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Tracer trajectories and displacement due to a micro-swimmer near a surface

Published online by Cambridge University Press:  27 May 2015

A. J. T. M. Mathijssen ,
D. O. Pushkin  and
J. M. Yeomans
A. J. T. M. Mathijssen*
Affiliation:
Rudolf Peierls Centre for Theoretical Physics, University of Oxford, 1 Keble Road, Oxford OX1 3NP, UK
D. O. Pushkin
Affiliation:
Rudolf Peierls Centre for Theoretical Physics, University of Oxford, 1 Keble Road, Oxford OX1 3NP, UK
J. M. Yeomans
Affiliation:
Rudolf Peierls Centre for Theoretical Physics, University of Oxford, 1 Keble Road, Oxford OX1 3NP, UK
*
†Email address for correspondence: [email protected]
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Abstract

We study tracer particle transport due to flows created by a self-propelled micro-swimmer, such as a swimming bacterium, alga or a microscopic artificial swimmer. Recent theoretical work has shown that as a swimmer moves in the fluid bulk along an infinite straight path, tracer particles far from its path perform closed loops, whereas those close to the swimmer are entrained by its motion. However, in biologically and technologically important cases tracer transport is significantly altered for swimmers that move in a run-and-tumble fashion with a finite persistence length, and/or in the presence of a free surface or a solid boundary. Here we present a systematic analytical and numerical study exploring the resultant regimes and their crossovers. Our focus is on describing qualitative features of the tracer particle transport and developing quantitative tools for its analysis. Our work is a step towards understanding the ecological effects of flows created by swimming organisms, such as enhanced fluid mixing and biofilm formation.

JFM classification

Biological Fluid Dynamics: Biological Fluid Dynamics Low-Reynolds-number flows: Low-Reynolds-number flows Biological Fluid Dynamics: Micro-organism dynamics
Type
Papers
Information
Journal of Fluid Mechanics , Volume 773 , 25 June 2015 , pp. 498 - 519
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Copyright
© 2015 Cambridge University Press 

