from 
layers on the walls AA. value in the c
ore to zero at the walls. In case (i) the flux deficit is O(M-1) 
The order-of-magnitude arguments also show how the conductivity oft the boundary l
ayer thickness is still O(M-l). This explains why the velocity 
walls affects the secondary currents which in turn affect the velocity distributi he boundary
 layer must be greater than its value in the core over part of the 
in the duct. In case (ii) when the walls BE are non-conducting, the seconda dary layer. The velocity profiles
 for case (i) are given by Hunt (1965). 
currents circulate in the duct and walls AA; they leave the walls AA in region (c 
and return through the Hartmann layer on walls BE, region (d) .This is similar ~e are grateful to Prof. J . A. She
rcliff for his helpful advice and criticism 
the way in which the core currents return through the Hartmann boundary laJlthe preparatio
n of this paper. J.C. R.H. is able to publish this work by per-
when the walls BE are non-conducting (figure 5(c)) . In case (i), when thewa ion of the Central Electricity Ge
nerating Board. 
BE are perfectly conducting, the secondary currents mainly return to the wa~ 
AA via the walls BE. Owing to the oscillatory nature of the boundary layer [ R
EFERENCES 
this case, some currents circulate solely in the duct and walls BE (see figure 4(b);lllG, c. c. & LUNDGREN, T. s. 1961 z
. angew. Math. Phys. 12, lOO. 
Thus in case (ii), because the secondary current has to return through a_na~o illERG, G. A. 1961 Appl. Math. and 
Mech. (PMM ), 25, 1536. 
layer of thickness O(M-1) on walls BE, the resistance of the currentpath1shig~ERG, G. A. 1962 Appl. Math. and 
Mech. (PMM), 26, 106. 
than in case (i); hence the secondary current, relative to the core velocity, !LE, F. J. & KERREBROCK, J. L. 1964 
AIAA J. 2, 461. 
lower. This means that the viscous stresses in the secondary boundary layer 81.'T, J.C. R. 19
65 J. Fluid Mech. 21, 577. 
less in case (ii) than case (i) . Since v8 has to rise from -ve to O as .Mt(x+LREBROCK, J. L. 1961 J. Aero/Space
 Sci. 28, 631. 
increases from O to 00, then the lower o2vs/ox2 and ovs/ox are, the greater will buitINGER, J. L. 
& MIGOTSKY, E. 1963 Phys. F luids, 6, 1164. 
Joo 
d fi •LER, E. J. & SEARS, W.R. 1958 J. Aero. Sci. 25, 235. 
v8 dx, i .e. the flux deficit. As we have already noted, in case (i), the flux e \r, N. & CHENG, H.K. 
1954 J. Rat. Mech. and Anal. 3, 357 . 
• 
0 h d b d l 1 . h fl . th . O(M-1) tunrROLIFF, J. A. 1962 The Theory of Elec
tromagnetic Flow lvleasurement. Cambridge 
m t e secon ary oun ary ayer re ative to t e ux m e core IS . University Pr
ess. 
as small as the flux deficit in case (ii). We see now that the explanation Iies:n!~aaLIFF, J. A. 1965 A Textbook of Mag
netohydrodynamics. Oxford: Pergamon Press. 
influence on the secondary currents of the conductivity of the walls BB in °"l'roN, G. w. & CA
RLSON, A. w. 1961 J. Fluid Mech. 11, 121. 
corner regions. 'LtAND, Y. S. 1
961 Sov . Phy8. Tech. Phys. 5, 1194. 
I 
11 
I 
Reprinted without change of pagination from the 
Proceedings of the Royal Society, A, volume 293, pp. 342-358, 1966 
the stability of parallel flows with parallel magnetic fields 
BY J. c. R. HUNTt 
Central Electricity Research Laboratories, Leatherhead 
(Communicated by K. Stewartson, F.R.S.-Received 15 November 1965) 
The first part of the paper is a physical discussion of the way in which a magnetic field affects 
the stability of a fluid in motion. Particular emphasis is given to how the magnetic field 
affects the interaction of the disturbance with the mean motion. 
The second part is an analysis of the stability of plane parallel flows of fluids with finite 
viscosity and conductivity under the action of uniform parallel magnetic fields. We show that, 
in general, three-dimensional disturbances are the most unstable, thus disagreeing with the 
conclusion of Michael (1953) and Stuart (1954). We show how results obtained for two-dimen-
sional disturbances can be used to calculate the most unstable three-dimensional disturbances 
and thence we prove that a parallel magnetic field can never completely stabilize a parallel 
flow. 
1. INTRODUCTION 
e stability of a fluid system is determined by the rate of growth or decay of 
turbances in any of its co.mponents, e.g. velocity, pressure, temperature, or 
gnetic field. Though there are exceptional cases, a perturbation of one component 
1ally causes perturbations in the others, so that if one type of disturbance is 
stable, in general all types are. Let us consider mechanical disturbances, as being 
nmon to all types of fluid systems. The forces which determine their rate of 
iwth or decay may be considered as falling into two main groups which are 
tinct though interdependent. The first consists of the forces acting on the distur-
nce directly, e.g. viscous, electromagnetic, or surface tension, and the second 
IBists of the forces due to the interaction of the disturbance with the fluid system, 
. centrifugal forces in rotating flows. (This is merely putting into words the 
nations for the forces acting on a small disturbance which contain terms inde-
ndent and dependent on values of the components of the undisturbed fluid 
,tern.) The effect of a magnetic field on the stability of various fluid systems is 
nilar in the sense that the electromagnetic forces acting on the disturbances 
ectly are similar but the forces due to the interaction of the disturbance and the 
iin fluid system are different in different fluid systems. Most explanations of the 
ect of a magnetic field on the stability of magnetohydrodynamic (m.h.d.) flows 
similar to those of its effects on the stability of static situations, being given in 
lrms of the fluid moving through or 'freezing' to the field lines (see, for example, 
andrasekhar 1961). This type of explanation does not demonstrate the essential 
lfferences between the effects of a magnetic field on the stability of dynamic and 
tic situations and is especially weak in dynamic situations where it only shows 
IV the magnetic field affects the first group of forces and not the second. 
In§ 2 of this paper we discuss the stability ofm.h.d. flows from a different physical 
int of view so as to demonstrate the effect of the magnetic field on the second 
oup of forces as well as the first. We first examine the currents induced by a 
t Seconded to School of Engineering Science, University of Warwick, Coventry. 
[ 342] 
Ii 
I 
343 On the stability of parallel flows with parallel magnetic fields J.C. R. Hunt 344 
disturbance, pointing out the important difference between electromagnetic andow consider a steady flow whose velocity vector is U(x, y, z) and vorticity vector 
mechanical disturbances-, and then examine the electromagnetic j x B force. (x,y,z). Let this steady flow be perturbed such that the velocity vector is now 
considering the rotationality of the j x B force we find under what conditions t U( ) ( ( t) 
· - · h d' W x,y,z + u x,y,z, , 
~agnetic field _affec~s the forces _ actmg on ~ e 1sturb~nce. e then consider t the vorticity vector is 
different ways m which a small disturbance mteracts with a mean flow and how the Q(x, Y, z) + w(x, Y, z, t). 
value of Rm, the magnetic Reynolds number, affects thi~_interaction. 188ume lu l ~ IUI and lw l ~ 121, 
For the rest of the paper we concentrate on the stab1hty of plane parallel flo~e magnetic field consists of a steady component, B0(x, y, z) and an unsteady 
for fluids with finite viscosity and ?onductivity,. unifor~ properties, and no fretponent, b(x, y, z, t). Let the steady current associated with B0 be J 0(x, y, z) and 
surfa~es. We prove that, when a umform magnetic fiel<l. rn parall.~l to the flow a perturbation current associated with b be j(x, y, z, t). Assume lbl ~ I B0 I, 
~uffic1ently large, the wavenum?er ~e.ctor of the ~ost ~nstabl~ disturbance is no !Joi · 
m general parallel to the flow, 1.e. it is a three-d1mens10nal disturbance. We thel(o examine the stability of the mean flow U under the action of the magnetic 
interpret this result physically using the concepts of§ 2. Our result invalidates t~B
0
, we examine the behaviour with time of the disturbances of the velocity, 
conclusion of Michael (1953) and Stuart (1954) who asserted that the wavenumb~dofthe magnetic field, b. We first consider the generation of electric currents 
vector of the most unstable disturbance was parallel to the flow, i.e. a two-dime the growth of b. 
sional disturbance. Using this erroneous assumption, Stuart (1954) , Velikhov (1959,.2. If we eliminate the mean values, Ohm's law, i.e. (2·2), for the disturbance 
and Tarasov (1960) examined the stability of plane Poiseuille flow with a paralltiimes : 
magnetic field. Drazin ( 1960) has examined some general aspects of the stabilizilli j = u(E+uxB0 +Uxb), (2·7) 
influence of a parallel magnetic field on a plane parallel flow, also considering onl ignore the second-order term u x b. Taking the curl of (2·7) and using (2·4) to 
two"dimensional disturbances. Wooler (196!) has examined the stability ofapla )we have: ob 
1 parallel flow when Rm~ 1 and when the magnetic field lies in the plane of the flo1 ~ + (U. V) b = (b . V) U + V x (u x B0 ) + -V2b. ut µu but is not parallel to it. He found that three-dimensional disturbances can be tht 
(2·8) 
most unstable; however, he did not point out that, even when the magnetic field ifbe terms in (2· 7) may be described physically as follows: the perturbation 
parallel to the flow, three-dimensional disturbances can still be the most unstabJeJcity u and the main magnetic field B0 induce a current j1 which produces its own 
For all values of R we show how results obtained for two-dimensional disturbanc~etic field bi, which with the mean flow U induces extra currents, j 2• These m 
can be modified to take into account three-dimensional disturbances and then(#lnts, j2, flow in such a direction that their own magnetic field , b 2, acts in the 
draw some general condusions about the stabilizing influence of a parallel magnet,IOsite direction to bi, by Lenz's law. j2 is modified by the growth of its own 
field on a plane parallel flow. In particular we prove that a parallel magnetic fie1'1letic field b 2, since j2 is induced by the mean flow U and the combined induced 
can never completely stabilize a flow, i .e. stabilize it for all values of the ReynoJ(jlletic fields (b1 + b 2). The relative directions of j 2 andj1 depend on the geometry 
number. lte mean flow and the magnetic field. 
