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Hydrodynamics of self-propulsion near a boundary: predictions and accuracy of far-field approximations

Published online by Cambridge University Press:  16 April 2012

Saverio E. Spagnolie  and
Eric Lauga
Saverio E. Spagnolie*
Affiliation:
School of Engineering, Brown University, 182 Hope Street, Providence, RI 02912, USA Department of Mechanical and Aerospace Engineering, University of California, San Diego, 9500 Gilman Drive, La Jolla, CA 92093, USA
Eric Lauga
Affiliation:
Department of Mechanical and Aerospace Engineering, University of California, San Diego, 9500 Gilman Drive, La Jolla, CA 92093, USA
*
†Email addresses for correspondence: Saverio_Spagnolie@brown.edu, elauga@ucsd.edu
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Abstract

The swimming trajectories of self-propelled organisms or synthetic devices in a viscous fluid can be altered by hydrodynamic interactions with nearby boundaries. We explore a multipole description of swimming bodies and provide a general framework for studying the fluid-mediated modifications to swimming trajectories. A general axisymmetric swimmer is described as a linear combination of fundamental solutions to the Stokes equations: a Stokeslet dipole, a source dipole, a Stokeslet quadrupole, and a rotlet dipole. The effects of nearby walls or stress-free surfaces on swimming trajectories are described through the contribution of each singularity, and we address the question of how accurately this multipole approach captures the wall effects observed in full numerical solutions of the Stokes equations. The reduced model is used to provide simple but accurate predictions of the wall-induced attraction and pitching dynamics for model Janus particles, ciliated organisms, and bacteria-like polar swimmers. Transitions in attraction and pitching behaviour as functions of body geometry and propulsive mechanism are described. The reduced model may help to explain a number of recent experimental results.

JFM classification

Low-Reynolds-number flows: Low-Reynolds-number flows Biological Fluid Dynamics: Micro-organism dynamics Biological Fluid Dynamics: Swimming/flying
Type
Papers
Information
Journal of Fluid Mechanics , Volume 700 , 10 June 2012 , pp. 105 - 147
Copyright
Copyright © Cambridge University Press 2012