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References

Alexander, G. P. & Yeomans, J. M. 2008 Dumb-bell swimmers. Europhys. Lett. 83 (3), 34006.CrossRefGoogle Scholar
Avron, J. E., Kenneth, O. & Oaknin, D. H. 2005 Pushmepullyou: an efficient micro-swimmer. New J. Phys. 7 (1), 234–242.Google Scholar
Berg, H. C. 2004 E. coli in Motion. Springer.CrossRefGoogle Scholar
Berg, H. C. & Brown, D. A. 1972 Chemotaxis in Escherichia coli analysed by three-dimensional tracking. Nature 239 (5374), 500–504.Google Scholar
Berke, A. P., Turner, L., Berg, H. C. & Lauga, E. 2008 Hydrodynamic attraction of swimming microorganisms by surfaces. Phys. Rev. Lett. 101 (3), 038102.Google Scholar
Blake, J. R. 1971a A note on the image system for a Stokeslet in a no-slip boundary. In Proceedings of the Cambridge Philosophical Society, vol. 70, pp. 303–310. Cambridge University Press.Google Scholar
Blake, J. R. 1971b A spherical envelope approach to ciliary propulsion. J. Fluid Mech. 46 (1), 199–208.Google Scholar
Costerton, J. W., Lewandowski, Z., Caldwell, D. E., Korber, D. R. & Lappin-Scott, H. M. 1995 Microbial biofilms. Annu. Rev. Microbiol. 49 (1), 711–745.CrossRefGoogle ScholarPubMed
Darwin, C. 1953 Note on hydrodynamics. In Mathematical Proceedings of the Cambridge Philosophical Society, vol. 49, pp. 342–354. Cambridge University Press.Google Scholar
Di Leonardo, R., Dell’Arciprete, D., Angelani, L. & Iebba, V. 2011 Swimming with an image. Phys. Rev. Lett. 106 (3), 038101.CrossRefGoogle ScholarPubMed
Drescher, K., Dunkel, J., Cisneros, L. H., Ganguly, S. & Goldstein, R. E. 2011 Fluid dynamics and noise in bacterial cell–cell and cell–surface scattering. Proc. Natl Acad. Sci. USA 108 (27), 10940–10945.Google Scholar
Drescher, K., Goldstein, R. E., Michel, N., Polin, M. & Tuval, I. 2010 Direct measurement of the flow field around swimming microorganisms. Phys. Rev. Lett. 105 (16), 168101.CrossRefGoogle ScholarPubMed
Dreyfus, R., Baudry, J., Roper, M. L., Fermigier, M., Stone, H. A. & Bibette, J. 2005 Microscopic artificial swimmers. Nature 437 (7060), 862–865.Google Scholar
Dunkel, J., Putz, V. B., Zaid, I. M. & Yeomans, J. M. 2010 Swimmer-tracer scattering at low Reynolds number. Soft Matt. 6 (17), 4268–4276.Google 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 (13), 6195–6199.Google Scholar
Golestanian, R. & Ajdari, A. 2008 Analytic results for the three-sphere swimmer at low Reynolds number. Phys. Rev. E 77 (3), 036308.Google Scholar
Gray, J. & Hancock, G. J. 1955 The propulsion of sea-urchin spermatozoa. J. Expl Biol. 32 (4), 802–814.Google Scholar
Hancock, G. J. 1953 The self-propulsion of microscopic organisms through liquids. Proc. R. Soc. Lond. A 217 (1128), 96–121.Google Scholar
Ishikawa, T., Locsei, J. T. & Pedley, T. J. 2010 Fluid particle diffusion in a semidilute suspension of model micro-organisms. Phys. Rev. E 82 (2), 021408.Google Scholar
Ishikawa, T., Simmonds, M. P. & Pedley, T. J. 2006 Hydrodynamic interaction of two swimming model micro-organisms. J. Fluid Mech. 568, 119–160.Google Scholar
Jeffrey, D. J. & Onishi, Y. 1984 Calculation of the resistance and mobility functions for two unequal rigid spheres in low-Reynolds-number flow. J. Fluid Mech. 139, 261–290.Google Scholar
Jepson, A., Martinez, V. A., Schwarz-Linek, J., Morozov, A. & Poon, W. C. K. 2013 Enhanced diffusion of nonswimmers in a three-dimensional bath of motile bacteria. Phys. Rev. E 88 (4), 041002.Google Scholar
Kamiya, R. & Hasegawa, E. 1987 Intrinsic difference in beat frequency between the two flagella of Chlamydomonas reinhardtii . Exp. Cell Res. 173 (1), 299–304.Google Scholar
Katija, K. 2012 Biogenic inputs to ocean mixing. J. Expl Biol. 215 (6), 1040–1049.Google Scholar
Katz, D. F., Overstreet, J. W., Samuels, S. J., Niswander, P. W., Bloom, T. D. & Lewis, E. L. 1986 Morphometric analysis of spermatozoa in the assessment of human male fertility. J. Androl. 7 (4), 203–210.Google Scholar
Kim, M. J. & Breuer, K. S. 2007 Controlled mixing in microfluidic systems using bacterial chemotaxis. Analyt. Chem. 79 (3), 955–959.CrossRefGoogle ScholarPubMed
Kim, S. & Karilla, S. 1991 Microhydrodynamics: Principles and Selected Applications. Butterworth Series of Chemical Engineering.Google Scholar
Kurtuldu, H., Guasto, J. S., Johnson, K. A. & Gollub, J. P. 2011 Enhancement of biomixing by swimming algal cells in two-dimensional films. Proc. Natl Acad. Sci. USA 108 (26), 10391–10395.Google Scholar
Lauga, E., DiLuzio, W. R., Whitesides, G. M. & Stone, H. A. 2006 Swimming in circles: motion of bacteria near solid boundaries. Biophys. J. 90 (2), 400–412.Google Scholar
Lauga, E. & Powers, T. R. 2009 The hydrodynamics of swimming microorganisms. Rep. Prog. Phys. 72 (9), 096601.Google Scholar
Leoni, M., Kotar, J., Bassetti, B., Cicuta, P. & Lagomarsino, M. C. 2009 A basic swimmer at low Reynolds number. Soft Matt. 5 (2), 472–476.Google Scholar
Leptos, K. C., Guasto, J. S., Gollub, J. P., Pesci, A. I. & Goldstein, R. E. 2009 Dynamics of enhanced tracer diffusion in suspensions of swimming eukaryotic microorganisms. Phys. Rev. Lett. 103 (19), 198103.Google Scholar
