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Boundary element methods for particles and microswimmers in a linear viscoelastic fluid

Published online by Cambridge University Press:  13 October 2017

Kenta Ishimoto Open the ORCID record for Kenta Ishimoto [Opens in a new window]  and
Eamonn A. Gaffney
Kenta Ishimoto*
Affiliation:
The Hakubi Center for Advanced Research, Kyoto University, Kyoto 606-8501, Japan Research Institute for Mathematical Sciences, Kyoto University, Kyoto 606-8502, Japan Wolfson Centre for Mathematical Biology, Mathematical Institute, University of Oxford, Oxford OX2 6GG, UK
Eamonn A. Gaffney
Affiliation:
Wolfson Centre for Mathematical Biology, Mathematical Institute, University of Oxford, Oxford OX2 6GG, UK
*
†Email address for correspondence: ishimoto@kurims.kyoto-u.ac.jp
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Abstract

The consideration of viscoelasticity within fluid dynamical boundary element methods has traditionally required meshing over the whole flow domain. In turn, a major advantage of the boundary element method is lost, namely the need to consider only surface boundary integrals. Here, using a generalised reciprocal relation and viscoelastic force singularities, a boundary element method is developed for linear viscoelastic flows. We proceed to explore finite-deformation microswimming in a linear Maxwell fluid. We firstly deduce a finite-amplitude generalisation of a previously reported result that the flow field is unchanged between a Newtonian and linear Maxwell fluid for prescribed small-amplitude deformations. Hence Purcell’s theorem holds for a linear Maxwell fluid. We proceed to consider deformation swimming in a linear Maxwell fluid given an external forcing. Boundary scattering trajectories for an exemplar squirmer approaching a surface are observed to exhibit a weak dependence on the Deborah number, while the trajectories of a sperm and monotrichous bacterium near a surface are predicted to be essentially unaffected at moderate Deborah number. In turn, the latter supports the common simplification of using Newtonian Stokes flows for studying flagellate swimming in linear Maxwell media. In addition, the motion of a magnetic helix under the influence of an external magnetic field is considered, and highlights that linear viscoelasticity can significantly impact the propagation of the helix, in turn demonstrating that even linear rheology is important to consider for forced swimmers. Finally, the presented framework requires minimalistic adjustments to Newtonian boundary element codes, enabling rapid implementation, and is more generally applicable, for instance to studies of particle interactions in active linear rheology on the microscale.

JFM classification

Biological Fluid Dynamics: Biological Fluid Dynamics Low-Reynolds-number flows: Boundary integral methods Non-Newtonian Flows: Viscoelasticity
Type
Papers
Information
Journal of Fluid Mechanics , Volume 831 , 25 November 2017 , pp. 228 - 251
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Copyright
© 2017 Cambridge University Press 

