Hostname: page-component-586b7cd67f-t7czq Total loading time: 0 Render date: 2024-11-25T22:28:50.375Z Has data issue: false hasContentIssue false
  • English
  • Français

Regularised non-uniform segments and efficient no-slip elastohydrodynamics

Published online by Cambridge University Press:  15 March 2021

B.J. Walker Open the ORCID record for B.J. Walker [Opens in a new window]  and
E.A. Gaffney
B.J. Walker*
Affiliation:
Wolfson Centre for Mathematical Biology, Mathematical Institute, University of Oxford, OxfordOX2 6GG, UK
E.A. Gaffney
Affiliation:
Wolfson Centre for Mathematical Biology, Mathematical Institute, University of Oxford, OxfordOX2 6GG, UK
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

The elastohydrodynamics of slender bodies in a viscous fluid have long been the source of theoretical investigation, being pertinent to the microscale world of ciliates and flagellates as well as to biological and engineered active matter more generally. Although recent works have overcome the severe numerical stiffness typically associated with slender elastohydrodynamics, employing both local and non-local couplings to the surrounding fluid, there is no framework of comparable efficiency that rigorously justifies its hydrodynamic accuracy. In this study, we combine developments in filament elastohydrodynamics with a recent regularised slender-body theory, affording algebraic asymptotic accuracy to the commonly imposed no-slip condition on the surface of a slender filament of potentially non-uniform cross-sectional radius. Further, we do this whilst retaining the remarkable practical efficiency of contemporary elastohydrodynamic approaches, having drawn inspiration from the method of regularised Stokeslet segments to yield an efficient and flexible slender-body theory of regularised non-uniform segments.