1, j2, b1 and b2 may be found from the following equations which derive from 2. PHYSICAL ASPECTS OF THE STABILITY OF M.H.D. FLOWS ) and (2·8): 
2· l. We shall only be interested in the stability of incompressible fluids witl 
uniform conductivity, density and viscosity. Then the relevant equations o: 
magneto-hydrodynamics are 
pDu/Dt = - Vp + j x B+17V2u, 
j = u(E + u x B), 
(2-l 
j = jl +j2, 
b = b1 + b2, 
j = (1/µ) V x B, 
V x E = - 8B/8t, 
v'.u = 0, 
V.B=O, 
(2·2, 
(2·3 
Hen Rm(= Ulµu) ~ 1, where U is a characteristic velocity of the mean flow and l 
( l characteristic length, for disturbances of interest 8b/8t is much smaller than 
(2· other terms in the above equations and we can then deduce the ratio of j1 to j2 
(2·8& crude order of magnitude argument, from the above equations. 
where u is velocity, p pressure, j current density, B magnetic flux density, E electrit 
field, andµ magnetic permeability, p density, 17 viscosity, and u conductivity;tbl 
latter four quantities are all constant. 1 
lb1 I "'O[lhl lµ] , 
Jj2I "'O[uJ DI Jb1 + b2JJ, 
Jb2 J "'OW2l lµ]. 
ii ' 
J.C. R. Hunt 346 345 On the stability of parallel flows with parallel magnetic fields 
Therefore when Rm~ 1, lj2I "' O[CTµll U I lj1IJ, and 
lj2lflj1I "'O[RmJ. (2 
Thus when R,,,_ ~ l,_ e.g. in a laboratory experiment with mercury, jj21 <{ lhl and 
a good approximation, Ohm's law becomes 
ihen the condition that b = 0 everywhere when t = 0 is not satisfied since the 
e of b varies with position. (For a particular example, see § 3). Therefore even 
·11) is satisfied, in general j =!= 0 fort > 0. However, if Rm~ 1 we can usually 
re ob/ot and using the result (2· 10) we obtain the good approximation to (2·8) 
0 = V x (u x B 0) + (1/µCT) V2b. t 
(2·10 
This approximation, which can be justified by more refined argu . ' X (u x B 0) = 0, this becomes (1//tCT) V2b = 0. Then if b = 0 on the boundaries, 
partic_ular problem, is used in all stability analyses when Rm~ 1 (see ~;~~sF~: 0 throughout the flui~. Thus it is only whe:1 condition (2· 11) i~ s~~isfied, R_m_ ~ 1 
equation (2·8) we can find out under what circumstances currents will b . d b = 0 on the boundaries, that we can say J = 0 whatever the m1tial conditions. 
j = CT(E+u x B 0 ). 
em uced li. h · 1 · t'fi t' f h' · · h "f 1 
a perturbation velocity, u. If u and Bo satisfy the condition -,rne p ys1ca JUS 1 ea ion o t 1s reasomng 1s t at 1 cur (u x B 0) = 0, the change 
iux of the main magnetic field , B 0, linked by a fluid loop due to the perturbed 
V x (u x B 0 ) = 0, 
equation (2·8) becomes: 
(2· l 'on, is zero and thereby no perturbation currents are induced. If a stray magnetic 
.I, b, exists, the change of flux of b linked by a fluid particle due to the mean 
(
2
, 1 tion is not necessarily zero, and hence perturbation currents can be induced 
~ther or not curl (u x B 0) = 0. Clearly, if curl (bx U) = 0, then this latter source 
The disturbances in the velocity and in the magnetic field are coupled. A distuiurrents also :'anishes. 
bance of one _leads to a dis~urbance of the other and an m.h.d. flow can be ma4l·3. A magn~tic field can o~ly affect ~he velocit~ of a disturb~nce by me~ns of the 
unst.able by either type of ?1sturba~ce. Equation (2· 12) is interesting in that it d~trom_a~netic body force. J x B, whrnh occurs m t~e equation of motion (2· l ( 
not mclude the perturbation velomty u, but only the mean velocity u. It sho~rly, if J = 0, the magnetrn field has no effect on a disturbance. However, even 1f 
how a disturbance in the magnetic field, b, can develop by its interaction with tb 0, the j x B force does not necessarily produce any change in the motion of the 
mean velocity, U, without being affected by u, provided u satisfies (2·ll). t'.11'bance. Shercliff (1965) has discussed how the j x B term in the equation of 
For b and U not to induce any currents they must satisfy the conditions t1on (2· l) only changes the velocity distribution of an incompressible flow when 
,rotational, i.e. curl (j x B) =!= 0, assuming the fluid has no free surfaces and has 
V x (U x b) = o. (2 1 !l 
· "llltant density. To examine the effect of curl (j x B) on the disturbance we con-
We now deduce three sets of conditions each of which is a sufficient condition fatr the curl of ( 2· l) ignoring second-order quantities: 
there to be no growth of b or j. 
(i) If conditions (2·11) and (2·13) are satisfied simultaneously equation (2·12 p[!_)w +(u.V)Q.-(Q..V)u-(w .V)U] 
becomes Dt 
ob = __!__ v2b· 
ot µCT ' 
(2· 14) 
which shows that any stray magnetic field will simply decay due to ohmi,rhus it follows that the rate of growth or decay of the vorticity of the disturbance 
dissipation. 1ot affected by the magnetic field if 
(2· 15) (ii) Consider the development of a small velocity u when condition (2·11) i, 
satisfied and let us postulate that at some time t = O, when the perturbation veloci~ 
u is generated there are no perturbation currents, j, in the fluid or in the regionPe are mainly interested in situations where B 0 is constant in space and does not 
surrounding the fluid. It follows from (2·3) and (2·6) that v2b = o. rywith time. Then J 0 = 0 and (2·11) becomes 
If, also, b-+ 0 as x, y, z-+ oo then b = 0 everywhere. Then from the form ofth• 
equation (2· 12), whatever the distribution of U or the value of Rm, the only solutio1 
for the equation when t > O is d (2· 15) becomes 
b = 0. 
(B0 • V)u = 0, 
(B0 . V)j = 0. 
(2·16) 
(2· 17) 
Hence if condition (2·11) is satisfied and ifj = 0 when t = O, for all t > O, rheimportant difference between conditions (2·16) and (2·17) is that (2·16) only pends on the disturbance velocity u and the magnetic field B 0 whereas, since j 
j = 0. 
. . . . . . . ' This approximation should be justified, a posteriori, in each case it is used. It is justifiable 
(m) Most studies of the stability ofm.h.d. flows use 'normal mode' analyses, witlillany cases for the disturbances which are most unstable, e.g. in Plane Poiseuille flow. 
u, b "' exp [i(ax - CTt)]. 
r very high frequency disturbances, which are highly damped and stable we could not 
0re 8b/8t. 
347 On the stability of parallel flows with parallel magnetic fields J.C. R. Hunt 348 
depends on U as well as u, (2· 17) depends on the mean velocity U a 1 [Bo = (0, 0, Bz) the flux linked by the fluid circuit ABOD is zero and
 does not 
disturbance velocity, u. However, when Rni ~ 1, we have shown 'th~t 8 we 1 as th,Dge with time. But now consider another fluid circuit PQRS at right angles to 
former one. Then, since the flow is incompressible, the area enclosed by this 
(2· l&uit is constant and so the flux linked by the field Bz remains constant. Hence, j = o-(E+u x B0), 
so that one can then ignore the effects of band the mean flow, U. Med there is no disturbance to the magnetic field, b, there are no induced 
Inde~ucing(2·17)weassumedJ0 = O. If,however,Rni ~ landauniformmagneti'ents, (Bz8/8z)u = 0, and therefore Bz does not affect such a flow. Thus a 
field B1 IS produced outside the region of flow and induces currents J 0 by means Jorm magnetic field does not affect a two-dimensional disturbance, if it is 
~he mean flow U, the field due to J 0, B 2 is very small compared to B1 (Shercliff 196 tllel to the vorticity of the disturbance. 
1.e. IB2I ~ IB1J, where 5 Ve now consider the mode of interaction between the mean flow and the distur-
Bo = B 1 + B 2 and J 0 = (1 /µ ) v x B 2. ce. This ~scussion is speculative since t~ere have be~n few 
previ?us attempts 
Th . £ 11 
liscuss this aspect of the problem physically. We will only consider here the 
. en it O ows that the terms on the right-hand side of (2·14) , (b. V)Jo, (J0.\7) 11tion where all the energy of the disturbance comes from the mean flow. 
(J. V) B 0 are much smaller than (B . V)j Thus if (B V)J0 - O it may b · · · 
. . o · o · - e a go()(,et us first consider plane parallel flows with parallel magnetic fields. When there 
approx1mat10n to say that the magnetic field has no eccect on a d1'sturban · · · · · 
. .
 . . w ce even Jio magnetic field such a flow 1s able to feed energy mto a disturbance by means 
Jo ~ o,_ provided R_ni ~ 1. However, this approximatwn would need careful inves~e Reynolds, or inertial , stresses ( -pux u ). Viscosity does not affect this inter-
gat10n many particular problem and would have to be 3'ust1'fied a poster· · . · · · Y • • • 
y 
c 
' iori. ion directly, but by the stresses 1t mduces 1t 1s able to change the disturbance 
ocities, and hence the Reynolds stresses. Thus, indirectly, viscous action can 
ke a flow stable or unstable. The electromagnetic force due to the current ji, i.e. 