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References

Berke, A. P., Turner, L., Berg, H. C. & Lauga, E. 2008 Hydrodynamic attraction of swimming microorganisms by surfaces. Phys. Rev. Lett. 101, 038102.CrossRefGoogle ScholarPubMed
Blake, J. R. 1971 A note on the image system for a Stokeslet in a no slip boundary. Proc. Camb. Phil. Soc. 70, 303–310.CrossRefGoogle Scholar
Boehme, G. & Mueller, A. 2015 Propulsion of axisymmetric swimmers in viscoelastic liquids by means of torsional oscillations. J. Non-Newtonian Fluid Mech. 224, 1–16.CrossRefGoogle Scholar
Bryers, J. D. 2008 Medical biofilms. Biotechnol. Bioengng 100, 1–18.CrossRefGoogle ScholarPubMed
Chaudhury, T. K. 1979 On swimming in a viscoelastic liquid. J. Fluid Mech. 95, 189–197.CrossRefGoogle Scholar
Corato, M. D., Greco, F. & Maffettone, P. L. 2015 Locomotion of a microorganism in weakly viscoelastic liquids. Phys. Rev. E 92, 053008.Google ScholarPubMed
Curtis, M. P. & Gaffney, E. A. 2013 Three-sphere swimmer in a non-linear viscoelastic medium. Phys Rev. E 87, 043006.Google Scholar
Denissenko, P., Kantsler, V., Smith, D. J. & Kirkman-Brown, J. 2012 Human spermatozoa migration in microchannels reveals boundary-following navigation. Proc. Natl Acad. Sci. USA 109.CrossRefGoogle ScholarPubMed
Elfring, G. J. & Goyal, G. 2016 The effect of gait on swimming in viscoelastic fluids. J. Non-Newtonian Fluid Mech. 234, 8–14.CrossRefGoogle Scholar
Feng, H., Cordoba, A., Hernandez, F., Indei, T., Li, S., Li, X. & Schieber, J. D. 2016 A boundary integral method for computing forces on particles in unsteady Stokes and linear viscoelastic fluids. Intl J. Numer. Meth. Fluids 82, 198–217.CrossRefGoogle Scholar
Frymier, P. D. & Ford, R. M. 1997 Analysis of bacterial swimming speed approaching a solid–liquid interface. AICHE J. 43, 1341–1347.CrossRefGoogle Scholar
Frymier, P. D., Ford, R. M., Berg, H. C. & Cummings, P. T. 1995 Three-dimensional tracking of motile bacteria near a solid planar surface. Proc. Natl Acad Sci. USA 92, 6195–6199.CrossRefGoogle Scholar
Fu, H. C., Powers, T. R. & Wolgemuth, C. W. 2007 Theory of swimming filaments in viscoelastic media. Phys. Rev. Lett. 99, 258101.CrossRefGoogle ScholarPubMed
Fu, H. C., Wolgemuth, C. W. & Powers, T. R. 2008 Beating patterns of filaments in viscoelastic fluids. Phys. Rev. E 78, 041913.Google ScholarPubMed
Fulford, G. R., Katz, D. F. & Powell, R. L. 1998 Swimming of spermatozoa in a linear viscoelastic fluid. Biorheology 35, 295–309.CrossRefGoogle Scholar
Fusco, S., Ullrich, F., Pokki, J., Chatzipirpiridis, G., Ozkale, B., Sivaraman, K. M., Ergeneman, O., Pane, S. & Nelson, B. J. 2014 Microrobots: a new era in ocular drug delivery. Expert Opin. Drug Deliv. 11, 1815–1826.CrossRefGoogle ScholarPubMed
Gagnon, D. A. & Arratia, P. E. 2016 The cost of swimming in generalized Newtonian fluids: experiments with C. elegans . J. Fluid Mech. 800, 753–765.CrossRefGoogle Scholar
Giacché, D., Ishikawa, T. & Yamaguchi, T. 2010 Hydrodynamic entrapment of bacteria swimming near a solid surface. Phys. Rev. E 82, 056309.Google Scholar
Godinez, F. A., Koens, L., Montenegro-Johnson, T. D., Zenit, R. & Lauga, E. 2015 Complex fluids affect low-Reynolds number locomotion in a kinematic-dependent manner. Exp. Fluids 56 (5), 97.CrossRefGoogle Scholar
Gomez, S., Godinez, F. A., Lauga, E. & Zenit, R. 2016 Helical propulsion in shear-thinning fluids. J. Fluid Mech. 812, R3.Google Scholar
Gomez-Solano, J. R. & Bechinger, C. 2015 Transient dynamics of a colloidal particle driven through a viscoelastic fluid. New J. Phys. 17, 103032.Google Scholar
Goto, T., Nakai, T. & Aoki, K. 2010 Boundary element analysis on transition of distance and attitude of a bacterium near a rigid surface. J. Biomech. Sci. Engng 5, 329–339.CrossRefGoogle Scholar
