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Analytical study for swimmers in a channel

Published online by Cambridge University Press:  24 October 2019

A. Farutin ,
H. Wu ,
W.-F. Hu ,
S. Rafaï ,
P. Peyla ,
M.-C. Lai  and
C. Misbah Open the ORCID record for C. Misbah [Opens in a new window]
A. Farutin*
Affiliation:
Univ. Grenoble Alpes, CNRS, LIPhy, F-38000 Grenoble, France
H. Wu
Affiliation:
Univ. Grenoble Alpes, CNRS, LIPhy, F-38000 Grenoble, France
W.-F. Hu
Affiliation:
Department of Applied Mathematics, National Chung Hsing University, 145, Xingda Road, Taichung City 402, Taiwan
S. Rafaï
Affiliation:
Univ. Grenoble Alpes, CNRS, LIPhy, F-38000 Grenoble, France
P. Peyla
Affiliation:
Univ. Grenoble Alpes, CNRS, LIPhy, F-38000 Grenoble, France
M.-C. Lai
Affiliation:
Department of Applied Mathematics, National Chiao Tung University, 1001, Ta Hsueh Road, Hsinchu 300, Taiwan
C. Misbah
Affiliation:
Univ. Grenoble Alpes, CNRS, LIPhy, F-38000 Grenoble, France
*
†Email address for correspondence: [email protected]
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Abstract

There is an overabundance of microswimmers in nature, including bacteria, algae, mammalian cells and so on. They use flagellum, cilia or global shape changes (amoeboid motion) to move forward. In the presence of confining channels, these swimmers exhibit often non-trivial behaviours, such as accumulation at the wall, navigation and so on, and their swimming speed may be strongly influenced by the geometric confinement. Several numerical studies have reported that the presence of walls either enhances or reduces the swimming speed depending on the nature of the swimmer, and also on the confinement. The purpose of this paper is to provide an analytical explanation of several previously obtained numerical results. We treat the case of amoeboid swimmers and the case of squirmers having either a tangential (the classical situation) or normal velocity prescribed at the swimmer surface (pumper). For amoeboid motion we consider a quasi-circular swimmer which allows us to tackle the problem analytically and to extract the equations of the motion of the swimmer, with several explicit analytical or semi-analytical solutions. It is found that the deformation of the amoeboid swimmer as well as a high enough order effect due to confinement are necessary in order to account for previous numerical results. The analytical theory accounts for several features obtained numerically also for non-deformable swimmers.

JFM classification

Biological Fluid Dynamics: Swimming/flying Biological Fluid Dynamics: Micro-organism dynamics
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 881 , 25 December 2019 , pp. 365 - 383
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Copyright
© 2019 Cambridge University Press 

