Skip to main content
    • Aa
    • Aa

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

  • Saverio E. Spagnolie (a1) (a2) and Eric Lauga (a2)

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.

Corresponding author
Email addresses for correspondence:,
Linked references
Hide All

This list contains references from the content that can be linked to their source. For a full set of references and notes please see the PDF or HTML where available.

1.J. Ainley , S. Durkin , R. Embid , P. Biondala & R. Cortez 2008 The method of images for regularized Stokeslets. J. Comput. Phys. 227, 46004616.

3.M. A. Bees & O. A. Croze 2010 Dispersion of biased swimming micro-organisms in a fluid flowing through a tube. Proc. R. Soc. Lond. A 466, 20572077.

4.H. C. Berg & L. Turner 1990 Chemotaxis of bacteria in glass capillary arrays. Biophys. J. 58, 919930.

5.A. P. Berke , L. Turner , H. C. Berg & E. Lauga 2008 Hydrodynamic attraction of swimming microorganisms by surfaces. Phys. Rev. Lett. 101, 038102.

7.J. R. Blake & A. T. Chwang 1974 Fundamental singularities of viscous flow. J. Engng Maths 8, 2329.

8.E. L. Bouzarth & M. L. Minion 2011 Modeling slender bodies with the method of regularized Stokeslets. J. Comput. Phys. 230, 39293947.

13.L. Cisneros , C. Dombrowski , R. E. Goldstein & J. O. Kessler 2007 Reversal of bacteria at an obstacle. Phys. Rev. E 73, 030901(R).

14.R. Cortez 2002 The method of regularized Stokeslets. SIAM J. Sci. Comput. 23, 12041225.

15.D. G. Crowdy 2011 Treadmilling swimmers near a no-slip wall at low Reynolds number. Intl J. Non-Linear Mech. 46, 577585.

16.D. G. Crowdy & Y. Or 2010 Two-dimensional point singularity model of a low-Reynolds-number swimmer near a wall. Phys. Rev. E 81, 036313.

17.R. Di Leonardo , L. Angelani , D. Dell’Arciprete , G. Ruocco , V. Iebba , S. Schippa , M. P. Conte , F. Mecarini , F. De Angelis & E. Di Fabrizio 2010 Bacterial ratchet motors. Proc. Natl. Acad. Sci. USA 107, 95419545.

18.R. Di Leonardo , D. Dell’Arciprete , L. Angelani & V. Iebba 2011 Swimming with an image. Phys. Rev. Lett. 106, 038101.

19.K. Drescher , J. Dunkel , L. H. Cisneros , S. Ganguly & R. E. Goldstein 2011 Fluid dynamics and noise in bacterial cell-cell and cell-surface scattering. Proc. Natl Acad. Sci. USA 108, 1094010945.

20.K. Drescher , R. E. Goldstein , N. Michel , M. Polin & I. Tuval 2010 Direct measurement of the flow field around swimming microorganisms. Phys. Rev. Lett. 105, 168101.

21.K. Drescher , K. C. Leptos , I. Tuval , T. Ishikawa , T. J. Pedley & R. E. Goldstein 2009 Dancing volvox: hydrodynamic bound states of swimming algae. Phys. Rev. Lett. 102, 168101.

22.R. Dreyfus , J. Baudry , M. L. Roper , M. Fermigier , H. A. Stone & J. Bibette 2005 Microscopic artificial swimmers. Nature 437, 862865.

23.J. Elgeti & G. Gompper 2009 Self-propelled rods near surfaces. Euro. Phys. Lett. 85, 38002.

24.J. Elgeti , U. B. Kaupp & G. Gompper 2010 Hydrodynamics of sperm cells near surfaces. Biophys. J. 99, 10181026.

25.A. Evans & E. Lauga 2010 Propulsion by passive filaments and active flagella near boundaries. Phys. Rev. E 82, 041915.

26.L. J. Fauci & A. McDonald 1995 Sperm motility in the presence of boundaries. Bull. Math. Biol. 57, 679699.

28.P. Galajda , J. Keymer , P. Chaikin & R. Austin 2007 A wall of funnels concentrates swimming bacteria. J. Bacteriol. 189, 87048707.

29.A. Ghosh & P. Fischer 2009 Controlled propulsion of artificial magnetic nanostructured propellers. Nanoletters 9, 22432245.

30.D. Giacché , T. Ishikawa & T. Yamaguchi 2010 Hydrodynamic entrapment of bacteria swimming near a solid surface. Phys. Rev. E 82, 056309.

