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Low-Reynolds-number swimming in a capillary tube

Published online by Cambridge University Press:  31 May 2013

L. Zhu ,
E. Lauga  and
L. Brandt
L. Zhu*
Affiliation:
Linné Flow Centre, KTH Mechanics, S-100 44 Stockholm, Sweden
E. Lauga
Affiliation:
Department of Mechanical and Aerospace Engineering, University of California San Diego, 9500 Gilman Drive, La Jolla CA 92093-0411, USA
L. Brandt
Affiliation:
Linné Flow Centre, KTH Mechanics, S-100 44 Stockholm, Sweden
*
†Email address for correspondence: [email protected]
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Abstract

We use the boundary element method to study the low-Reynolds-number locomotion of a spherical model microorganism in a circular tube. The swimmer propels itself by tangential or normal surface motion in a tube whose radius is of the order of the swimmer size. Hydrodynamic interactions with the tube walls significantly affect the average swimming speed and power consumption of the model microorganism. In the case of swimming parallel to the tube axis, the locomotion speed is always reduced (respectively, increased) for swimmers with tangential (respectively, normal) deformation. In all cases, the rate of work necessary for swimming is increased by confinement. Swimmers with no force dipoles in the far field generally follow helical trajectories, solely induced by hydrodynamic interactions with the tube walls, and in qualitative agreement with recent experimental observations for Paramecium. Swimmers of the puller type always display stable locomotion at a location which depends on the strength of their force dipoles: swimmers with weak dipoles (small $\alpha $) swim in the centre of the tube while those with strong dipoles (large $\alpha $) swim near the walls. In contrast, pusher swimmers and those employing normal deformation are unstable and end up crashing into the walls of the tube. Similar dynamics is observed for swimming into a curved tube. These results could be relevant for the future design of artificial microswimmers in confined geometries.

JFM classification

Biological Fluid Dynamics: Biological Fluid Dynamics Low-Reynolds-number flows: Boundary integral methods Low-Reynolds-number flows: Low-Reynolds-number flows Biological Fluid Dynamics: Swimming/flying
Type
Papers
Information
Journal of Fluid Mechanics , Volume 726 , 10 July 2013 , pp. 285 - 311
Copyright
©2013 Cambridge University Press 