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References

1. Ainley, J., Durkin, S., Embid, R., Biondala, P. & Cortez, R. 2008 The method of images for regularized Stokeslets. J. Comput. Phys. 227, 4600–4616.CrossRefGoogle Scholar
2. Armitage, J. P. & Macnab, R. M. 1987 Unidirectional, intermittent rotation of the flagellum of rhodobacter sphaeroides. J. Bacteriol. 169, 514–518.CrossRefGoogle ScholarPubMed
3. Bees, M. A. & Croze, O. A. 2010 Dispersion of biased swimming micro-organisms in a fluid flowing through a tube. Proc. R. Soc. Lond. A 466, 2057–2077.CrossRefGoogle Scholar
4. Berg, H. C. & Turner, L. 1990 Chemotaxis of bacteria in glass capillary arrays. Biophys. J. 58, 919–930.CrossRefGoogle ScholarPubMed
5. 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
6. Blake, J. R. 1971 A spherical envelope approach to ciliary propulsion. J. Fluid Mech. 46, 199–208.CrossRefGoogle Scholar
7. Blake, J. R. & Chwang, A. T. 1974 Fundamental singularities of viscous flow. J. Engng Maths 8, 23–29.CrossRefGoogle Scholar
8. Bouzarth, E. L. & Minion, M. L. 2011 Modeling slender bodies with the method of regularized Stokeslets. J. Comput. Phys. 230, 3929–3947.CrossRefGoogle Scholar
9. Brady, J. F. & Bossis, G. 1985 The rheology of concentrated suspensions of spheres in simple shear flow by numerical simulation. J. Fluid Mech. 155, 105–129.CrossRefGoogle Scholar
10. Brenner, H. 1961 Effect of finite boundaries on the Stokes resistance of an arbitrary particle. J. Fluid Mech. 8, 35–48.Google Scholar
11. Childress, S. 1981 Mechanics of Swimming and Flying. Cambridge University Press.CrossRefGoogle Scholar
12. 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, 787–815.CrossRefGoogle Scholar
13. Cisneros, L., Dombrowski, C., Goldstein, R. E. & Kessler, J. O. 2007 Reversal of bacteria at an obstacle. Phys. Rev. E 73, 030901(R).CrossRefGoogle Scholar
14. Cortez, R. 2002 The method of regularized Stokeslets. SIAM J. Sci. Comput. 23, 1204–1225.CrossRefGoogle Scholar
15. Crowdy, D. G. 2011 Treadmilling swimmers near a no-slip wall at low Reynolds number. Intl J. Non-Linear Mech. 46, 577–585.CrossRefGoogle Scholar
16. 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.CrossRefGoogle ScholarPubMed
17. Di Leonardo, R., Angelani, L., Dell’Arciprete, D., Ruocco, G., Iebba, V., Schippa, S., Conte, M. P., Mecarini, F., De Angelis, F. & Di Fabrizio, E. 2010 Bacterial ratchet motors. Proc. Natl. Acad. Sci. USA 107, 9541–9545.CrossRefGoogle ScholarPubMed
18. Di Leonardo, R., Dell’Arciprete, D., Angelani, L. & Iebba, V. 2011 Swimming with an image. Phys. Rev. Lett. 106, 038101.CrossRefGoogle ScholarPubMed
19. 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
20. 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.CrossRefGoogle ScholarPubMed
21. 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.CrossRefGoogle ScholarPubMed
22. Dreyfus, R., Baudry, J., Roper, M. L., Fermigier, M., Stone, H. A. & Bibette, J. 2005 Microscopic artificial swimmers. Nature 437, 862–865.CrossRefGoogle ScholarPubMed
23. Elgeti, J. & Gompper, G. 2009 Self-propelled rods near surfaces. Euro. Phys. Lett. 85, 38002.CrossRefGoogle Scholar
24. Elgeti, J., Kaupp, U. B. & Gompper, G. 2010 Hydrodynamics of sperm cells near surfaces. Biophys. J. 99, 1018–1026.CrossRefGoogle ScholarPubMed
25. Evans, A. & Lauga, E. 2010 Propulsion by passive filaments and active flagella near boundaries. Phys. Rev. E 82, 041915.CrossRefGoogle ScholarPubMed
26. Fauci, L. J. & McDonald, A. 1995 Sperm motility in the presence of boundaries. Bull. Math. Biol. 57, 679–699.CrossRefGoogle ScholarPubMed
27. Fournier-Bidoz, S., Arsenault, A. C., Manners, I. & Ozin, G. A. 2005 Synthetic self-propelled nanorotors. Chem. Commun. 4, 441–443.CrossRefGoogle Scholar
28. Galajda, P., Keymer, J., Chaikin, P. & Austin, R. 2007 A wall of funnels concentrates swimming bacteria. J. Bacteriol. 189, 8704–8707.CrossRefGoogle Scholar
29. Ghosh, A. & Fischer, P. 2009 Controlled propulsion of artificial magnetic nanostructured propellers. Nanoletters 9, 2243–2245.CrossRefGoogle ScholarPubMed