B0, can act in a similar way, as shown by Drazin (1960) . The main difference 
ween these two types of damping forces is that the electromagnetic damping of 
isturbance depends on its direction relative to B0 , i.e. it is anisotropic. Drazin did 
differentiate between j 1 and j 2, since he was only interested in the case of Rni ~ 1, 
enj2 is negligible. When Rni ~ 1 so that effects due to j 2 are negligible, Drazin 
wed that if B 0 is sufficiently great the dissipative forces due to j 1 x B 0 are so 
p p' mg that at a given finite value of the Reynolds number any plane parallel flow 
~ 1be stabilized by a parallel magnetic field. (We show in§ 3 that this result is not 
B / inged by considering three-dimensional disturbances). 
z 
FmunE 1 A typical disturbance t 11 . . th dir t . 0 Th ,,,, f h .In other flows where the destabilizing agents are inert
ial forces, surface tension, 
· rave 1ng 1n e ec ion x. e eu ect o t e ma.gneti . . . . . . . 
field on the disturbance is seen by examining the change of flux linked by the fluid circnirVIty, etc., and where the act10n of vrncos1ty IS primarily to reduce the rate of 
ABCD and PQRS with the imposed magnetic fields Bx and B,. (The induced magnetilwth of disturbances, then when Rrn ~ 1 the effect of a magnetic field, i.e. the 
fields and currents are not shown ) B r · · · r d · Oh dr kh ' 
· 
i O iorce, 1s also dampmg. Examples of this effect to be ioun m an ase ar s 
Let us now consider a pa t' 1 1 t d t t th h . 1 . ilk, are Couette flow between rotating cyli
nders and capillary instability of jets. 
r 1cu ar examp e o em ons ra e e p ys1ca meanmg o . . . . . 
the conditions we have derived for the magnetic field to affect a disturbance (2.16us by means of the currents Ji, the energy of the disturbance 1s dissipated, but 
and (2· 17) Consider a sim 1 t d' · 1 d' t b f th ' velocity distribution of the disturbance may be altered in such a way that the 
. p e wo- 1mens10na 1s ur ance o e type 1 
. 
urbance can grow more rapidly. 
u = [ u(y) exp [i(ax - o-t)] , v(y) exp [i(ax- o-t)] , OJ, \Ve have already described how the currents j 2 are induced by the perturbation 
imposed upon a mean flow in the x direction, as shown in figure 1. The disturbano'gnetic field b 1 and the mean flow U. By means of these currents, h, the mean 
moves in the Ox direction, and its amplitude varies in the Oy direction. At iW can feed energy into the perturbation field b and by means of the j 2 x B 0 force 
certain time, t, a circuit drawn in the fluid lies on the points ABOD. At a latertilllcan feed energy into the perturbation velocity, u. Thus the current, j 2, leads to 
t + ~t the particles on the circuit have reached A' B' O' D'. th the electromagnetic and mechanical energy being transferred electromagneti-
If B0 = (Bx, 0, 0) the flux linked by the fluid circuit has changed in the tirne d/yt as distinct from the mechanical or inertial way it is transferred when the 
and consequently currents are induced in the z direction. The currents which a turbance and the mean flow exchange energy by inertial stresses. An important 
induced will affect the flow because j varies with x and hence (j x B ) is rotational i.f 
z 
O 
' t This mechanism may also be interpreted as the interaction of the mean flow with the 
l (, B ) _ (B 8) . \gnetic shear stresses (bxb 11/µ). See, for instance, the equation for the energy of a small 
cur J X o - x· ox Jz =I= 0. urbance in a plane parallel flow deduced by Stuart (1954). 
11 
I 
349 On the stability of parallel flows with parallel magnetic fields 
difference between these two methods of energy transfer is that the former is lo 
range and the latter short range, i.e. a small localized disturbance can exchanl let 
energy electromagnetically with a part of the mean flow some distance from it. 
When Rm<{ 1, lj2I <{ lj1 I (equation (2·9)), and consequently the main means 0: 
U( ) _ Uo(Y) 
y - '(] ' 
J.C. R. Hunt 
* Vl 1' = yl ' A= ,J(a2 + y2), v = R, 
energy transfer between the disturbance and the mean flow is mechanical. As R , _ /l d */ * = tan e 
II y -y , an ya , increases j 2 and b 2 increase and the means of energy transfer between the main flo\'I . . . h 
350 
and the disturbance is as much electromagnetic as mechanical. ire '(] is a characteristic velocity of the flow and l is a characteristic lengt . · 
· · · · f th 1 't and magnetic field are inserted m the In the presence of a magnetic field some of the kmetic energy of a mechanicaif the above express10ns or e ve 0?1 Y . . . . 
disturbance is converted to electrical and magnetic energy. This energy is botlli,d. equations (2· l) to (2·6), and if d1fferentiat10ns _are earned out with respect 
dissipated as Joule heating and stored in electric and magnetic fields. The energJY', then Stuart (1954) has shown that the equat10ns can be reduced to two 
which is stored can change back into mechanical energy. This alternation betweeJlations in v and ifr, namely . 
mechanical and electromagnetic energy provides the mechanism for a disturbanCE · (U -c) (v" -A2v) - vU" _ Bo_ (ijf" _ )..2ijf) = -\viv _ 2A2v" + A4v), 
to propagate along the field lines as Alfven waves. At the present time there is littlE µpU aR 
(3·3) 
physical understanding of how this mechanism affects the stability of flows B v _ i 
especially since the normal mode analysis does not reveal much about how al (U-c)ifr- V = aRm (ifr"-A2ifr). 
disturbance propagates. Velikov (1959) has shown that when Rm~ oo and R-+ '1vh R <{ 1 (3·4) reduces to 
a magnetic field can stabilize all two-dimensional disturbances in a plane flow en m 
parallel to it if 
_ B~v = -i (ifr" -A2ifr).t 
A Bo 1 U aRm 
(3·4) 
(3·5) 
= ,J(µp) um> ' 
rhis approximation is only valid for disturbances ~h?se w~velength is O[l] and 
where Um is the maximum velocity of the flow. He gave no physical explanation fo1quency o[U /l ]. These are not very restrictive cond1t10ns smce they are us~ally 
his result. isfied by the disturbances of interest, i.e. the most uns~ab~e: However, ~s pomted 
Thus there are three new factors which arise on considering the stability offlowetin § 2, this approximation should be justified, a posteriori, many particular case 
in the presence of magnetic fields: ing the values of a and c found in the analysis. 
(1) An anisotropic dissipative force acting on the disturbance alone. rhen from (3·5), (3·3) becomes 
(2) A new, long range, mechanism for the interaction of the disturbance and the icr B5 lav - i ( iv 2) 2 ,, + ) 4v) (u )( ,, )..2)-vU"+ = -V - 1tV it. mean flow. 
-c v - v pU aR 
(3·6) 
(3) A mechanism for the propagation of the disturbance. 
In general we expect the two latter effects to be unimportant when Rm<{ 1. 3·2. Three-dimensional disturbances 
3. STABILITY OF PLANE PARALLEL FLOWS WITH ALINED MAGNETIC 
FIELDS-LOW MAGNETIC REYNOLDS NUMBER 
3· l. General equations 
Equation (3·6) describes the motion of a disturbance travelli~g at. an angle Oto 
,and B
0
• Now the motion of such a disturbance can ~e described m ter~s.of an 
1uivalent one for which e = o. Let aR = AR and R = R cos O where R 1s the 
eynolds number for the equivalent disturbance. 
Consider a steady flow whose velocity is U 0 = (U0(y), 0, 0) and an infinitesiroaJLet 
disturbance velocity 
u = (u(y) exp [i(a*x + y*z - ,Bt]) , v(y) exp [i(a*x + y*z - ,Bt)], liere 
crB5l 
-=-- = q0 and aq0 = Xq, pU 
_ crB5 l e q=pUCOS. 
w(y) exp [i(a*x + y*z- ,Bt)]). (3·lfen (3·6) becomes 
(U-c) (v" -A2v)-vU" +iAqv = -(i f.AR) (viv_2A2v" +A4v). (3·7) 
There is an alined magnetic field with flux density B 0 = (B0 , 0, 0) and an infinite· . . 
simal perturbation magnetic field t As pointed out by Hains (1965), (3·5) may not be the corr~ct first:rd~~ app~o;im~i~; 
' hen R <{ 1 if the perturbed magnetic field is externally applied rat er an m uce 
b = {<]S(y)exp[i(a*x+y*z-,Bt)], ijf(y)exp[i(a*x+y*z-,Bt)], tlocitymperturbations. But for low frequency oscillations of interest, the currents due.to 
eh a perturbation produce an irrotational j x B force and hence do not affect the velocity 
x(y) exp [i( a*x + y*z - ,Bt)]}. (3·2lirturbat,ions . 
351 On the stability of parallel flows with parallel magnetic field
s J.C. R. Hunt 352
 
_Assuming the existen_ce of suitable boundary co
nditions for v which do not varject due to the magnetic field actin
g on the induced currents, h, as() increases to 
with the an~le () of the disturbance, we now have an eigenval
ue relation between thi• (Bo. v') u-+ O. It was shown in§ 2 that whe
n Rm~ 1, (B0 . v') u = O and there are 
parameters m (3·7) of the form 
!applied currents on the boundaries, j = O in the fluid. Hence as
()-+ f7T, j x B 0 
F(c, i\., q, R) = 0. (3-sicreases. However, t
he inertial terms also decrease as() increases. The reason wh
y 
This relation is the same whatever the angle
 () chosen for the disturban '1e rat io of electromagnetic to i
nertial forces for a disturbance with given wave
-
Equation (3·8) may be rewritten 
ce.ngth, i\., i.e. q, decreases as () increases, is because the elec
tromagnetic forces 
)[11Bi J u J cos2 ()J) decrease proportionally to cos2 () and the inertial forc
es 
G(c,i\.,e,%,R) = 0. (3·9) (O[p JUJ cos() JuJ/i\.]) 
The main information we want to derive from 
relations (3·8) and (3·9) is th d . 11 
() 
1 ·t f h fl · 
· 
~y ecrease proport10na y to cos . 
ve om y o t e md at whrnh the flow becomes
 unstable, given the value of the 
applied magnetic field and the values of the fluid 
properties. In terms of the mathe-
. . 
matical problem we want to know, given% or so
me other parameter involving B 
3·3. General considerations for low Rm flows 
what is the lowest value of R,Rerit., for which th
e flow becomes unstable, i.e. whe:As a result of sho
wing that Squire's theorem cannot be extended to
 plane parallel 
ci = 0. To do this we calculate R for various val
ues of er, i\. and () and look for it,eOWS with a parallel magnetic 
field in § 3·2 we can draw some general conclusions 
lowest value. 
oout the stabilizing influence of a magnetic field. 