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References

Berg, H. C. 2000 Motile behaviour of bacteria. Phys. Today 53, 24–29.Google Scholar
Berg, H. C. & Turner, L. 1979 Movement of microorganisms in viscous environments. Nature 278, 349–351.Google Scholar
Berke, A. P., Turner, L., Berg, H. C. & Lauga, E. 2008 Hydrodynamic attraction of swimming microorganisms by surfaces. Phys. Rev. Lett. 101, 038102.Google Scholar
Biondi, S. A., Quinn, J. A. & Goldfine, H. 1998 Random mobility of swimming bacteria in restricted geometries. AIChE J. 44, 1923–1929.CrossRefGoogle Scholar
Blake, J. R. 1971 A spherical envelope approach to ciliary propulsion. J. Fluid Mech. 46, 199–208.CrossRefGoogle Scholar
Brennen, C. & Winet, H. 1977 Fluid mechanics of propulsion by cilia and flagella. Annu. Rev. Fluid Mech. 9, 339–398.Google Scholar
Cortez, R., Fauci, L. & Medovikov, A. 2005 The method of regularized Stokeslets in three dimensions: analysis, validation, and application to helical swimming. Phys. Fluids 17, 031504.CrossRefGoogle Scholar
Crowdy, D. & Samson, O. 2011 Hydrodynamic bound states of a low-Reynolds-number swimmer near a gap in a wall. J. Fluid Mech. 667, 309–335.CrossRefGoogle Scholar
Crowdy, D. & Yizhar, O. 2010 Two-dimensional point singularity model of a low-Reynolds-number swimmer near a wall. Phys. Rev. E 81, 036313.Google Scholar
Dombrowski, C., Cisneros, L., Chatkaew, S., Goldstein, R. E. & Kessler, J. O. 2004 Self-concentration and large-scale coherence in bacterial dynamics. Phys. Rev. Lett. 93, 098103.Google Scholar
Doostmohammadi, A., Stocker, R. & Ardekani, A. M. 2012 Low-Reynolds-number swimming at pycnoclines. Proc. Natl Acad. Sci. USA 109, 3856–3861.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, 10940–10945.CrossRefGoogle ScholarPubMed
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, 168101.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, 168101.Google Scholar
Durham, W. M., Kessler, J. O. & Stocker, R. 2009 Disruption of vertical motility by shear triggers formation of thin phytoplankton layers. Science 323, 1067–1070.CrossRefGoogle ScholarPubMed
Elfring, G., Pak, O. S. & Lauga, E. 2010 Two-dimensional flagellar synchronization in viscoelastic fluids. J. Fluid Mech. 646, 505–515.CrossRefGoogle Scholar
Evans, A. A., Ishikawa, T., Yamaguchi, T. & Lauga, E. 2011 Orientational order in concentrated suspensions of spherical microswimmers. Phys. Fluids 23, 111702.Google Scholar
Fauci, L. J. & Dillon, R. 2006 Biofluidmechanics of reproduction. Annu. Rev. Fluid Mech. 38, 371–394.Google Scholar
Fauci, L. J. & Mcdonald, A. 1995 Sperm mobility in the presence of boundaries. Bull. Math. Biol. 57, 679–699.Google Scholar
Felderhof, B. U. 2009 Swimming and peristaltic pumping between two plane parallel walls. J. Phys.: Condens. Matter 21, 204106.Google Scholar
Felderhof, B. U. 2010 Swimming at low Reynolds number of a cylindrical body in a circular tube. Phys. Fluids 22, 113604.Google Scholar
Fu, H., Powers, T. R. & Wolgemuth, C. W. 2008 Theory of swimming filaments in viscoelastic media. Phys. Rev. Lett. 99, 258101.Google Scholar
Giacché, D. & Ishikawa, T. 2010 Hydrodynamic interaction of two unsteady model microorganisms. J. Theor. Biol. 267, 252–263.CrossRefGoogle ScholarPubMed
Goldman, A. J., Cox, R. G. & Brenner, H. 1967 Slow viscous motion of a sphere parallel to a plane wall—I Motion through a quiescent fluid. Chem. Engng Sci 22, 637–651.CrossRefGoogle Scholar
Goto, T., Nakata, K., Baba, K., Nishimura, M. & Magariyama, Y. 2005 A fluid-dynamic interpretation of the asymmetric motion of singly flagellated bacteria swimming close to a boundary. Biophys. J. 89, 3771–3779.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, 168102.Google Scholar
Hamel, A., Fisch, C., Combettes, L., Dupuis-Williams, P. & Baroud, C. N. 2011 Transitions between three swimming gaits in paramecium escape. Proc. Natl Acad. Sci. USA 108, 7290–7295.Google Scholar
Higdon, J. J. L. & Muldowney, G. P. 1995 Resistance functions for spherical particles, droplets and bubbles in cylindircal tubes. J. Fluid Mech. 298, 193–210.Google Scholar
Hill, J., Kalkanci, O., McMurry, J. L. & Koser, H. 2007 Hydrodynamic surface interactions enable Escherichia coli to seek efficient routes to swim upstream. Phys. Rev. Lett. 98, 068101.Google Scholar
Huang, Q. & Cruse, T. A. 1993 Some notes on singular integral techniques in boundary element analysis. Intl J. Numer. Meth. Engng 36, 2643–2659.Google Scholar
Ishikawa, T. 2009 Suspension biomechanics of swimming microbes. J. R. Soc. Interface 6, 815–834.Google Scholar
Ishikawa, T. & Hota, M. 2006 Interaction of two swimming paramecia. J. Expl Biol. 209, 4452–4463.Google Scholar
Ishikawa, T. & Pedley, T. J. 2008 Coherent structures in monolayers of swimming particles. Phys. Rev. Lett. 100, 088103.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