30. Giacché, D., Ishikawa, T. & Yamaguchi, T. 2010 Hydrodynamic entrapment of bacteria swimming near a solid surface. Phys. Rev. E 82, 056309.CrossRefGoogle Scholar
31. Goldman, A. J., Cox, R. G. & Brenner, H. 1966 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
32. Golestanian, R., Liverpool, T. B. & Adjari, A. 2007 Designing phoretic micro- and nano-swimmers. New J. Phys. 9, 126.CrossRefGoogle Scholar
33. 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.CrossRefGoogle ScholarPubMed
34. Gray, J. & Hancock, G. J. 1955 The propulsion of sea-urchin spermatozoa. J. Expl. Biol. 32, 802–814.CrossRefGoogle Scholar
35. Guasto, J. S., Johnson, K. A. & Gollub, J. P. 2010 Oscillatory flows induced by microorganisms swimming in two dimensions. Phys. Rev. Lett. 105, 168102.CrossRefGoogle ScholarPubMed
36. Happel, J. & Brenner, H. 1965 Low Reynolds Number Hydrodynamics. Prentice Hall.Google Scholar
37. Harkes, G., Dankert, J. & Feijen, J. 1992 Bacterial migration along solid surfaces. Appl. Environ. Microbiol. 58, 1500–1505.CrossRefGoogle ScholarPubMed
38. Harshey, R. M. 2003 Bacterial motility on a surface: many ways to a common goal. Ann. Rev. Microbiol. 57, 249–273.CrossRefGoogle ScholarPubMed
39. Harvey, R. W. & Young, L. Y. 1980 Enumeration of particle-bound and unattached respiring bacteria in the salt marsh environment. Appl. Environ. Microbiol. 40, 156–160.CrossRefGoogle ScholarPubMed
40. Hernandez-Ortiz, J. P., Stoltz, C. G. & Graham, M. D. 2005 Transport and collective dynamics in suspensions of confined swimming particles. Phys. Rev. Lett. 95, 204501.CrossRefGoogle ScholarPubMed
41. Hernandez-Ortiz, J. P., Underhill, P. T. & Graham, M. D. 2009 Dynamics of confined suspensions of swimming particles. J. Phys.: Condens. Matter 21, 204107.Google ScholarPubMed
42. 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.CrossRefGoogle ScholarPubMed
43. Hohenegger, C. & Shelley, M. J. 2010 Stability of active suspensions. Phys. Rev. E 81, 046311.CrossRefGoogle ScholarPubMed
44. Ishikawa, T. & Pedley, T. J. 2007 Diffusion of swimming model micro-organisms in a semi-dilute suspension. J. Fluid Mech. 588, 437–462.CrossRefGoogle Scholar
45. Ishikawa, T., Simmonds, M. P. & Pedley, T. J. 2006 Hydrodynamic interaction of two swimming model micro-organisms. J. Fluid Mech. 568, 119–160.CrossRefGoogle Scholar
46. Jiang, H.-R., Yoshinaga, N. & Sano, M. 2010 Active motion of a Janus particle by self-thermophoresis in a defocused laser beam. Phys. Rev. Lett. 105, 268302.CrossRefGoogle Scholar
47. Johnson, R. E. & Brokaw, C. J. 1979 Flagellar hydrodynamics. A comparison between resistive-force theory and slender-body theory. Biophys. J. 25, 113–127.CrossRefGoogle ScholarPubMed
48. Kanevsky, A., Shelley, M. J. & Tornberg, A.-K. 2010 Modelling simple locomotors in Stokes flow. J. Comput. Phys. 229, 958–977.CrossRefGoogle Scholar
49. Kaya, T. & Koser, H. 2009 Characterization of hydrodynamic surface interactions of Escherichia coli cell bodies in shear flow. Phys. Rev. Lett. 103, 138103.CrossRefGoogle ScholarPubMed
50. Keller, S. R. & Wu, T. Y. T. 1977 A porous prolate-spheroidal model for ciliated micro-organisms. J. Fluid Mech. 80, 259–278.CrossRefGoogle Scholar
51. Kim, S. & Karrila, S. J. 1991 Microhydrodynamics: Principles and Selected Applications. Dover.Google Scholar
52. Klapper, I. & Dockery, J. 2010 Mathematical description of microbial biofilms. SIAM Rev. 52, 221–265.CrossRefGoogle Scholar
53. Kolter, R. & Greenberg, E. P. 2006 The superficial life of microbes. Nature 441, 300–302.CrossRefGoogle ScholarPubMed
54. 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
55. Lauga, E. & Powers, T. 2009 The hydrodynamics of swimming microorganisms. Rep. Prog. Phys. 72, 096601.CrossRefGoogle Scholar
56. Li, G., Tam, L.-K. & Tang, J. X. 2008 Accumulation of microswimmers near a surface mediated by collision and rotational Brownian motion. Phys. Rev. Lett. 103, 078101.CrossRefGoogle Scholar
57. Li, G. & Tang, J. X. 2009 Amplified effect of Brownian motion in bacterial near-surface swimming. Proc. Natl Acad. Sci. USA 105, 18355–18359.CrossRefGoogle Scholar