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References

Acemoglu, A. & Yesilyurt, S. 2014 Effects of geometric parameters on swimming of micro organisms with single helical flagellum in circular channels. Biophys. J. 106 (7), 1537–1547.Google Scholar
Alouges, F., Desimone, A. & Heltai, L. 2011 Numerical strategies for stroke optimization of axisymmetric microswimmers. Math. Models Meth. Appl. Sci. 21, 361–388.Google Scholar
Aoun, L., Negre, P., Farutin, A., Garcia-Seyda, N., Rivzi, M. S., Galland, R., Michelot, A., Luo, X., Biarnes-Pelicot, M., Hivroz, C. et al. 2019 Mammalian amoeboid swimming is propelled by molecular and not protrusion-based paddling in lymphocytes. Preprint.10.1101/509182Google Scholar
Avron, J. E., Gat, O. & Kenneth, O. 2004 Optimal swimming at low Reynolds numbers. Phys. Rev. Lett. 93, 186001.Google Scholar
Bae, A. J. & Bodenschatz, E. 2010 On the swimming of Dictyostelium amoebae. Proc. Natl Acad. Sci. USA 107, E165–E166.Google Scholar
Barry, N. P. & Bretscher, M. S. 2010 Dictyostelium amoebae and neutrophils can swim. Proc. Natl Acad. Sci. USA 107, 11376–11380.10.1073/pnas.1006327107Google Scholar
Bergert, M., Erzberger, A., Desai, R. A., Aspalter, I. M., Oates, A. C., Charras, G., Salbreux, G. & Paluch, E. K. 2015 Force transmission during adhesion-independent migration. Nat. Cell Biol. 17, 524–529.10.1038/ncb3134Google Scholar
Bilbao, A., Wajnryb, E., Vanapalli, S. A. & Blawzdziewicz, J. 2013 Nematode locomotion in unconfined and confined fluids. Phys. Fluids 25 (8), 081902.Google Scholar
Blake, J. R. 1971 A spherical envelope approach to ciliary propulsion. J. Fluid Mech. 46, 199–208.10.1017/S002211207100048XGoogle Scholar
Crowdy, D. G. & Or, Y. 2010 Two-dimensional point singularity model of a low-Reynolds-number swimmer near a wall. Phys. Rev. E 81, 036313.Google Scholar
Danker, G., Biben, T., Podgorski, T., Verdier, C. & Misbah, C. 2007 Dynamics and rheology of a dilute suspension of vesicles: higher-order theory. Phys. Rev. E 76 (4), 041905.Google Scholar
Drescher, K., Leptos, K. C., Tuval, I., Ishikawa, T., Pedley, T. J. & Goldstein, R. E. 2009 Dancing volvox: hydrodynamic bound states of swimming algae. Phys. Rev. Lett. 102 (16), 168101.10.1103/PhysRevLett.102.168101Google Scholar
Ezhilan, B. & Saintillan, D. 2015 Transport of a dilute active suspension in pressure-driven channel flow. J. Fluid Mech. 777, 482–522.10.1017/jfm.2015.372Google Scholar
Farutin, A., Rafaï, S., Dysthe, D. K., Duperray, A., Peyla, P. & Misbah, C. 2013 Amoeboid swimming: A generic self-propulsion of cells in fluids by means of membrane deformations. Phys. Rev. Lett. 111, 228102.Google Scholar
Felderhof, B. U. 2010 Swimming at low Reynolds number of a cylindrical body in a circular tube. Phys. Fluids 22 (11), 113604.Google Scholar
Garcia, M., berti, S., Peyla, P. & Rafaï, S. 2011 Random walk of a swimmer in a low-Reynolds-number medium. Phys. Rev. E 83, 035301.Google Scholar
Giacché, D., Ishikawa, T. & Yamaguchi, T. 2010 Hydrodynamic entrapment of bacteria swimming near a solid surface. Phys. Rev. E 82, 056309.Google Scholar
Guasto, J. S., Johnson, K. A. & Gollub, J. P. 2010 Oscillatory flows induced by microorganisms swimming in two dimensions. Phys. Rev. Lett. 105 (16), 168102.Google Scholar
Hawkins, R. J., Piel, M., Faure-Andre, G., Lennon-Dumenil, A. M., Joanny, J. F., Prost, J. & Voituriez, R. 2009 Pushing off the walls: a mechanism of cell motility in confinement. Phys. Rev. Lett. 102 (5), 058103.Google Scholar