When q0 = 0 Squire (1933) showed that disturbances for whi
ch () > O (three-From the relation (3·8) we can deduce a relatio
n between .Rerit. and q of the form 
dimensional disturbances) become unstable at higher valu
es of R than those for 
R . = f(- ) (3.12) 
h . h () O (t d' · 1 d' b 
-
ent q · 
w 1c = wo- 1mens10na IStur ances) . If the lowest
 value of R for which t " (3 12) b "tt 
· 
. -
. 
qua 10n · may e rewn en 
ci = 0 is Rerit., then for a disturbance at an angle
(), Rerit. = f (q0 cos ())/cos() (3· 13) 
Rerit. = .Rerit.f cos(), which is greater than .Rerit. · 
_ 
1d for () = 0 becomes Rerit. = f (q0). 
(3· 14) 
The lowest value of Rerit. occurs when () = 0 and R .t = R 
.t . Hence the theorem 
is proved. 
en · en· ossow
 (1959) and Stuart (1954) have obtained relations of the type 
(3·14) in 
When q0 > O if the lowest value o
f _Rat which c . = o for some value of q is R . 1amining the stabiliz
ing influence of a parallel magnetic field on bou
ndary layer 
then, _ 
' 
cnt.JWS and plane Poiseuille flow. They did not 
consider disturbances for () > 0. 
Rerit. = Rerit. and % = _.1_ . (3·10)typicalj(q) is s
hown in figure 2. Using this figure we show how t
o find the value of 
cos() cos() which gives th
e minimum value of Rerit. for any given%· (We assume() is po
sitive, 
If () = 0, Rerit. = .Rerit. and q0 = q. 
(3·ll)'ough it could equally be negative.) 
OF' K ' is a tangent drawn from the origin Oto the cu
rve off (x). OP' = x' is the 
For the same value ofq, we have the same result as 
before that the most unstabletdinate of F'. 
disturban~e occurs w~en () = 0. But for given q0, i.e. a given f
low and given magnetic Let % = x1 < x' and for som
e value of() choose x2 such that x2/x1 = cos(). There-
field, as () mcreases, q decreases and, depending on the valu
e of() and %, this m&Y,re -q = q cos() = x and R . = f(x ) = ~ F. on the di
agram. 
1 d t · d 
· R · d 
O 2 er1t. 2 2 2 
ea o an mcrease or ecrease m 
erit.· Hence 1t is, in general, incorrect to conclu esin
ce 
Rerit. = .Rerit.fcos (), 
that the most unstable disturbances are those
 whose wavenumber vectors are 
parallel to the flow and magnetic field, i.e. those fo
r which() = 0. Michael ( 19 53) and 
Stuart (1954) did not appreciate that the magnetic interaction te
rm q varied wilh 
()forgiven B 0 and U and erroneously concluded that
 Squire's theorem couldbe'II the diagram. But for 
() = 0, Rerit. = P1F1 and for any () > 0, Rerit. > P1F1. 
extended to apply to flows with parallel magneti
c fields. lterefore Rerit. is gr
eater when() < 0 than when() = 0. Hence when%< x', Rer
it. 
Our result can easily be seen in physical terms. W
hen there is no magnetic field a minimum when () = 0. 
there are two competing effects, the viscous dam
ping of the disturbance, andtnelfnow q0 = x3 ( > x'
) and w_e choose() such that: x4/x3 = cos e,. then q = x 4 , and 
inertial interaction between the Reynolds stresse
s and the velocity gradient ofln&crit. = f (x4) = P4 F4 on the diagram. _Th
en Rerit. = Pa K 3 on the diagram. . 
mean flow. The latter effect, which provides the
 mechanism for the mean flow But P 3 K 3 < P 3F 3 an
d hence Rerit. 1s least for a value of() > 0. To find this value 
feed energy into the disturbance decreases as 
() increases whereas the former' 0 we choose() such that P 3K 3 
is least. P 3K 3 is least if x4 = x' and K 3 = K'. The
n, 
independent of e. Hence the most unstable disturban
ces are those for which()"' general, for qo > x'' the minimum
 value of Rerit. is given by 
When there is a magnetic field parallel to the flow
, B0, and Rm~ 1, there is a tb' 
Rerit. , min. = P3 K'. 
(3·15) 
353 On the stability of parallel flows with parallel magnetic fields J.C. R. Hunt 354 
The value of e for which Rcrit. is a minimum is given by Thirdly if the curve of Rcrit. against q0 for two-dimensional disturbances is of 
Je form of curve O in figure 3, then the curves of A against R for which q0 is a 
cose = x'/xa = x'/qo. (3·16) Jnstant, ci = O, and e = Oare closed curves and when q0 = Q, the closed curve 
Note that, as%-+ oo, 8-+ f7T, since x' is fixed, and also e = O when%= x'. 1generates to a point. It follows that similar curves of A against R when e > 0 are 
Thus any relation between Rcrit. and q0 calculated by considering disturbances ~o closed curves and that for a given value of e, = Ov there is some value of q0 , = Qi, 
for which e = O can be converted to one giving the minimum value of R . Jf which the curve of A against Ron which% = Q1, ci = 0, and e = Ov degenerates 
R ·t . for given q0 by drawing a tangent from the origin to thee = O curve Tchnt., 1a point. But, whatever the value of q0, there is always some value of e, < f7T, for cr1 ., min., • en 
the curve of Rcrit., min. against q0 follows the e = 0 curve from F0 to F' (see figure 3) ·hich the A - R curve has not degenerated to a point. 
t 
f(x) t 
R 
X3 Q ffo--
FIGURE 2. F. F' F. is a typical curve of R . against q for two-dimensional disturbances. 'mURE 3. The effect of tJ:µ,ee-dimensional disturbances in a hypothetical flow for which all 
Here q0 ~ x !nd J (x) = Rmit. · The mi:i'i~um valu: of Rc,it.• taking into account three- two-dimensional di~turba~ces ar_e stable when q0 > Q. ~, Unst3:ble fo~ all di~turbances; 
dimensional disturbances, is shown to lie on the curve FoF' and then the line F' K'. The ~, stable for two-~nnens10nal disturbances, unstable for three-d1mens10nal disturbance; 
dashed line is a typical Rc,it. - q0 curve at a constant value of () > O. lilll, stable for all disturbances. 
, . . Fourthly, Drazin (1960) has shown that a sufficient condition for stability can be 
and for qo > x it follows the tang~nt. The curve may also ~e described as the leduced for the flows we are considering. He only considered disturbances for which 
envelope of the curves of Rcrit. agamst q0 at constant 8; a typical curve for O > 01= 0 and obtained the result that if 
is shown in figure 2. Should it be possible to draw more than one tangent from the 
origin to the e = 0 curve, then the curve of Rcrit., min. against q0 is a little more 
(3·17) 
complicated to construct. Some general consequences follow from this methodofrhere t is the maximum value of du/dy', then the flow is stable. Now the curve 
constructing the curve. I= 8%/t2 is a straight line and no tangent can be drawn to it from the origin. Hence 
First, we need only calculate the curve fore = 0 and for q0 ~ x ' to know the value his relation is equally valid for disturbances for which e > 0. See figure 3. 
of Rcrit., min. at all values of%· Thus we can state an upper limit and a lower limit to the extent that a magnetic 
Secondly, we can now prove that, whatever the value of q0 , there is some finite ield can stabilize a flow, but we still cannot say whether a parallel magnetic field 
value of R for which the flow is unstable, provided there is some finite value of Rin lower the critical Reynolds number of a flow or not, though no instance of this 
for which it is unstable when q0 = 0. If there is any flow which is completely stable1~s yet been found. 
for all two-dimensional disturbances when q0 > Q then the curve of Rcrit. agains\qo 
fore = 0 will touch a line parallel to the ordinate and cutting the abscissa at q0 "'Q, 4. ARBITRARY MAGNETIC REYNOLDS NUMBER 
e.g. the curve O in figure 3. Now it is possible to draw a tangent from the origin to If R d . f th d't' th t R 1 t · l'f (3 4) t • • • • • U l oes not sat1s y e con 1 10n a ~ we canno s1mp 1 y · o 
any curve which cuts the ordinate and touches a hne to the right of 1t and para e bt . m . £ 1 . (3 6) F (3m4) h 
t ·t Th h fR . b . h 1. £ Jue am an equatwn or v on y, 1.e. · . rom · we ave o 1 . en t e curve o crit., min. agamst % ecomes a stra1g t me or some va 
of q0 > (, (q' < Q), and hence for any value of% there is some value of R for which v = U [( U _ c) ij, + _ i _ ( ij," _ A 2v,)] . 
the flow 1s unstable. B0 aRm 
355 On the stability of parallel flows with parallel magnetic fields J.C. R. Hunt 356 
Substituting this value of v into (3·3) we obtain an equation in i/r, which is pro b Th "d d th di t b d t h "t £ rm
 as n ari"es 
- ~tur ance. en, prov1 e e s ur ance oes no c ange 1 s o u v , 
portional to the induced field in they direction: 
· d s· b 1· · th I 
,8 increases V x (u x B0) and hence Ji and b1 ecrease. mce 1 1es m e pane 
[( U _ c )2-A 2 _ i U" ] (if
"_ A2if) + 2 U' ( U _ c) if,' yk, 'v x (U x b1) and hence j 2 and b2 decrease
 as () increases. Therefore the iz x B0 
aRm ,well as the j1 x B0 force decreases with e. Thus, as well as in the inertial interaction, 
2 ) 2 2 ( u ) (D2 ) 2)2 '''] 1 ne electromagnetic interaction between the disturbance and the mean flow also 
= -1· [(D - 1l) (U-c)'''+ -c -{l 'f' + - 2- - (D2-A2)a,,,, 
R 'f' R RR 'f' (4·1) .ecreases as e increases. a a m a m 
_ 
We can draw some general conclusions about the stabilizing influence of a 
where D = d/dy', and the Alfven number A = B 0/ ,J(µp) U. This equation describes Jagnetic field at arbitrary Rm. From (4·3) we deduce a relation of the
 form 
the motion of a disturbance travelling at an angle Oto U 0 and B 0 ; asin § 3 the motion R . = f(R A) 
of such a three-dimensional disturbance can be described in t erms of an equivalent cn
t. m, ' 
two-dimension_al one. - - - rhich may be rewritten R 
- f(Rm cos(), A) 
Id 
crit. - cos e . 
