dc.contributor.author Pesci, AI dc.contributor.author Goldstein, RE dc.contributor.author Shelley, MJ dc.date.accessioned 2018-09-08T06:30:49Z dc.date.available 2018-09-08T06:30:49Z dc.date.issued 2020 dc.identifier.issn 0010-3640 dc.identifier.uri https://www.repository.cam.ac.uk/handle/1810/279800 dc.description.abstract A classical problem in fluid mechanics is the motion of an axisymmetric vortex sheet evolving under the action of surface tension, surrounded by an inviscid fluid. Lagrangian descriptions of these dynamics are well-known, involving complex nonlocal expressions for the radial and longitudinal velocities in terms of elliptic integrals. Here we use these prior results to arrive at a remarkably compact and exact Eulerian evolution equation for the sheet radius $r(z,t)$ in an explicit flux form associated with the conservation of enclosed volume. The flux appears as an integral involving the pairwise mutual induction formula for vortex loop pairs first derived by Helmholtz and Maxwell. We show how the well-known linear stability results for cylindrical vortex sheets in the presence of surface tension and streaming flows [A.M. Sterling and C.A. Sleicher, $J.~Fluid~Mech.$ ${\bf 68}$, 477 (1975)] can be obtained directly from this formulation. Furthermore, the inviscid limit of the empirical model of Eggers and Dupont [$J.~Fluid~Mech.$ $\textbf{262}$ 205 (1994); $SIAM~J.~Appl.~Math.$ ${\bf 60}$, 1997 (2000)], which has served as the basis for understanding singularity formation in droplet pinchoff, is derived within the present formalism as the leading order term in an asymptotic analysis for long slender axisymmetric vortex sheets, and should provide the starting point for a rigorous analysis of singularity formation. dc.description.sponsorship This work was supported in part by Established Career Fellowship EP/M017982/1 from the EPSRC (REG & AIP). REG and AIP are grateful to the I.H.E.S., and especially Patrick Gourdon, for hospitality during an extended visit supported by the Schlumberger Visiting Professorship (REG). dc.publisher Wiley dc.title A Compact Eulerian Representation of Axisymmetric Inviscid Vortex Sheet Dynamics dc.type Article prism.endingPage 256 prism.issueIdentifier 2 prism.publicationDate 2020 prism.publicationName Communications on Pure and Applied Mathematics prism.startingPage 239 prism.volume 73 dc.identifier.doi 10.17863/CAM.27170 dcterms.dateAccepted 2018-03-26 rioxxterms.versionofrecord 10.1002/cpa.21879 rioxxterms.licenseref.uri http://www.rioxx.net/licenses/all-rights-reserved rioxxterms.licenseref.startdate 2020-02-01 dc.contributor.orcid Goldstein, Raymond [0000-0003-2645-0598] dc.identifier.eissn 1097-0312 rioxxterms.type Journal Article/Review pubs.funder-project-id Engineering and Physical Sciences Research Council (EP/M017982/1) pubs.funder-project-id Engineering and Physical Sciences Research Council (EP/I036060/1) cam.issuedOnline 2019-12-17 rioxxterms.freetoread.startdate 2019-09-06
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