Hiraiwa, T., Shitara, K. & Ohta, T. 2011 Dynamics of a deformable self-propelled particle in three dimensions. Soft Matt. 7 (7), 3083–3086.Google Scholar
Kantsler, V., Dunkel, J., Polin, M. & Goldstein, R. E. 2013 Ciliary contact interactions dominate surface scattering of swimming eukaryotes. Proc. Natl Acad. Sci. USA 110 (4), 1187–1192.10.1073/pnas.1210548110Google Scholar
Lauga, E. & Powers, T. R. 2009 The hydrodynamics of swimming microorganisms. Rep. Prog. Phys. 72, 096601.Google Scholar
Ledesma-Aguilar, R. & Yeomans, J. M. 2013 Enhanced motility of a microswimmer in rigid and elastic confinement. Phys. Rev. Lett. 111, 138101.10.1103/PhysRevLett.111.138101Google Scholar
Lighthill, M. J. 1952 On the squirming motion of nearly spherical deformable bodies through liquids at very small Reynolds numbers. Commun. Pure Appl. Maths 5, 109–118.10.1002/cpa.3160050201Google Scholar
Liu, Y.-J., Le Berre, M., Lautenschläger, F., Maiuri, P., Callan-Jones, A., Heuzé, M., Takaki, T., Voituriez, R. & Piel, M. 2015 Confinement and low adhesion induce fast amoeboid migration of slow mesenchymal cells. Cell 160 (4), 659–672.10.1016/j.cell.2015.01.007Google Scholar
Loheac, J., Scheid, J.-F. & Tucsnak, M. 2013 Controllability and time optimal control for low Reynolds numbers swimmers. Acta Appl. Math. 123, 175–200.10.1007/s10440-012-9760-9Google Scholar
Lushi, E., Kantsler, V. & Goldstein, R. E. 2017 Scattering of biflagellate microswimmers from surfaces. Phys. Rev. E 96, 023102.Google Scholar
Ohta, T. & Ohkuma, T. 2009 Deformable self-propelled particles. Phys. Rev. Lett. 102, 154101.Google Scholar
Pinner, S. & Sahai, E. 2008 Imaging amoeboid cancer cell motility in vivo. J. Microsc. 231, 441–445.Google Scholar
Ranganathan, M., Farutin, A. & Misbah, C. 2018 Effect of cytoskeleton elasticity on amoeboid swimming. Biophys. J. 115, 1316–1329.Google Scholar
Saintillan, D. & Shelley, M. 2012 Emergence of coherent structures and large-scale flows in motile suspensions. J. R. Soc. Interface 9, 571–585.10.1098/rsif.2011.0355Google Scholar
Shapere, A. & Wilczek, F. 1987 Self-propulsion at low Reynolds number. Phys. Rev. Lett. 58, 2051.Google Scholar
Shum, H., Gaffney, E. A. & Smith, D. J. 2010 Modelling bacterial behaviour close to a no-slip plane boundary: the influence of bacterial geometry. Proc. R. Soc. Lond. A 466 (2118), 1725–1748.Google Scholar
Smith, D. J., Gaffney, E. A., Blake, J. R. & Kirman-Brown, J. C. 2009 Human sperm accumulation near surfaces: a simulation study. J. Fluid Mech. 621, 289–320.Google 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.Google Scholar
Throndsen, J. 1969 Flagellates of Norwegian coastal waters. Norw. J. Bot. 16, 161–216.Google Scholar
Vilfan, A. 2012 Optimal shapes of surface slip driven self-propelled microswimmers. Phys. Rev. Lett. 109, 128105.Google Scholar
Wu, H., Farutin, A., Hu, W.-F., Thiebaud, M., Rafai, S., Peyla, P., Lai, M.-C. & Misbah, C. 2016 Amoeboid swimming in a channel. Soft Matt. 12, 7470–7484.Google Scholar
Wu, H., Thiébaud, M., Hu, W.-F., Farutin, A., Rafaï, S., Lai, M.-C., Peyla, P. & Misbah, C. 2015 Amoeboid motion in confined geometry. Phys. Rev. E 92, 050701.Google Scholar
Zhang, P., Jana, S., Giarra, M., Vlachos, P. P. & Jung, S. 2015 Paramecia swimming in viscous flow. Eur. Phys. J. Spec. Top. 224 (17–18), 3199–3210.Google Scholar
Zhu, L., Lauga, E. & Brandt, L. 2013 Low-Reynolds-number swimming in a capillary tube. J. Fluid Mech. 726, 285–311.Google Scholar