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A regularised slender-body theory of non-uniform filaments

Published online by Cambridge University Press:  14 July 2020

B. J. Walker*
Wolfson Centre for Mathematical Biology, Mathematical Institute, University of Oxford, OxfordOX2 6GG, UK
M. P. Curtis
Wolfson Centre for Mathematical Biology, Mathematical Institute, University of Oxford, OxfordOX2 6GG, UK Hampton School, Hanworth Road, Hampton, MiddlesexTW12 3HD, UK
K. Ishimoto
Research Institute for Mathematical Sciences, Kyoto University, Kyoto606-8502, Japan
E. A. Gaffney
Wolfson Centre for Mathematical Biology, Mathematical Institute, University of Oxford, OxfordOX2 6GG, UK
Email address for correspondence:


Resolving the detailed hydrodynamics of a slender body immersed in highly viscous Newtonian fluid has been the subject of extensive research, applicable to a broad range of biological and physical scenarios. In this work, we expand upon classical theories developed over the past fifty years, deriving an algebraically accurate slender-body theory that may be applied to a wide variety of body shapes, ranging from biologically inspired tapering flagella to highly oscillatory body geometries with only weak constraints, most significantly requiring that cross-sections be circular. Inspired by well known analytic results for the flow around a prolate ellipsoid, we pose an ansatz for the velocity field in terms of a regular integral of regularised Stokes-flow singularities with prescribed, spatially varying regularisation parameters. A detailed asymptotic analysis is presented, seeking a uniformly valid expansion of the ansatz integral, accurate at leading algebraic order in the geometry aspect ratio, to enforce no-slip boundary conditions and thus analytically justify the slender-body theory developed in this framework. The regularisation within the ansatz additionally affords significant computational simplicity for the subsequent slender-body theory, with no specialised quadrature or numerical techniques required to evaluate the regular integral. Furthermore, in the special case of slender bodies with a straight centreline in uniform flow, we derive a slender-body theory that is particularly straightforward via use of the analytic solution for a prolate ellipsoid. We evidence the validity of our simple theory with explicit numerical examples for a wide variety of slender bodies, and highlight a potential robustness of our methodology beyond its rigorously justified scope.

JFM Papers
© The Author(s), 2020. Published by Cambridge University Press

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