JFM classification

Mathematical Foundations: Computational methods Low-Reynolds-number flows: Slender-body theory
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 915 , 25 May 2021 , A51
Check for updates
Copyright
© The Author(s), 2021. Published by Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Ainley, J., Durkin, S., Embid, R., Boindala, P. & Cortez, R. 2008 The method of images for regularized Stokeslets. J. Comput. Phys. 227 (9), 4600–4616.CrossRefGoogle Scholar
Chwang, A.T. & Wu, T.Y.-T. 1975 Hydromechanics of low-Reynolds-number flow. Part 2. Singularity method for Stokes flows. J. Fluid Mech. 67 (4), 787–815.CrossRefGoogle Scholar
Cortez, R. 2001 The method of regularized Stokeslets. SIAM J. Sci. Comput. 23 (4), 1204–1225.CrossRefGoogle Scholar
Cortez, R. 2018 Regularized Stokeslet segments. J. Comput. Phys. 375, 783–796.CrossRefGoogle Scholar
Cortez, R. & Nicholas, M. 2012 Slender body theory for Stokes flows with regularized forces. Commun. Appl. Maths Comput. Sci. 7 (1), 33–62.CrossRefGoogle Scholar
Cox, R.G. 1970 The motion of long slender bodies in a viscous fluid Part 1. General theory. J. Fluid Mech. 44 (04), 791–810.CrossRefGoogle Scholar
Curtis, M.P., Kirkman-Brown, J.C., Connolly, T.J. & Gaffney, E.A. 2012 Modelling a tethered mammalian sperm cell undergoing hyperactivation. J. Theor. Biol. 309, 1–10.CrossRefGoogle ScholarPubMed
Gillies, E.A., Cannon, R.M., Green, R.B. & Pacey, A.A. 2009 Hydrodynamic propulsion of human sperm. J. Fluid Mech. 625, 445–474.CrossRefGoogle Scholar
Gray, J. 1928 Ciliary Movement. Cambridge University Press.Google Scholar
Gray, J. & Hancock, G.J. 1955 The propulsion of sea-urchin spermatozoa. J. Exp. Biol. 32 (4), 802–814.Google Scholar
Guglielmini, L., Kushwaha, A., Shaqfeh, E.S.G. & Stone, H.A. 2012 Buckling transitions of an elastic filament in a viscous stagnation point flow. Phys. Fluids 24 (12), 123601.CrossRefGoogle Scholar
Hall-McNair, A.L., Montenegro-Johnson, T.D., Gadêlha, H., Smith, D.J. & Gallagher, M.T. 2019 Efficient implementation of elastohydrodynamics via integral operators. Phys. Rev. Fluids 4 (11), 1–24.CrossRefGoogle Scholar
Hancock, G.J. 1953 The self-propulsion of microscopic organisms through liquids. Proc. R. Soc. Lond. A 217 (1128), 96–121.Google Scholar
Ishimoto, K. & Gaffney, E.A. 2018 An elastohydrodynamical simulation study of filament and spermatozoan swimming driven by internal couples. IMA J. Appl. Maths 83 (4), 655–679.CrossRefGoogle Scholar
Johnson, R.E. 1980 An improved slender-body theory for Stokes flow. J. Fluid Mech. 99 (2), 411–431.CrossRefGoogle Scholar
Keller, J.B. & Rubinow, S.I. 1976 Slender-body theory for slow viscous flow. J. Fluid Mech. 75 (4), 705–714.CrossRefGoogle Scholar
Lighthill, J. 1976 Flagellar hydrodynamics. SIAM Rev. 18 (2), 161–230.CrossRefGoogle Scholar
Moreau, C., Giraldi, L. & Gadêlha, H. 2018 The asymptotic coarse-graining formulation of slender-rods, bio-filaments and flagella. J. R. Soc. Interface 15 (144), 20180235.CrossRefGoogle ScholarPubMed
Neal, C.V., Hall-McNair, A.L., Kirkman-Brown, J., Smith, D.J. & Gallagher, M.T. 2020 Doing more with less: the flagellar end piece enhances the propulsive effectiveness of human spermatozoa. Phys. Rev. Fluids 5 (7), 073101.CrossRefGoogle Scholar
Olson, S.D., Lim, S. & Cortez, R. 2013 Modeling the dynamics of an elastic rod with intrinsic curvature and twist using a regularized Stokes formulation. J. Comput. Phys. 238, 169–187.CrossRefGoogle Scholar
Pozrikidis, C. 1992 Boundary Integral and Singularity Methods for Linearized Viscous Flow. Cambridge University Press.CrossRefGoogle Scholar
Pozrikidis, C. 2010 Shear flow over cylindrical rods attached to a substrate. J. Fluids Struct. 26 (3), 393–405.CrossRefGoogle Scholar
Roper, M., Dreyfus, R., Baudry, J., Fermigier, M., Bibette, J. & Stone, H.A. 2006 On the dynamics of magnetically driven elastic filaments. J. Fluid Mech. 554, 167–190.CrossRefGoogle Scholar
du Roure, O., Lindner, A., Nazockdast, E.N. & Shelley, M.J. 2019 Dynamics of flexible fibers in viscous flows and fluids. Annu. Rev. Fluid Mech. 51 (1), 539–572.CrossRefGoogle Scholar
Schoeller, S.F. & Keaveny, E.E. 2018 From flagellar undulations to collective motion: predicting the dynamics of sperm suspensions. J. R. Soc. Interface 15 (140), 20170834.CrossRefGoogle ScholarPubMed
Shampine, L.F. & Reichelt, M.W. 1997 The MATLAB ODE suite. SIAM J. Sci. Comput. 18 (1), 1–22.CrossRefGoogle Scholar
Simons, J., Fauci, L. & Cortez, R. 2015 A fully three-dimensional model of the interaction of driven elastic filaments in a Stokes flow with applications to sperm motility. J. Biomech. 48 (9), 1639–1651.CrossRefGoogle Scholar
Smith, D.J. 2009 A boundary element regularized Stokeslet method applied to cilia- and flagella-driven flow. Proc. R. Soc. Lond. A 465 (2112), 3605–3626.Google Scholar
Smith, D.J., Montenegro-Johnson, T.D. & Lopes, S.S. 2019 Symmetry-breaking cilia-driven flow in embryogenesis. Annu. Rev. Fluid Mech. 51 (1), 105–128.CrossRefGoogle Scholar
Tornberg, A.K. & Shelley, M.J. 2004 Simulating the dynamics and interactions of flexible fibers in Stokes flows. J. Comput. Phys. 196 (1), 8–40.CrossRefGoogle Scholar
Walker, B.J., Curtis, M.P., Ishimoto, K. & Gaffney, E.A. 2020 a A regularised slender-body theory of non-uniform filaments. J. Fluid Mech. 899, A3.CrossRefGoogle Scholar
Walker, B.J., Ishimoto, K. & Gaffney, E.A. 2020 b Efficient simulation of filament elastohydrodynamics in three dimensions. Phys. Rev. Fluids 5 (12), 123103.CrossRefGoogle Scholar
Walker, B.J., Ishimoto, K., Gadêlha, H. & Gaffney, E.A. 2019 Filament mechanics in a half-space via regularised Stokeslet segments. J. Fluid Mech. 879, 808–833.CrossRefGoogle Scholar