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The boundary integral formulation of Stokes flows includes slender-body theory

Published online by Cambridge University Press:  02 July 2018

Lyndon Koens Open the ORCID record for Lyndon Koens [Opens in a new window]  and
Eric Lauga
Lyndon Koens
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, UK
Eric Lauga*
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, UK
*
†Email address for correspondence: e.lauga@damtp.cam.ac.uk
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Abstract

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.

JFM classification

Low-Reynolds-number flows: Boundary integral methods Low-Reynolds-number flows: Low-Reynolds-number flows Low-Reynolds-number flows: Slender-body theory
Type
JFM Rapids
Information
Journal of Fluid Mechanics , Volume 850 , 10 September 2018 , R1
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Copyright
© 2018 Cambridge University Press 

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References

Barta, E. & Liron, N. 1988 Slender body interactions for low Reynolds numbers. Part I: body–wall interactions. SIAM J. Appl. Maths 48, 992–1008.Google Scholar
Batchelor, G. K. 1970 Slender-body theory for particles of arbitrary cross-section in Stokes flow. J. Fluid Mech. 44, 419–440.Google Scholar
Chwang, A. T. & Wu, T. Y. 1975 Hydromechanics of low-Reynolds-number flow. Part 2. Singularity method for Stokes flows. J. Fluid Mech. 67, 787–815.Google Scholar
Clarke, N. S. 1972 The force distribution on a slender twisted particle in a Stokes flow. J. Fluid Mech. 52, 781–793.Google Scholar
Cox, R. G. 1970 The motion of long slender bodies in a viscous fluid. Part 1. General theory. J. Fluid Mech. 44, 791–810.Google Scholar
Götz, T.2000 Interactions of fibers and flow: asymptotics, theory and numerics. PhD thesis, University of Kaiserslautern, Kaiserslautern, Germany.Google Scholar
Gradshteyn, I. S., Ryzhik, I. M., Jeffrey, A. & Zwillinger, D. 2000 Table of Integrals, Series, and Products. Academic Press.Google Scholar
Guo, H., Fauci, L., Shelley, M. & Kanso, E. 2018 Bistability in the synchronization of actuated microfilaments. J. Fluid Mech. 836, 304–323.Google Scholar
Hinch, E. J. 1991 Perturbation Methods. Cambridge University Press.Google Scholar
Johnson, R. E. 1979 An improved slender-body theory for Stokes flow. J. Fluid Mech. 99, 411–431.Google Scholar
Keller, J. B. & Rubinow, S. I. 1976 Slender-body theory for slow viscous flow. J. Fluid Mech. 75, 705–714.Google Scholar
Kim, S. & Karrila, S. J. 2005 Microhydrodynamics: Principles and Selected Applications. Courier Corporation.Google Scholar
Koens, L. & Lauga, E. 2014 The passive diffusion of Leptospira interrogans. Phys. Biol. 11, 066008.Google Scholar
Koens, L. & Lauga, E. 2016 Slender-ribbon theory. Phys. Fluids 28, 013101.Google Scholar
Koens, L. & Lauga, E. 2017 Analytical solutions to slender-ribbon theory. Phys. Rev. Fluids 2, 084101.Google Scholar
Lighthill, J. 1976 Flagellar hydrodynamics: the John von Neumann lecture, 1975. SIAM Rev. 18, 161–230.Google Scholar
Myerscough, M. & Swan, M. 1989 A model for swimming unipolar spirilla. J. Theor. Biol. 139, 201–218.Google Scholar
Pozrikidis, C. 1992 Boundary Integral and Singularity Methods for Linearized Viscous Flow. Cambridge University Press.Google Scholar
Sellier, A. 1999 Stokes flow past a slender particle. Proc. R. Soc. Lond. A 455, 2975–3002.Google Scholar
Smith, D. J., Gaffney, E. A., Blake, J. R. & Kirkman-Brown, J. C. 2009 Human sperm accumulation near surfaces: a simulation study. J. Fluid Mech. 621, 289–320.Google Scholar
Tornberg, A. & Gustavsson, K. 2006 A numerical method for simulations of rigid fiber suspensions. J. Comput. Phys. 215, 172–196.Google Scholar
Tornberg, A. & Shelley, M. J. 2004 Simulating the dynamics and interactions of flexible fibers in Stokes flows. J. Comput. Phys. 196, 8–40.Google Scholar