Jana, S., Um, S. H. & Jung, S. 2012 Paramecium swimming in capillary tube. Phys. Fluids 24, 041901.Google Scholar
Katz, D. F. 1974 On the propulsion of micro-organisms near solid boundaries. J. Fluid Mech. 64, 33–49.Google Scholar
Katz, D. F. 1975 On the movement of slender bodies near plane boundaries at low Reynolds number. J. Fluid Mech. 72, 529–540.Google Scholar
Kaya, T. & Koser, H. 2012 Direct upstream motility in Escherichia coli . Biophys. J. 102, 1514–1523.CrossRefGoogle ScholarPubMed
Lauga, E. 2007 Propulsion in a viscoelastic fluid. Phys. Fluids 19, 083104.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, 400–412.CrossRefGoogle ScholarPubMed
Lauga, E. & Powers, T. R. 2009 The hydrodynamics of swimming microorganisms. Rep. Prog. Phys. 72, 096601.Google Scholar
Leonardo, R. Di, Dell’Arciprete, R., Angelani, L. & Iebba, V. 2011 Swimming with an image. Phys. Rev. Lett. 106, 038101.CrossRefGoogle ScholarPubMed
Leshansky, A. M. 2009 Enhanced low-Reynolds-number propulsion in heterogeneous viscous environments. Phys. Rev. E 80, 051911.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, 109–118.CrossRefGoogle Scholar
Lighthill, J. 1975 Mathematical Biofluiddynamics. SIAM.Google Scholar
Lighthill, J. 1976 Flagellar hydrodynamics – the John von Neumann lecture, 1975. SIAM Rev. 18, 161–230.CrossRefGoogle Scholar
Lin, Z., Thiffeault, J. & Childress, S. 2011 Stirring by squirmers. J. Fluid Mech. 669, 167–177.CrossRefGoogle Scholar
Liu, B., Powers, T. R. & Breuer, K. S. 2011 Force-free swimming of a model helical flagellum in viscoelastic fluids. Proc. Natl Acad. Sci. USA 108, 19516–19520.Google Scholar
Magar, V., Goto, T. & Pedley, T. J. 2003 Nutrient uptake by a self-propelled steady squirmer. Q. J. Mech. Appl. Maths 56, 65–91.CrossRefGoogle Scholar
Magariyama, Y. & Kudo, S. 2002 A mathematical explanation of an increase in bacterial swimming speed with viscosity in linear-polymer solutions. Biophys. J. 83, 733–739.Google Scholar
Nakamura, S., Adachi, Y., Goto, T. & Magariyama, Y. 2006 Improvement in motion efficiency of the spirochete Brachyspira pilosicoli in viscous environments. Biophys. J. 90, 3019–3026.Google Scholar
Nguyen, H., Ortiz, R., Cortez, R. & Fauci, L. 2011 The action of waving cylindrical rings in a viscous fluid. J. Fluid Mech. 671, 574–586.CrossRefGoogle 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 24, 487–490.Google Scholar
Pozrikidis, C. 2002 A Practical Guide to Boundary Element Methods with the Software Library BEMLIB, 1st edn. CRC Press.Google Scholar
Pozrikidis, C. 2005 Computation of stokes flow due to motion or presence of a particle in a tube. J. Engng Maths 53, 1–20.Google Scholar
Purcell, E. M. 1977 Life at low Reynolds number. Am. J. Phys. 45, 3–11.Google Scholar
Ramia, M., Tullock, D. L. & Phan-Thien, N. 1993 The role of hydrodynamic interaction in the locomotion of microorganisms. Biophys. J. 65, 755–778.CrossRefGoogle ScholarPubMed
Shen, X. N. & Arratia, P. E. 2011 Undulatory swimming in viscoelastic fluids. Phys. Rev. Lett. 106, 208101.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. A 466, 1725–1748.Google Scholar
Smith, D. J. 2009 A boundary element regularized stokeslet method applied to cilia- and flagella-driven flow. Proc. R. Soc. A 465, 3605–3626.CrossRefGoogle 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
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
Stocker, R. & Durham, W. M. 2009 Tumbling for stealth? Science 24, 400–402.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, 248101.Google Scholar
Wen, P. H., Aliabadi, M. H. & Wang, W. 2007 Movement of a spherical cell in capillaries using a boundary element method. J. Biomech. 40, 1786–1793.Google Scholar
Winet, H. 1973 Wall drag on free-moving ciliated micro-organisms. J. Expl Biol. 59, 753–766.Google Scholar
Yates, G. T. 1986 How microorganisms move through water: the hydrodynamics of ciliary and flagellar propulsion reveal how microorganisms overcome the extreme effect of the viscosity of water. Am. Sci. 74, 358–365.Google Scholar
Yizhar, O. & Richard, M. M. 2009 Dynamics and stability of a class of low Reynolds number swimmers near a wall. Phys. Rev. E 79, 045302.Google Scholar
Zhu, L., Do-Quang, M., Lauga, E. & Brandt, L. 2011 Locomotion by tangential deformation in a polymeric fluid. Phys. Rev. E 83, 011901.CrossRefGoogle Scholar
Zhu, L., Lauga, E. & Brandt, L. 2012 Self-propulsion in viscoelastic fluids: pushers vs. pullers. Phys. Fluids 24, 051902.Google Scholar
Zöttl, A. & Stark, H. 2012 Nonlinear dynamics of a microswimmer in Poiseuille flow. Phys. Rev. Lett. 108, 218104.Google Scholar

Zhu et al. supplementary movie

'A neutral squirmer swimming in a curved pipe'.

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Zhu et al. supplementary movie

'A neutral squirmer following the helical trajectory in a straight pipe'

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