Gray, J. & Hancock, G. J. 1955 The propulsion of sea urchin spermatozoa. J. Expl Biol. 32, 802–814.CrossRefGoogle Scholar
Helander, H. F. & Fandriks, L. 2014 Surface area of the digestive tract – revisited. Scand. J. Gastroenterol. 49, 681–689.CrossRefGoogle ScholarPubMed
Ishikawa, T., Simmons, M. P. & Pedley, T. J. 2006 Hydrodynamic interaction of two-swimming model micro-organisms. J. Fluid Mech. 568, 119–160.CrossRefGoogle Scholar
Ishimoto, K., Cosson, J. & Gaffney, E. A. 2016 A simulation study of sperm motility hydrodynamics near fish eggs and sphere. J. Theor. Biol. 389, 187–197.CrossRefGoogle Scholar
Ishimoto, K. & Gaffney, E. A. 2013 Squirmer dynamics near a boundary. Phys. Rev. E 88, 062702.Google Scholar
Ishimoto, K. & Gaffney, E. A. 2014a A study of spermatozoan swimming stability near a surface. J. Theor. Biol. 360, 187–199.CrossRefGoogle ScholarPubMed
Ishimoto, K. & Gaffney, E. A. 2014b Swimming efficiency of spherical squirmers: beyond the Lighthill theory. Phys. Rev. E 90, 012704.Google ScholarPubMed
Ishimoto, K. & Gaffney, E. A. 2015 Fluid flow and sperm guidance: a simulation study of hydrodynamic sperm rheotaxis. J. R. Soc. Interface 12, 20150172.Google ScholarPubMed
Ishimoto, K. & Gaffney, E. A. 2016 Mechanical tuning of mammalian sperm behaviour by hyperactivation, rheology and substrate adhesion: a numerical exploration. J. R. Soc. Interface 13, 20160633.CrossRefGoogle ScholarPubMed
Ishimoto, K. & Yamada, M. 2012 A coordinate-based proof of the scallop theorem. SIAM J. Appl. Maths 72, 1686–1694.CrossRefGoogle Scholar
Kantsler, V., Dunkel, J. & Goldstein, M. P. R. E. 2013 Ciliary contact interactions dominate surface scattering of swimming eukaryotes. Proc. Natl Acad. Sci. USA 110, 1187–1192.CrossRefGoogle ScholarPubMed
Kim, S., Lee, S., Lee, J., Nelson, B. J., Zhang, L. & Choi, H. 2016 Fabrication and manipulation of ciliary microrobots with non-reciprocal magnetic actuation. Sci. Rep. 3, 30713.CrossRefGoogle Scholar
Klein, J. D., Clapp, A. R. & Dickinson, R. B. 2003 Direct measurement of interaction forces between a single bacterium and a flat plate. J. Colloid Interface Sci. 261, 379–385.CrossRefGoogle Scholar
Krieger, M. S., Spagnolie, S. E. & Powers, T. 2015 Microscale locomotion in a nematic liquid crystal. Soft Matt. 11 (47), 9115–9125.CrossRefGoogle Scholar
Lauga, E. 2009 Life at high Deborah number. Europhys. Lett. 86, 64001.CrossRefGoogle Scholar
Lauga, E. 2014 Locomotion in complex fluids: integral theorems. Phys. Fluids 26, 081902.CrossRefGoogle Scholar
Miki, K. & Clapham, D. E. 2013 Rheotaxis guides mammalian sperm. Curr. Biol 443–452.CrossRefGoogle ScholarPubMed
Montenegro-Johnson, T. D., Gadelha, H. & Smith, D. J. 2015 Spermatozoa scattering by a microchannel feature: an elastohydrodynamic model. R. Soc. Open Sci. 2, doi:10.1098/ rsos.140475.CrossRefGoogle ScholarPubMed
Nosrati, R., Driouchi, A., Yip, C. M. & Sinton, D. 2015 Two-dimensional slither swimming of sperm within a micrometre of a surface. Nat. Commun. 6, 8703.CrossRefGoogle ScholarPubMed
Okabe, M. 2013 The cell biology of mammalian fertilization. Development 140, 4471–4479.CrossRefGoogle ScholarPubMed
Patteson, A. E., Gopinath, A. & Arratia, P. E. 2016 Active colloids in complex fluids. Curr. Opin. Colloid Interface Sci. 21, 86–96.CrossRefGoogle Scholar
Pedley, T. J. & Kessler, J. O. 1992 Hydrodynamic phenomena in suspensions of swimming microorganisms. Annu. Rev. Fluid Mech. 24, 313–358.CrossRefGoogle Scholar
Phan-Thien, N. & Fan, X.-J. 2002 Viscoelastic mobility problem using a boundary element method. J. Non-Newtonian Fluid Mech. 105, 131–152.CrossRefGoogle Scholar
Phan-Thien, N., Tran-cong, T. & Ramia, M. 1987 A boundary-element analysis of flagellar propulsion. J. Fluid Mech. 184, 533–549.CrossRefGoogle Scholar
Pozrikidis, C. 1992 Boundary Integral and Singularity Methods for Linearized Viscous Flow. Cambridge University Press.CrossRefGoogle Scholar