Let aR = AR and aRm = AR11., where Rand Rm are the Reyno s number and 
(4·6) 
magnetic Reynolds number for the equivalent disturbance. lore= O, (4·6) becomes 
(4·7) 
Then R = Rcose, Rm= Rmcose. 
Equation ( 4· l) becomes 
1'arasov (1960) has examined the stabilizing influence of a parallel magnetic field 
n plane Poiseuille flow for arbitrary Rm by only considering disturbances for which 
:c 0. He obtained results for Rcrit. at various values of A and Rm. The shape of a 
[(U - c)2-A2- if!.."] (if" - A2if) + 2U'(U -c) if' 
urve of Rcrit. against Rm for a constant value of A, plotted from Tarasov's results 
A.Rm 1similar to that of curve O in figure 3. 
__ . [(D2 - A 2)2 ( U - c) if, ( U - c) (D2 - A 2) if] 1 (D2 - A 2)3 if Since A does not vary with e, we can apply the argument of § 3·3 to deduce the 
-
1 AR + AR + A2RR · (4·Z) ninimum value of Rcrit. when all values of e are considered. Then we find as before m m 
. 
nat for Rm greater than some value, the Rcrit.-Rm curve for constant A becomes a 
Note that A is the same for any e. traight line, namely the tangent from th
e origin to the Rcrit.-Rm curve, i.e. the 
Assuming the existence of suitable boundary conditions for if which do not vary 1 . -R curve for constant A for two-dimensional disturbances. 
with the angle e of the _disturbance, we once more have an eigenvalue relation cTtis ;ill be the ultimate form ofthe
Rcrit.-Rm curve for constant A for all velocity 
between the parameters m (4·2) of the form rofiles at all values of A. It follows, therefore, that there is n
o parallel magnetic 
F(c A A R R) = o (4-3) ield which will completely stabilize a flow for finite values of Rm. In other words, 
' ' ' m, . rhatever the value of A or Rm, there is always some value of R for which the flow is 
Note that we have one more parameter than in (3·8). This relation is the same forall IIlstable. 
values of e. Equation ( 4·3) may be rewritten: By only considering disturbances for which e = 0, Tarasov er
roneously concluded 
G(c, A, e, A, Rr,,, R) = O. (4-4) uat t here was always some value of A which could stabilize the flow. 
Velikhov (1959) followed Michael (1953) in assuming that the most unstable 
As in § 3, we are_ interested in findi~g the lowest v~lue of R for whic~ the flow !tsturbances occur fore = 0 and thence deduced a sufficient condition 
for a parallel 
becomes unstable, I.e. ci = 0. We examm~ the role of d1sturbanc!s for "'."h1ch () > 0. nagnetic field to stabilize a flow when R and Rm-+ oo. If dis
turbances for which 
- For some value of A > 0 and Rm > 0, if the lowest value of Rat which ci = 01s I> 0 are considered, the basis of his ar
guments has to be changed, e.g. if 
R ·t then 
_ 
en · ' R . = Rcrit. and R = Rm (4·5) e = f7T-O(R;;;,1], Rm= O[l], 
cnt. cos e m cos e' 1nd the dissipative terms in (4·2) could not be ignored. The dissipative terms can 
and if() = 0, Rcrit. = Rcrit. and Rm= Rm. inly be ignored if Rm-+ oo and R-+ oo. Then Velikhov's resu
lt, that if A > 1, all 
For a given flow and given magnetic field, i.e. a given value of A, as e increases lows are stabilized, is correct. In gene
ral, therefore, a careful examination is 
Rm decreases. It is generally found that the lower Rm the less the stabilizing effert ieeded of the limiting processes R-+
 oo or Rm-+ oo when three-dimensional distur-
of a magnetic field and so we expect that the most unstable disturbance occurs for 1ances are considered. 
some e > 0. Anyway it is incorrect to conclude that the most unstable disturbantts CO
NCLUSION 
are t~ose for which e = 0, .whatever the value of Rm. . . . ur· It was stated in§ 3 ~hat the main purpose of the sta~ilit
y analysis was to find ~he 
This result may be seen m terms of the concepts of§ 2. Consider a velocity d1st owest velocit at which the flow becomes unstable, given th
e value of the a pp bed 
bance, u, which lies in the plane Oyk where Ok is the direction of propagation of tie nagnetic fielJ and the values of th
e fluid properties. It is not very instructive, 
357 On the stability of parallel flows with parallel magnetic fields J.C. R. Hunt 358 
physically, to consider Rcrit. as a function of q0 = er Bi l/pU as we did in§ 3 or as a sponent' /3 = Uaci. Michael (1961) showed that, if we regard U as the parameter to 
function of A and Rm as in §4since q0,A, andRmarenotindependentvariablesifone evaried so that R oc U, then 
considers B0 and '[J as the parameters to be varied in an experiment: We considered /3 = .!:__ ac. R = .!:__A.Re. oc A.Re. 
these particular non-dimensional parameters since they appear m the stability £ 2 i £2 i i· 
equations. From the graphs already shown one could plot o~her graphs of RcrH. 1ote that the value of this expression is independent of e. 
against M = Bol(cr/17)Hor values of constant P, = vµcr, say. This would be a simple Consider the case of Rm~ l; then a curve of a against R, at constant values of 
operation sin?e !R<\ and %, is the same as that of an equivalent two-dimensional disturbance for 
M = ,J(Rqo) = ,J(A 2RRm) · rhich A is plotted against Rat constant values of A.Rei and q. If Rcrit. is the lowest 
Replotting. Stuart's results in this way indicates that the form of the Rcrit.-M ·alue of R for a given q and A.Rei, and we then plot Rcrit. against q for constant 
curve P ~ 1 is similar to that of the Rcrit.-q0 curve for Rm~ 1. (See figure 4.) Itis !Rei, we can calculate the minimum value of Rcrit. for given q0 and A.Rei, using the 
easily proved that for sufficiently high values of M the Rc,it.-M curve becomes a onstruction of§ 3. Thus, as with the case of ci = 0, when 1o becomes sufficiently 
straight line, which is a tangent from the origin to the curve for a two-dimensional reat the curves become straight lines. In experimental terms this means that we 
disturbance. It is interesting to note that when M = 50 the angle at which the most an calculate the minimum velocity of the mean flow for a given amplification rate 
unstable disturbance travels to the direction of flow is 60°. Thus in an experiment fthe disturbance and given magnetic field. 
with mercury the effect of three-dimensional disturbances will become apparent Though we have only examined the stability of fluids with uniform properties 
at values of the magnetic field of about 2000 Gin a duct whose width is about 1 cm. nder the action of a uniform magnetic field, by virtue of the physical reasons given 
Rcrit. for two-dimensional disturbances 
>} 20 
I 
0 
...... 
0 20 
,,,,,,""' 
/. 
40 
/ 
/ 
/ 
/ 
/ , 
/ 
; , 
60 
tangent from 
origin to curve 
M 
FIGURE 4. Rc,it. against 11i( = B 0 l(0"/7J)!) for plane Poiseuille flow when µ0"8~ l, 
based on the theory of Stuart (1954). 
n § 2 it is likely that, in general, three-dimensional disturbances will be the most 
lllstable in practical situations, m.h.d. generators for instance, where magnetic 
.elds are applied to the flow of fluids with non-uniform properties. For an example 
f some of the other interesting kinds of instability which can occur under such 
ircumstances see McCune (1965). 
I should like to thank Dr J. A. Shercliff for the helpful advice he has given me in 
he preparation of this paper. This work is published by permission of the Central 
!lectricity Generating Board. 
REFERENCES 
handrasekhar, S. 1961 Hydrodynamic and hydromagnetic stability. Oxford University Press. 
lrazin, P. G. 1960 Stability of parallel flow in a parallel magnetic field at small magnetic 
Reynolds numbers. J. Fluid Mech. 8, 130. 
:lobe, S. 1961 The effect of a longitudinal magnetic field on pipe flow of mercury. Trans. 
Amer. Soc. Mech. Engrs. 83, (C}, 445. 
lains, F. D. 1965 Stability diagrams for magnetogasdynamic channel flow. Phys. Fluids, 
8, 2014. 
lichael, D. H. 1953 The stability of plane parallel flows of electrically conducting fluids. 
Proc. Oamb. Phil. Soc. 49, 166. 
. . . . lie! lichael, D. H . 1961 Note on the stability of plane parallel flows. J. Fluid Mech. 10, 525. 
The only experimental work done on the stab1hty of parallel flows with pa~a lcCune, J.E. 1965 Non-linear effects of fluctuations on MHD performance. Sixth Symp. on 
magnetic fields has been done with flows in circular tubes (Globe 1961 ). There is no Eng. Aspects of Magnetohydrodynamics. Pittsburgh, 1965. 
satisfactory theory for the onset of instability of such a flow with or without a lossow, V. J · 1959 Boundary-layer stability diagrams for electrically conducting fluids in 
. the presence of a magnetic field. N.A.S.A. Tech. Rep. R-37. 
magnet10 field. lquire, H. B. 1933 On the stability for three-dimensional disturbances of viscous fluid flow 
In experiments on the stability of plane Poiseuille flows with no magnetic fields between parallel walls. Proc. Roy. Soc. A, 142, 621. 
it has not proved possible to obtain a critical Reynolds number in agreement with lhercliff, J. A. 1965 A textbook of magnetohydrodynamics. Pergamon. 
th t d' t d b r . d theor It is likely to be equally difficult in experiments tuart, J. T. 1954 On ~he stability of viscous flow between parallel planes in the presence of 
_a pre IC e Y 1nea:ize Y· . . . . . nts , a coplanar magnetw field. Proc. Roy. Soc. A, 221 , 189. 
with a parallel magnetic field. There IS more hkehhood of success with expenme arasov, Yu. A. 1960 Stability of plane Poiseuille flow of a plasma with finite conductivity 
on the stability of boundary layers with parallel magnetic fields . . in a magnetic field. Soviet. Phys. JETP, 10, 1209 (in English). 
