The boundary integral formulation of Stokes flows includes slender-body theory
Journal of Fluid Mechanics
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Koens, L. M., & Lauga, E. L. (2018). The boundary integral formulation of Stokes flows includes slender-body theory. Journal of Fluid Mechanics, 850 R11-R112. https://doi.org/10.1017/jfm.2018.483
© 2018 Cambridge University Press. The incompressible Stokes equations can classically be recast in a boundary integral (BI) representation, which provides a general method to solve low-Reynolds-number problems analytically and computationally. Alternatively, one can solve the Stokes equations by using an appropriate distribution of flow singularities of the right strength within the boundary, a method that is particularly useful to describe the dynamics of long slender objects for which the numerical implementation of the BI representation becomes cumbersome. While the BI approach is a mathematical consequence of the Stokes equations, the singularity method involves making judicious guesses that can only be justified a posteriori. In this paper, we use matched asymptotic expansions to derive an algebraically accurate slender-body theory directly from the BI representation able to handle arbitrary surface velocities and surface tractions. This expansion procedure leads to sets of uncoupled linear equations and to a single one-dimensional integral equation identical to that derived by Keller & Rubinow (J. Fluid Mech., vol. 75, 1976, p. 705) and Johnson (J. Fluid Mech., vol. 99, 1979, p. 411) using the singularity method. Hence, we show that it is a mathematical consequence of the BI approach that the leading-order flow around a slender body can be represented using a distribution of singularities along its centreline. Furthermore, when derived from either the single-layer or the double-layer modified BI representation, general slender solutions are only possible in certain types of flow, in accordance with the limitations of these representations.
ECH2020 EUROPEAN RESEARCH COUNCIL (ERC) (682754)
External DOI: https://doi.org/10.1017/jfm.2018.483
This record's URL: https://www.repository.cam.ac.uk/handle/1810/285447