Pozrikidis, C. 2002 A Practical Guide to Boundary Element Method with Software Library BEMLIB. CRC Press.CrossRefGoogle Scholar
Purcell, E. M. 1977 Life at low Reynolds number. Am. J. Phys. 45, 3–11.CrossRefGoogle Scholar
Qiu, F. & Nelson, B. J. 2015 Magnetic helical micro- and nanorobots: toward thier biomedical applications. Engineering 1, 21–26.CrossRefGoogle Scholar
Qiu, T., Lee, T.-C., Mark, A. G., Morozov, K. I., Münster, R., Mierka, O., Turek, S., Leshansky, A. M. & Fischer, P. 2014 Swimming by reciprocal motion at low Reynolds number. Nat. Commun. 5, 5119.CrossRefGoogle ScholarPubMed
Ribet, D. & Cossart, P. 2015 How bacterial pathogens colonize their hosts and invade deeper tissues. Microbes Infection 17, 173–183.CrossRefGoogle ScholarPubMed
Riley, E. E. & Lauga, E. 2015 Small-amplitude swimmers can self-propel faster in viscoelastic fluids. J. Theor. Biol. 382, 345–355.CrossRefGoogle ScholarPubMed
Salazar, D., Roma, A. M. & Ceniceros, H. D. 2016 Numerical study of an inextensible, finite swimmer in Stokesian viscoelastic flow. Phys. Fluids 28 (6).CrossRefGoogle Scholar
Shum, H., Gaffney, E. A. & Smith, D. J. 2010 Modelling bacteria behaviour close to a no-slip plane boundary: the influence of bacterial geometry. Proc. R. Soc. Lond. A 466, 1725–1748.Google Scholar
Shum, H., Tripathi, A., Yeomans, J. M. & Balazs, A. C. 2013 Active ciliated surfaces expel model swimmers. Langmuir 29, 12770–12776.CrossRefGoogle ScholarPubMed
Smith, D. J., Gaffney, E. A., Blake, J. R. & Kirkman-Brown, J. C. 2009a Human sperm accumulation near surfaces: a simulation study. J. Fluid Mech. 621, 289–320.CrossRefGoogle Scholar
Smith, D. J., Gaffney, E. A., Gadêlha, H., Kapur, N. & Kirkman-Brown, J. C. 2009b Bend propagation in the flagella of migrating human sperm, and its modulation by viscosity. Cell Motil. Cytoskel. 66, 220–236.CrossRefGoogle ScholarPubMed
Spagnolie, S. E. 2015 Complex Fluids in Biological Systems. Springer.CrossRefGoogle Scholar
Spagnolie, S. E. & Lauga, E. 2012 Hydrodynamics of self-propulsion near a boundary predictions and accuracy of far-field approximations. J. Fluid Mech. 700, 105–147.CrossRefGoogle Scholar
Suarez, S. S. 2016 Mammalian sperm interactions with female reproductive tract. Cell Tissue Res. 363, 185–194.CrossRefGoogle ScholarPubMed
Suarez, S. S. & Pacey, A. A. 2006 Sperm transport in the female reproductive tract. Hum. Reprod. Update 12, 23–37.CrossRefGoogle ScholarPubMed
Teran, J., Fauci, L. & Shelley, M. 2010 Viscoelastic fluid response can increase the speed and efficiency of a free swimmer. Phys. Rev. Lett. 104, 038101.CrossRefGoogle ScholarPubMed
Thomases, B. & Guy, R. D. 2014 Mechanisms of elastic enhancement and hindrance for finite-length undulatory swimmers in viscoelastic fluids. Phys. Rev. Lett. 113 (9).CrossRefGoogle ScholarPubMed
Xu, K., Forest, M. G. & Klapper, I. 2007 On the correspondence between creeping flows of viscous and viscoelastic fluids. J. Non-Newtonian Fluid Mech. 145, 150–172.CrossRefGoogle Scholar
Yazdi, S., Ardekani, A. M. & Borhan, A. 2014 Locomotion of microorganisms near a no-slip boundary in a viscoelastic fluid. Phys. Rev. E 90, 043002.Google Scholar
Yazdi, S., Ardekani, A. M. & Borhan, A. 2015 Swimming dynamics near a wall in a weakly elastic fluid. J. Nonlinear Sci. 25, 1153–1167.CrossRefGoogle Scholar
Zhang, Z., Liu, J., Meriano, J., Ru, C., Xie, S., Luo, J. & Sun, Y. 2016 Human sperm rheotaxis: a passive physical process. Sci. Rep. 6.Google ScholarPubMed
Zheng, R. & Phan-Thien, N. 1992 A boundary element simulation of the unsteady motion of a sphere in a cylindrical tube containing a viscoelastic fluid. Rheol. Acta 31, 323–332.CrossRefGoogle Scholar
Zhu, L., Lauga, E. & Brandt, L. 2012 Self-propulsion in viscoelastic fluids: pushers versus pullers. Phys. Fluids 24, 051902.CrossRefGoogle Scholar