Though we have only discussed neutrally stable disturbances (ci = 0), growing le[~ov, E.~- ~959 The st~bility of a _plane Poiseuille flow of _an idea~ly conducting fluid 
. • . . The Ill a long1tudma l magnetic field. Soviet Phys. JEPT, 9, 848 (m English). 
disturbances also become three-d1mens10nal as the m~gnet~c field 1,ncrea~es. . \Tooler, P. T. 1961 Instability of flow between parallel planes with a co-planar magnetic 
rate of growth of a disturbance depends on the d1mens10nless amphficat1on field. Phys. Fluids, 4, 24. 
... 
...... 
... 
INTERNATIONA L ATOMIC ENERGY AGENCY 
EUROPEAN -NUCLEAR ENERGY AGENCY ~ -
INTERNATIONAL SYMPOSIUM ON 
MAGNETOHYDRODYNAMIC EU:CTRICAL POWER ~ENERATION 
Salzburg, A ustria, 4-8 July 1966 
SM-74/13 
REVISED ABSTRACT 
ON SOME FLUID DYNAMIC EFFECTS IN LARGE SCALE M. H. D. GENERATORS 
J. C.R. HUNT - UNIVERSITY OF WARWICK, COVENTRY 
\lt the - present time we are unable to carry out a complete 
analysis of the fluid dynamics and electrodynamics of an M. H. D. 
generator. However, various aspects of the behaviour of an 
M. H. D. generatot may be examined by the use of simplified models 
e.g. 
(i) One - dimensional gas dynamics. (e.g. Louis et al, 1964). 
(ii) 
(iii) 
( i _v) 
The current distribution can be found if the veiocity is 
·assumed const_ant across the duct. (e.g. Witalis, 1965). 
The s_kig f_riction · and heat transfer to . the walls can be. 
calculated by boundary layer analysis if the flow is 
assumed to _ be laminar . (Kerrebrock, _ 1961), 
A complete ·description of the velocity and current 
distribution across the duct can be given if the flow is 
assumed to be uniform, laminar, incompres_sible and not- . 
varying in the flow direction, (Hunt & Stewartson, 1965), 
Taken together; these and other ·models will enable us to describe 
most of the effect's in an M, H. D. generator, 
In this pap.er another simplification is considered in which the 
electroma'gnetic ·forces a_re assumed to be much 1arger than the 
inertial forces, The ratio of these two forces is measured by the 
-parameter, S ;trB,. .... d../eU.., where er is the conducti-vity, Bo the 
magnetic field, d. the width of hhe duct, e the density andU the 
niean velocity. Thus S >.:-- /. We also assume that the magnetic --
Reynolds number is very much less than one. In · the largest 
· experimental . generators how being built S""l. . . Thus, though . the _ 
results of this model a re not immediately applicable, they should 
indicate the· effects of increasing the magnetic field strength 
and the size of M. H. D. generators. · 
When S >> 1, one can consider the duct" to ·be divided int? 
2 regions: 
(i) A core region where electromagnetic forces are balanced 
by the pressure gradient and where inertiil a s well as 
viscous- forces are negligibl e, 
(ii) Boundary layers ·on the walls where again inertial forces 
are negligi,ble but where the viscous, electroma,gnetic 
and pre·ssure forces are of the same order, 
We show how it is then possible to cal~ulate the core flow in 
diver~ing ducts and .in ducts with non-uniform magnetic fields, with 
the Hall effect and compiessibility included, and obtain approximate 
answers for these otherwise very difficult problems. We also 
demonstrate the simplifications in the analysis of the boundary 
l ayers which result from this approximation, 
3 ON SOME FLUID DYNAMIC EFFECTS IN LARGE SCALE M. I{. D. GENERATORS 
J. ·c . R. HUNT - UNIVERSITY OF WARWICK, COVENTRY, ENGLAND 
I. INTRODUCTION 
At the present time we.are unabie to carry out a complete analysis 
~f the fliid dynamics and electrodynami~s of a n M. H. D. Generator. 
Howev.er variqus aspects of the behaviour of an M. H. D. Generator 
may be examined by the use of simp li f i ed models: the greater the 
s implific ation the . more complete the description, 
. . 
Laminar incompressible flow in a rectangular duit h as been exami~ed 
in (1] and laminar compressible boundary layer flows in (2] and (3] 
but most generator calculations use ·the method of one-dimensional 
gas dynamics, [4] and [5], which is based upon the assumptions that 
the variations of flow and fluid properties are small across the 
generator duct and that transverse velocities are small. 
Calc~lations based on this approximat ion have pro ved satisfactory 
for invisc id flow regions outside boundary layers, e,i, in the 
entrance region .of a duct or for fully developed turbulent flow 
in a due~. In the former case _ the effects of heat transfer and 
viscous friction are mainly :felt in the boundary layers; they only 
affect the flow outside the boundary layers by varying the thickness 
of the boundary layers. In the latter case, though there are large 
velocity gradie~ts at the wall , the wall effects are diffused right 
across the duct"; The electroma&._net ic effects are considered by 
adding the electromagnetic rx lr'body force · to the momentum equat ion; 
and the work d9.n~ by the :.eleetric field,~- '?-: to - the energy . ; 
equation . When . the Hall effe'c:t ' and ·the problem of segmented : 
electrodes are -Considered it is found that the current varies across : 
the duct. In that case the cur-rent distribution is usually 
calculated by assuming the velocity is constant across the duct, e.g~ 
(9] but ignorin~ the effects on the flow of J x B varying across the 
duct, Thus the~e calculations are valid f~r highly turbulent.flows 
id ducts with small changes in cross-sectional area and where the 
electromagnet i c for_ces ~re small compared to the_ inertial forces, 
In this paper we cons ider another mode l for flow in a M~ H. D. 
generator duct in which we assume the electromagnetic body forces 
are very . much larger than the inertial forces.~ The ratio of these 
two forces is measured by the parameter S ; tr a: J/e U:, where er is 
the conductivity, B0 the ma gn~tic flux density, d.. a characteristic 
duct width, t the density and U a characteristic ve locity, the valu~ 
ofcrbeing assumed to . be _typical for the central regions o~ the duct, 
We also assume the magnetic Reynolds number, Rm, is very much less 
than one, Hence 
·H2 /Re = S >?' l; Rm q.. 1 ; where Re ·is the Reynolds number and H (=Bod .( .r/"I ) 2, where '1. is the viscosity) is the Hartmann number, In the largest experimentaf gener,tors now being built: 
H~l.03,_ Re~ 5 x 105 and hence S~2. 
The most important physical effect of raising Sis the damping of the turbulence _ in the flow. .Murgatroyd (.6) showed that wh e n _Re/H<.225 the .turbulence · in a .flat channel "is damped o·ut; Jloweve.r, for values 
of Re/H greater . than 225 the turbulence is damped in the centre of the duct where the velocity gradients are small and as the magnetic field is further increase d the last remaining turbulence is d amped from the boundary layers •. Thus, since in existing generators the turbulence is highly damped and the electromagnetic forces are of the same order 
or greater than t he iner tia l forces, the basic asswnptions of most 
existing generator calculatious are not satisfied. Therefore, though the results of this model are not inunediate ly · applicable, they should indicate the effects of increasing the magnetic field strength and the [
1
J
1 
I sizi· of M. H, D, gen.erators, as well as demonstrating some of the _ .. defects of exi_sting mode.ls. 
,•:, <, · , .. ,_ ... c.,, 
• !..!. ~ : ~ l". • When S >>l, one can consider the duct to be divided into .2 regio.11.s, .~,: .. ,. ,J.,·. 
, : I ~. -
(i). A core region where electromagnetic forc es are balanced by 
the pressure gradient and where inertial as we ll as visco.us forces · a-re negligible, 
,.,_ ·~ 1 . ~ ... (' J . _ _; .!. ~· 
(ii) Boundary layers on the walls _where again ine rtial forces are 
.-~ : • ' .:" ~ 'j j . • ~ ; : . 
negligible bl.\t where _the vis.cous ,. _electromagneti_c and pressu;i;e_.,.,_,·.1.: 1, 0 -;~ :o, i.< forces are of the same ordeJ',, The thickne_ss of th.e J _ayers, '"' ,-,·~t- ,;\· is assumed to be very much smaller than the duct widt:h, · .' · . .'.·"· 1 ,,;:; 
.- ._, r i. 0-In many exi ·sting M. H. D, generators, the generator ducts. have,:larg.e , 
. changes in their cross-sectional area, but becaµse _ the calculations 
. ' 
are based on one-dimensional ga_s dynamics the effects of ,large ... transverse velocities caused by the are~ change are not considered; . We show how flow in such ducts can be calculated exactly when .S 1. Another problem ·n.ot suitable for solving by one-dimensional gas· dynamics is that of . calculating the effect of variations ' in the. " 
magnetic field both along and across the duct, · whereas in our approxi-
mation a fairly simple method eiists for this c~iculation. · 
·By means of our boundary layer approximations we examine the effects 
of viscous friction and heat transfer at the wall when S increases, We conclude that owing to the damping of the turbulence which occurs 
when S-:>>l and the formation of slowly growing boundary layers the 
effects on the core flow of the wall . shear stress and hea·t transfer become progressively less as S increases. 