Li, G., Bensson, J., Nisimova, L., Munger, D., Mahautmr, P., Tang, J. X., Maxey, M. R. & Brun, Y. V. 2011 Accumulation of swimming bacteria near a solid surface. Phys. Rev. E 84 (4), 041932.CrossRefGoogle Scholar
Li, G. & Tang, J. X. 2009 Accumulation of microswimmers near a surface mediated by collision and rotational Brownian motion. Phys. Rev. Lett. 103 (7), 078101.Google 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 (2), 109–118.Google Scholar
Lin, Z., Thiffeault, J.-L. & Childress, S. 2011 Stirring by squirmers. J. Fluid Mech. 669, 167–177.Google Scholar
Miño, G. L., Dunstan, J., Rousselet, A., Clément, E. & Soto, R. 2013 Induced diffusion of tracers in a bacterial suspension: theory and experiments. J. Fluid Mech. 729, 423–444.Google Scholar
Mino, G., Mallouk, T. E., Darnige, T., Hoyos, M., Dauchet, J., Dunstan, J., Soto, R., Wang, Y., Rousselet, A. & Clement, E. 2011 Enhanced diffusion due to active swimmers at a solid surface. Phys. Rev. Lett. 106 (4), 048102.Google Scholar
Molaei, M., Barry, M., Stocker, R. & Sheng, J. 2014 Failed escape: solid surfaces prevent tumbling of Escherichia coli . Phys. Rev. Lett. 113 (6), 068103.Google Scholar
Morozov, A. & Marenduzzo, D. 2014 Enhanced diffusion of tracer particles in dilute bacterial suspensions. Soft Matt. 10 (16), 2748–2758.Google Scholar
Najafi, A. & Golestanian, R. 2004 Simple swimmer at low Reynolds number: three linked spheres. Phys. Rev. E 69 (6), 062901.Google Scholar
Paxton, W. F., Kistler, K. C., Olmeda, C. C., Sen, A., St. Angelo, S. K., Cao, Y., Mallouk, T. E., Lammert, P. E. & Crespi, V. H. 2004 Catalytic nanomotors: autonomous movement of striped nanorods. J. Am. Chem. Soc. 126 (41), 13424–13431.Google Scholar
Polin, M., Tuval, I., Drescher, K., Gollub, J. P. & Goldstein, R. E. 2009 Chlamydomonas swims with two ‘gears’ in a eukaryotic version of run-and-tumble locomotion. Science 325 (5939), 487–490.Google Scholar
Porter, J. R. 1976 Antony van Leeuwenhoek: tercentenary of his discovery of bacteria. Bacteriol. Rev. 40 (2), 260–269.CrossRefGoogle Scholar
Purcell, E. M. 1977 Life at low Reynolds number. Am. J. Phys. 45, 3–11.Google Scholar
Pushkin, D. O., Shum, H. & Yeomans, J. M. 2013 Fluid transport by individual microswimmers. J. Fluid Mech. 726, 5–25.Google Scholar
Pushkin, D. O. & Yeomans, J. M. 2013 Fluid mixing by curved trajectories of microswimmers. Phys. Rev. Lett. 111 (18), 188101.Google Scholar
Pushkin, D. O. & Yeomans, J. M. 2014 Stirring by swimmers in confined microenvironments. J. Stat. Mech. 2014 (4), P04030.Google Scholar
Rothschild, Lord 1963 Non-random distribution of bull spermatozoa in a drop of sperm suspension. Nature 198 (488), 1221–1222.Google Scholar
Sokolov, A., Goldstein, R. E., Feldchtein, F. I. & Aranson, I. S. 2009 Enhanced mixing and spatial instability in concentrated bacterial suspensions. Phys. Rev. E 80 (3), 031903.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
Stokes, G. G. 1846 On the friction of fluids in motion, and the equilibrium and motion of elastic solids. In Mathematical and Physical Papers, vol. 1. Cambridge University Press.Google Scholar
Suarez, S. S. & Pacey, A. A. 2006 Sperm transport in the female reproductive tract. Hum. Reprod. Update 12 (1), 23–37.Google Scholar
Tam, D. & Hosoi, A. E. 2007 Optimal stroke patterns for Purcell’s three-link swimmer. Phys. Rev. Lett. 98 (6), 068105.Google Scholar
Taylor, G. 1951 Analysis of the swimming of microscopic organisms. Proc. R. Soc. Lond. A 209 (1099), 447–461.Google Scholar
Thiffeault, J.-L.2014 Short-time distribution of particle displacements due to swimming microorganisms. arXiv:1408.4781.Google Scholar
Thiffeault, J.-L. & Childress, S. 2010 Stirring by swimming bodies. Phys. Lett. A 374 (34), 3487–3490.Google Scholar
Thutupalli, S. 2014 Swimming droplets: artificial squirmers. In Towards Autonomous Soft Matter Systems, pp. 79–94. Springer.Google Scholar
Turner, L., Ryu, W. S. & Berg, H. C. 2000 Real-time imaging of fluorescent flagellar filaments. J. Bacteriol. 182 (10), 2793–2801.Google Scholar
Underhill, P. T., Hernandez-Ortiz, J. P. & Graham, M. D. 2008 Diffusion and spatial correlations in suspensions of swimming particles. Phys. Rev. Lett. 100 (24), 248101.Google Scholar
Van Loosdrecht, M. C., Lyklema, J., Norde, Wt. & Zehnder, A. J. 1990 Influence of interfaces on microbial activity. Microbiol. Rev. 54 (1), 75–87.Google Scholar
Vladescu, I. D., Marsden, E. J., Schwarz-Linek, J., Martinez, V. A., Arlt, J., Morozov, A. N., Marenduzzo, D., Cates, M. E. & Poon, W. C. K. 2014 Filling an emulsion drop with motile bacteria. Phys. Rev. Lett. 113 (26), 268101.Google Scholar
Witman, G. 2009 The Chlamydomonas Sourcebook: Cell Motility and Behavior. Academic.Google Scholar
Wu, X.-L. & Libchaber, A. 2000 Particle diffusion in a quasi-two-dimensional bacterial bath. Phys. Rev. Lett. 84 (13), 3017–3020.Google Scholar
Yeomans, J. M., Pushkin, D. O. & Shum, H. 2014 An introduction to the hydrodynamics of swimming microorganisms. Eur. Phys. J. Special Topics 223 (9), 1771–1785.Google Scholar
Zöttl, A. & Stark, H. 2012 Nonlinear dynamics of a microswimmer in Poiseuille flow. Phys. Rev. Lett. 108 (21), 218104.Google Scholar
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