58. Liao, Q., Subramanian, G., DeLisa, M. P., Koch, D. L. & Wu, M. 2007 Pair velocity correlations among swimming Escherichia coli bacteria are determined by force-quadrupole hydrodynamic interactions. Phys. Fluids 19, 061701.CrossRefGoogle Scholar
59. 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
60. Lighthill, M. J. 1996 Helical distributions of Stokeslets. J. Engng Maths 30, 35–78.CrossRefGoogle Scholar
61. Lin, Z., Thiffeault, J.-L. & Childress, S. 2011 Stirring by squirmers. J. Fluid Mech. 669, 167–177.CrossRefGoogle Scholar
62. Llopis, I. & Pagonabarraga, I. 2011 Hydrodynamic interactions in squirmer motion: Swimming with a neighbour and close to a wall. J. Non-Newtonian Fluid Mech. 165, 946–952.CrossRefGoogle Scholar
63. Lynch, J. F., Lappin-Scott, H. M. & Costerton, J. W. 2003 Microbial Biofilms. Cambridge University Press.Google Scholar
64. Magnaudet, J., Takagi, S. & Legendre, D. 2003 Drag, deformation and lateral migration of a buoyant drop moving near a wall. J. Fluid Mech. 476, 115–157.CrossRefGoogle Scholar
65. Michelin, S. & Lauga, E. 2010 Efficiency optimization and symmetry-breaking in a model of ciliary locomotion. Phys. Fluids 22, 111901.CrossRefGoogle Scholar
66. Otoole, G., Kaplan, H. B. & Kolter, R. 2000 Biofilm formation as microbial development. Annu. Rev. Microbiol. 54, 49–79.CrossRefGoogle ScholarPubMed
67. Pak, O. S., Gao, W., Wang, J. & Lauga, E. 2011 High-speed propulsion of flexible nanowire motors: theory and experiments. Soft Matt. 7, 8169–8181.CrossRefGoogle Scholar
68. 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, 13424.CrossRefGoogle ScholarPubMed
69. Poortinga, A. T., Bos, R., Norde, W. & Busscher, H. J. 2002 Electric double layer interactions in bacterial adhesion to surfaces. Surf. Sci. Rep. 47, 1–32.CrossRefGoogle Scholar
70. Power, H. & Miranda, G. 1987 Second kind integral equation formulation of Stokes’ flows past a particle of arbitrary shape. SIAM J. Appl. Maths 47, 689–698.CrossRefGoogle Scholar
71. Pozrikidis, C. 1992 Boundary Integral and Singularity Methods for Linearized Viscous Flow. Cambridge University Press.CrossRefGoogle Scholar
72. Purcell, E. M. 1977 Life at Low Reynolds number. Am. J. Phys. 45, 3–11.CrossRefGoogle Scholar
73. Rothschild, L. J. 1963 Non-random distribution of bull spermatazoa in a drop of sperm suspension. Nature (London) 198, 122–1222.CrossRefGoogle Scholar
74. Rückner, G. & Kapral, R. 2007 Chemically powered nanodimers. Phys. Rev. Lett. 98, 150603.CrossRefGoogle ScholarPubMed
75. Saintillan, D. & Shelley, M. J. 2008 Instabilities and pattern formation in active particle suspensions: kinetic theory and continuum simulation. Phys. Rev. Lett. 100, 178103.CrossRefGoogle Scholar
76. 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, 1725–1748.CrossRefGoogle Scholar
77. Smith, D. J. & Blake, J. R. 2009 Surface accumulation of spermatozoa: a fluid dynamic phenomenon. Math. Sci. 34, 74–87.Google Scholar
78. 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.CrossRefGoogle Scholar
79. Spagnolie, S. E. & Lauga, E. 2010 Jet propulsion without inertia. Phys. Fluids 22, 081902.CrossRefGoogle Scholar
80. Swan, J. W. & Khair, A. S. 2008 On the hydrodynamics of ‘slip–stick’ spheres. J. Fluid Mech. 606, 115–132.CrossRefGoogle Scholar
81. Taylor, G. I. 1951 Analysis of the swimming of microscopic organisms. Proc. R. Soc. Lond. A 209, 447–461.Google Scholar
82. Tuval, I., Cisneros, L., Dombrowski, C., Wolgemuth, C. W., Kessler, J. O. & Goldstein, R. E. 2005 Bacterial swimming and oxygen transport near contact lines. Proc. Natl. Acad. Sci. USA 102, 2277–2282.CrossRefGoogle ScholarPubMed
83. Van Loosdrecht, M. C. M., Lyklema, J., Norde, W. & Zehnder, A. J. B. 2003 Influence of interfaces on microbial activity. Microbiol. Rev. 54, 75–87.CrossRefGoogle Scholar
84. Wang, J. 2009 Can man-made nanomachines compete with nature biomotors? ACS Nano 3, 4–9.CrossRefGoogle ScholarPubMed
85. Zargar, R., Najafi, A. & Miri, M. 2009 Three-sphere low-Reynolds-number swimmer near a wall. Phys. Rev. E 80, 026308.CrossRefGoogle ScholarPubMed
86. 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