· 
/ 
2. _ COllE PLO\f 
_In this s ection we consider the flow in the core and the.refore we 
wi ll ignore viscous forces. Th e governing e quations for th·e · steady 
flow of a conducting gas are taken to be: 
Mome ntum: 
Continuity: 
Energy: 
Ohm's Law: 
Maxwell's 
Equat ions: ,· -~ 
e ( 1t . v) 1t , ... - v P -i' 1 ~ ~ , ( 2. 1) 
-(e·v)t!+(it.v)e O (2.2) 
e(U: . v)(h.+\i!/·)~ E. j+V.(I\.VT), (2.3) 
7~ ($' Ct _+ 11)( a)-(i..,-r/U3\)(f,:s) (2.4) 
... 
v. 8 : 0 -, 
v-,. g__ = 0 
I 
~ -?0-
V x 8 -= fJ 
(2.5) 
(2 .6 ) 
(2.7) 
Where t he symbol·s have th·eir usual meaning. _~ · In this paper we 
consider the magnetic _fie.Id. t _o be produced by external coil_s and we 
assume that the magnetic Reynolds number , Rm is sufficiently small 
for us to ignore the field produc e d -by the induced currents,~ 
Let the imposed· __ fij ld be B0 , _- then in the generator _ duct 
-1: 
(2.8) 
We now co nsider the equations when the magnetic field is .uniform and · 
S >>l. Then, · using the co-ordinate system of Fig.I, equations (2.2) 
~nd (2.3) becomei 
0:. ·- ~p/hc. - Ja Bo, 
oz -op/~~ (2.9) 
0 ~ - °op/~'l + J:.. Bo I 
I 
+ Uj ~ + lL'l i)" E,.J,,+ ~J:,+_E"'lJ'l+ V-(h.V~ (2 .10) 
As is to be expected this approximation ieads to our neglecting the 
inertial- terms in (2.9) and (2.10). Hen~e taking the curl of (~.9) 
we have: / 
= ~':1. = i)J~ = 0 (2.11) 
::, ~ 
We now show by an order ·of magnitude argument . that when S >> , -
and the flow is compressible we can ignore the heat cond u ct ion terms 
in 2.10. From (2.9 ) we see _ that t he characteristic distance in 
which pchanges in the x - direct i on is O (s-1) a nd hence T c hanges 
in a simi l ar distance~ Th e n 
) 
and,. si-nce in most cases Re':>'), S and Pr = 0(1),. we can ignore the heat 
conduction ·terms. Physically this means that in the _core the _hea,t , 
generat~d by ohmic dissipation is cohvected rather th~n ci~duct~d 
away. 
"> ...• . [ .• .)';_ .. /\~ 1 '; t .'i~~·I) 
For the symmetrical gen~r~ior shown i~_-Fig. (1.), ·J~' = 0 on the ·c·e'ntre 
line and hence from (2.11), ~ 
\: - . ' i ~ .' ·""' JJ - ~~ ." .. ) • 
_ J y =· Ey = 0 
throughout the duet. ' ' 'Th'ence, 
; I -· . l.°1 
from (~.}). -
~ E',1. -= °?J1, ~ 'I; \' -r: -a, 0, ·! ~ ~-
_.-.-.-~-
!. 
.i 
;_:.:_.--:,: 
( :! • 7 ; 
~-- . . . . - . -
and Ex, E ·, J x, J~ are alT . fu'.ncti'ons 'of 'x ·and ·: onTy •. ,,~, Then ;· sinc'e :;:,,pc,e w0 
6 
we can ig;ore .the c'onduction ternis in (2'.1o)'we :·have:- ·2J. · :;~; :';"..-:r,,,:1 _,, :i_i l: ,,_,,, 
· • • · • • · - . ,; · : · ·• · ""' ..., ; -. • 1 l~ .:'; ' • i - ·--. - -; '. · "' · / 
i.) c e ( lL)(._°h. + u1 ~ +. u.i! ~) ]·-'· -=- c/~ .. T ••• - :: , :c2n1;-i ,~:,.~:;~;-, t' · · ~ l. . , ,;x .· _ c i .. _ .:.., ··, ,- ,::0 :,::, ,., ',, l .luct • -
Since Jr/h. = o (S) '>> }T/J:, and by considering 'i>J /~ we . can formally . 
show that _ the solution: 1 · .. ' ~ ,j I:. ··' ' 
-~Th ~ "" · ~ u~ /)~ = i>Ua 0:, - >.e)J'I =- 0 ,. (2.'13) , -: • ,, : : , 
. f. ~-- . ' . ~ ~ . 
satisfies all · the equations inc"luding (2.12). Also -we can construct 
a ~olution to fit the bounda_ry coudi'tior:is. whi_ch satisfies (2.13)~ , 
The comi'equence .of the 1 imi ting solution ( 2 .13) is that all fluid 
_properties as well as the , · current and electric field 
become functions of x and s only. In an incompressible flow this 
would not occur whatever the value of . S, because there is no coupling 
between the velocity and temperature, but in a compressible flow the 
velocity and fluid properti es are coupled so that when S >> 1 and do = 
',)j"'~::, = 0 the coupling necessitates that all ·t_l,_e other propertie~are 
also uniform in they- direction. 
--;r- · .... 
If now we c~·nsid~r the 'flow in a ~egmented M, ·H. D. generator in whi~h_ 
the electrodes are connected so that J,.,-= 0 , then ~p/~l = O and we can · 
show by an or_der of magni fude argwnent in considering the boundary 
layers (see 3) that, if the duct width in the~- direction is 
. constant, lli! << ·IJ.~ in the core. Then ~T/)1- = O and eqilations (2.9) -(2.10) may be .written: · '-
dp/h:. = -o-· u.,,., B.i. (1-k) , 
U o~/h : - ·er, (L.,..'l.. 8.,t. k {{-k) I e )< . • 
(2.14) 
! 
I I 
fi· 
7 where .K .; (- Es/Ux B0 ). Since Ux and e do not vary in the y or s -
, direction; Ux ·e A ; constant. (2.16) 
where A (x) _is the cross - sectional area. Thus we have arrived at 
a sei of equations with a one - dimensiona l form, yet we can satisfy 
the boundary conditions . on Uy in a dur.t with large variation of the . 
duct .11•idth~for Uy may be calculated fr_om (2,2) once Ux · and e are 
r°ound . from (2,14) - 2.i6). Si.nee Ux · satisfies ( 2.16), it i ,s . 
easi ly seen that Uy satisfies the condit io,1 that: 
. U-~ / (.(..._ : <if / tk_ Wh.~" ·:1 ~ tl><) ·. lS<>e f.) · i). 
The pr~pertie~ of this set of equations has not _yet been studied 
bu°t some observations can be made at this time.f 
Firstly, . if 1 Ux is. c·anstant acro .ss the duct, the total velocity, (ux2 + uy2 )2 is clearly greatest near the · walls in a diverging duct. 
This implies an increased drag on the · wal l s and reduced tendency _to 
separate. · 
Secondly, these ~iua~ions become invalid where large gradients of 
velocity oc.cur such as near . shocks or near the ·exit of a duct. 
Thirdly, if ixfo;· large transverse pressure gradients exist and 
hence l arge chang~s i _n fluid properties across the duct. Then it 
is not possible to calculate the current distribution near segmented 
electrodes in the manner of Wi t -alis (9J without considering the · 
changes in velocity, conductivity and Hall effect across the .duct. -~ 
To illustrate thi§, - let us writejx ;·dflk_,iy ; 0, j
0
_=s ~ ~'t'/~::x. . 
Then there is a simple integral to equations (2 .9 a and 2.9 c),l-~. 
ri-Bi,'/'; constant • .-~.- Thus the pressure is constant along the current 
.lines and large c~anges in ~luid properties across the duct will 
o·ccur. 
*• . Fourthly, . the variables . in these equatio~s only depend on boundary 
conditions at values of x ~ . The conditions in the ' walls at 
y; !f (x) affect the 6urrent in· the boundary l aye~s which in turn 
affects the current in the core. Thu.s we d.o not ·have a direct 
interaction between the wal ls · and the cor·e flow as in ordinary one-
dimensiona l gas dynamics but an :indirect electromagne.tic interact ion . 
Fifthly, we c•n consider a ll the variab les in the equations as the 
_ first · term of an expansion in . decreasing powers ·of S1 where S1 is 
the value of Sat the entrance, e,g. 
Knowing Uxa from our ser9th approximation we can calculate the first 
approx imatio n and thus we can see the e.ffect of the inertial terms . 
This approach has _been u sed by Hunt and Leibovitch [8] to examine 
in compressible flow and found to.be much simpler than expanding in 
ascending powers of S. 
~ L.~. ("2..1"- 2.16). 
t l-,v,J~fi"' Jr tt ~~ tv) r~ +~ J~ 
lt;t --J- ~ (;)'._ s .. ~,c1. w l..a.v.__ l'-12-<< , . 
-~ 
r·.-
L 
,"II 
I h 
I 
I 
L~stly, it is of interest to note that equations (2.14 - 2,16) are 
exactly the · same as those for . a frictionless constant velocity duct (r]. ·The differences ir: this approac h and that of one-dimensional 
gas dynamics will app~ar most marked when the hiiher order approxi~ 
mations, · as discussed .in t:JP. preceding paragraph, are considered 
As explsined in§ l, our aim in thi~ se~tion is to consider the 
boundary layers on the walls perpendicular to the magnetic fields 
wheu S >>l. We assume the walls to be electrically insulating and 
then - the approximate equations writ ten in terms of the bound;fry 
layer co - ordinate.s s, n and i5 are: 
0 :. op/os . - J;J Ro~"'- + 'o!an [71 cl~/&M.), 
°op h "' + Ji! B,, .su. o(. 
0 "' - ?>p/?J~ + J~ fso C.~ o( +- "o/c)l'\. (.,, ~Ll,i!/ol'I.), . 
e{Us oh/~+_ U,, Jt../~n + Lt~ ?>l.(o~)-: d/on (R ~T/o") + 
)f'~.,.._(,~jj,.(U./·-tu.:·+11-;)/1.) + J,£,. -+-J1:£''t · 
! 
a) 
) 
(3.1) 
b) (3,2) 
. ) _ .. _ 
c) ., .. 
Js =. I+ %1: c.O;I~)'- r Er.s + <-<l't" ~ ... ~ t + Bo etnoi.(-U.z.-t ~t' ~~IJ~)°J,e3 :4 )~ 
)r-:: 1+[,i-eo,~)Ltc:~ - l.>'C ~IC'.£'s~-+ &,~~cu~~ W't'lD"lO(U~) f ~' ( · 
t.Jh.e~ +M-1~ : J.f/J..cc_. (f.·3 . .:t) • . , 3.5) 
For typical generator loading ~·ond i tions K · =-Ei5/UB . ""' a/1 , and 
then the current density in t hese l ayers is very muc.h greater than 
that in the core. ~owever , the conductivity is usually low niar .· 
the wall and si nc e the velocity gradients there are large th~ l~vel 
of t~rbulence will b~ gr~ater in these layers than in th~ core and 
ou:r approximat!.;in less valid. When H»l an d S::vi. the boundary layer 
thickness; S, is O {H- 1) as may br. seen;,. (3.2 a), For the rea'Sons 
discussed iri} 2 the velocity anrl temperature variations are coupled 
and consequently the velocity . and temperature boundary layer ar~ 
identical. 
If the wall is kept at a constant temperature while the core is at Tc 
we can sho~ that in (3.3) th e convective terms are very much smaller 
than the conducting . tert11s parti cul ctrly when ( 1 - K)« 1. In the core 
the convective terms are of the same order as the"tE term and m~ch · 
grJater tha~ the_conductio~ terms, but in the bounrlary ~ay~rsJ-E is . 
o(~K - 1 )) times 1 ts vi:.l ~~ _1n: the co re and the viscous d1" s1pa t1on term 
is O (i Us{cor e) H2/a2) which is O (1/K(K-l) times the _value of "7 E 
in the core. Thus in t he boundary layers, since the convective terms 
are of the same order as in the core, most of the extra heat dis s ipat-
ion is too great for c·onvection a nd i" c.:ondu.cted away through the walls, 
,,. 
1' -; .. 
.•• ; j l 
11 
IC 
9 
.to 
., 
'- · ,-
"' 
If there is no heat loss through the walls, the ras in the boundary 
layer hea ts up as it _progresses'. th~ he~t ?eing_convected alon'.);, 
Clearly in this case our approximation is invalid • 
. -" The approximate equations 'liay now be wr itten in the following way, 
using the values of pressure gradient in the core and igno~ing terms I'll 
of O (S), 
~(ells)+ "c)~(eu .. ) + t)~(elL-=) = o., (3.1) 
'.3/dl\. C 11 )u.sh") - 0~ - j~ c.ou. )so = o., (3. 6) 
o/dYI. ( "'] ou;~f\) -t- (J.s -.k co..e) Bo = 0, (3:+) 
:is ~ + J~ Fr .,.-¥wi h-b ( 11.x,1. ;- IJ.,/)/2.) + ~/Jn{ k °;)Tfdn)= O ( 3. s) 
, oE:s /t)t\ = "H:~ 1J ri = o ., (3. g) 
wh~'i.e . thii suffiX: . •core.·1 refers to . the value of the variable in the 
core·, We can -deduce simple re.suits by integrating (3.5) and (3',7) 
with respect · to . n from n = 0 to n =O, We obtain 
! , . '" ' 
,. ;; ;80 ,:_~/i1:-_/l-'°.N..__) di\:. -/J ~u.s/Jnt .. 0 : -7:,,, { (3,_io) 
, ~8., fo ( ,&_- J~eo,c ) dn f i dU.r~n/ ncD = -C- 'l: J 
wl:ieri 'ts / 't~ are the components of' ski~ friction in s and i, directions, 
Wli.eu K~l, j 2 · and j.s become very much greater than their value in the 
. c·ore. In .. this special case · . · . . .· ' 
-. . f6 . J . r' d \~ ~TI - Gl 
.. E.._$ ,- o .JS Q.t'\ + EE- Jo Jil P\: h\ t'\•O - J 
wheri . Q is the heat leaving the walls, From (3,10)
1 
' 
Thus we h;ve a relation b e tween 'i" a~d Q which is independent of th.e 
fluid properties but which is only valid when the generator is almost 
on open ~circu"i t. 
Let us now consider the. special case of J;ic.~O in the core in order to 
find the order of magnitude of the swirl velocity, u~, induced by the , 
Hall effe_ct. We assume Us>> u~ ·in -the core and then s h ow that this 
is true, From(3.4)Jin the core: . . . - ( ) 
. . .. ( B II' . . ) ,..., - W'?:' G~ .,_c(. ll.s Bo l- k , E& ~ - "-' 7: lOS o<. Eqi:-,. o I.ls Ge::.,/.. - . . . 
Hence in the boundary layer, . where Us is small n ear the wall: 
1 N " .0: (- U.s Bo w 't' ~ ... cl - ll-... Bo u,t)O(.) ,- ~ ~~ . c 
I +(w7; '-'" r,,i . . • · 
Sub stitutin g this a ppro ximate value for Jg . in (3_., +) we have: 
... • 
[
tr I.ls C.:,~ °B;- Wt:'. (.C)Sl.ol 01.J : 
Ui!! =-0 I ·1- (i..,'t' c.os a(.\'- - . 
since S is O (fC1 ) and wt: is assumed to be O (1 ). 
Thus the swirl velocity is of the same order as the core velocity 
in the boundary layer. But an equal quantity of ga~ must return in 
the opposite direction in the core and since the flow properties do 
not vary in they - direction, we have: 
ll~ ::.-0 -. [ 6 
. C.o.e. o.. 
Thus we can say that the swirl velocity U,1 is small in t _he core when 
jx = 0, Since U,1 =-0 (Us) in the boundary layer it means that in 
this case't,1 =-0 ('t's)• 
-We now estimate the order of magnitude. of 1: s and thence examine the 
effect on the core flow of the boundary layers. 
'ts : 0 [ 7lw U.s C.0'4. /s] .< o [ 11t.o>e Lls~ J 
where~~ is the value bf viicosity at the wall and if the wall is 
cooled j < 11 • · Hence 
. "" /C,o,e_ 
1:5 ;$ O · [ U."' '°~ 80 (sr 'l )'l•J / 
where er and 7) are taken at their core values, 
current leaving the generator is I, by adding 
boundary layers to that in the core we have: 
I: er U.xc.,,4. Bo(l-k) -'ts/Bo ~ 
/ 
Then if the total 
the current in the 
Thus it . is only when (1-K) !::!.. (irl) that the effect of the wa lls on the 
core flow is appreciable, Therefore, whatever the roughnes s or the 
heat withdrawn from the· walls perpe ndicular to the field when (1-K) 
>>H-1 the core flow is unaffected. This is an important re s ult as 
it shows that even if the walls are very roug~ as_ is oft_en the case 
in M. H. D. generators, _ the flow in the core will not be greatly 
affected. Thus t~e situation is very different from a norma l 
compressible duct flow where turbulence distributes th~ she ar 
stresses and the heat input at the wall ~ver the duc t Gn~these 
effects have to be considered in the core, 
4 • . NON-UNIFORM MAGNETIC FIELDS 
-A~ a simpl e e~ample of the ~ethod for examining the effect of non -
unifo~m magnetic fields .we conside r the flow · in "the core of a 
constant area duct wh e n th e re is no ilall effe;t and the magn e tic 
10 
f::· l·l; 
I. 
I 
11 field, Bo, only varies in the y and ·~ directions . Since B0 
satisfies (2.9) when Rm»l.,we can write the components of Bo as: 
B.,._=O, ~:.-~, Si!:c ~~ (4,1)· 
where A satisfies, · V1. A "" O, 
and the momentum equatio ns for our special case as: 
o .• _ ?Jph-.x:. -(J.,, 8; - J'O Bi ) 
0 ..,, -op/?J-a ... - op/~~ 
Thence (3.3) becomes: 
where J:a."" o, Jij: -}lf'fe~, J~c }ipfi~ 
1 
· o.,,.d. p.,. "cp,!b~. 
The general solution to (3,5) may be written as 
(4.2) 
( 4. a-) 
(4.4) 
(4.5) 
. . . -t-P d.1;1 . 
'f' = J(tl) 1. ("b ~ /H) 1 (4.6) 
where the integral is a line integral taken along a 1line of constant A, i,e,.a field line; In the case of a rectangular duct in which · 
the magnetic field is symmetric about the centre line in both 
directions, the current distribution is symmetric and 'r' = ·o at y 
Then (3 .• 6)becomes: _ ' o. 
P Yr A~ 
oJ<.A i:onst.:..t} O> A- /lr ) (4,7) 
:_ As · a ve.ry siJDple- exampl e consider 
' they directi~n which has a small 
directions.' Then: 
1 
a mainly uniform ma g~etic f ie ld in 
parabolic variatiori in the t and~ 
I 
A [fr0 i: -+ fs., ; 3 - lr. e ';jl.J I 
· where lr0 _>?> lr, . The , s~lution for 'f to first order in ~ /lr0 is: 
f~ t: [ I- 6- r2. ~ + ;t] I: 
Thence Ji - ~[ I- ~ 'i:2. + 6- l. J = ~ ~.. 
and J~ :. i_p(,. ~ ~ &- 2.. 
0 
This method can be extended to more complicated situai ions, Als~ it 
enables us to find the . velocity distribution in the core, though th e 
boundary layer structure the n has to be taken into acco unt, 
' 
-· 
c-,· 
., 
\' 't 
!, 
\ .. 1; 
.:· ( 
-REFERENCES 
( 1) Hunt, J. C.R. & Stewartson, K. 1965, J, Fluid_ Mech, 23, 563 
(2) ·Kerrebrock, J. L. 1961, J. Aero/Space Sci. 28·, 631 
(3) Hal_e, F. J, & Kerrebrock J, L. 1964. AIAA _g_, _461 
· (4) Swift-Hook, D, T, & Wrig1:t· J:. K. 1963. J, Fluid Mech • .!.§_, 97 
,(5) Louis, J. F. Etal. 1964, Phys. Fluids 2, 362 
(6) Murgatroyd, W, 1953, Phil. Mag, 44, 1348 
(7) Carter; C. etal, 1964, MHD Confere nce, Paris, "Paper 85 
(8) Hunt ,' J, c. -R. & Leillovitch, S. 1966 
submitted for publicition to J. Fluid Me ch, 
(9) Witalis, E.A. {965 ·. J; Nuc. _Energy. C. 2, 235. 
J . 
' -. / • 1 
; ; .... 
t~ . 
0---lp ,~. --o~u:. i! ,%. . . . • ' . ' :,:.. 
i 
....... :.. / i ( ·-,' . '' ,' 
r:. 
Fig. 1. Notation for the analysis of the core and boundary 
·1ayer flows. 
I f 
r~ ' _.,. . 
12 
-..:,_l I: 
'.,L -