31.A. J. Goldman , R. G. Cox & H. Brenner 1966 Slow viscous motion of a sphere parallel to a plane wall – I Motion through a quiescent fluid. Chem. Engng Sci. 22, 637651.

32.R. Golestanian , T. B. Liverpool & A. Adjari 2007 Designing phoretic micro- and nano-swimmers. New J. Phys. 9, 126.

33.T. Goto , K. Nakata , K. Baba , M. Nishimura & Y. Magariyama 2005 A fluid-dynamic interpretation of the asymmetric motion of singly flagellated bacteria swimming close to a boundary. Biophys. J. 89, 37713779.

35.J. S. Guasto , K. A. Johnson & J. P. Gollub 2010 Oscillatory flows induced by microorganisms swimming in two dimensions. Phys. Rev. Lett. 105, 168102.

38.R. M. Harshey 2003 Bacterial motility on a surface: many ways to a common goal. Ann. Rev. Microbiol. 57, 249273.

40.J. P. Hernandez-Ortiz , C. G. Stoltz & M. D. Graham 2005 Transport and collective dynamics in suspensions of confined swimming particles. Phys. Rev. Lett. 95, 204501.

42.J. Hill , O. Kalkanci , J. L. McMurry & H. Koser 2007 Hydrodynamic surface interactions enable Escherichia Coli to seek efficient routes to swim upstream. Phys. Rev. Lett. 98, 068101.

43.C. Hohenegger & M. J. Shelley 2010 Stability of active suspensions. Phys. Rev. E 81, 046311.

44.T. Ishikawa & T. J. Pedley 2007 Diffusion of swimming model micro-organisms in a semi-dilute suspension. J. Fluid Mech. 588, 437462.

46.H.-R. Jiang , N. Yoshinaga & M. Sano 2010 Active motion of a Janus particle by self-thermophoresis in a defocused laser beam. Phys. Rev. Lett. 105, 268302.

47.R. E. Johnson & C. J. Brokaw 1979 Flagellar hydrodynamics. A comparison between resistive-force theory and slender-body theory. Biophys. J. 25, 113127.

48.A. Kanevsky , M. J. Shelley & A.-K. Tornberg 2010 Modelling simple locomotors in Stokes flow. J. Comput. Phys. 229, 958977.

49.T. Kaya & H. Koser 2009 Characterization of hydrodynamic surface interactions of Escherichia coli cell bodies in shear flow. Phys. Rev. Lett. 103, 138103.

51.S. Kim & S. J. Karrila 1991 Microhydrodynamics: Principles and Selected Applications. Dover.

52.I. Klapper & J. Dockery 2010 Mathematical description of microbial biofilms. SIAM Rev. 52, 221265.

53.R. Kolter & E. P. Greenberg 2006 The superficial life of microbes. Nature 441, 300302.

54.E. Lauga , W. R. DiLuzio , G. M. Whitesides & H. A. Stone 2006 Swimming in circles: Motion of bacteria near solid boundaries. Biophys. J. 90, 400412.

55.E. Lauga & T. Powers 2009 The hydrodynamics of swimming microorganisms. Rep. Prog. Phys. 72, 096601.

56.G. Li , L.-K. Tam & J. X. Tang 2008 Accumulation of microswimmers near a surface mediated by collision and rotational Brownian motion. Phys. Rev. Lett. 103, 078101.

57.G. Li & J. X. Tang 2009 Amplified effect of Brownian motion in bacterial near-surface swimming. Proc. Natl Acad. Sci. USA 105, 1835518359.

58.Q. Liao , G. Subramanian , M. P. DeLisa , D. L. Koch & M. Wu 2007 Pair velocity correlations among swimming Escherichia coli bacteria are determined by force-quadrupole hydrodynamic interactions. Phys. Fluids 19, 061701.

59.M. J. Lighthill 1952 On the squirming motion of nearly spherical deformable bodies through liquids at very small Reynolds numbers. Commun. Pure Appl. Maths 5, 109118.

60.M. J. Lighthill 1996 Helical distributions of Stokeslets. J. Engng Maths 30, 3578.

62.I. Llopis & I. Pagonabarraga 2011 Hydrodynamic interactions in squirmer motion: Swimming with a neighbour and close to a wall. J. Non-Newtonian Fluid Mech. 165, 946952.

64.J. Magnaudet , S. Takagi & D. Legendre 2003 Drag, deformation and lateral migration of a buoyant drop moving near a wall. J. Fluid Mech. 476, 115157.

65.S. Michelin & E. Lauga 2010 Efficiency optimization and symmetry-breaking in a model of ciliary locomotion. Phys. Fluids 22, 111901.

66.G. Otoole , H. B. Kaplan & R. Kolter 2000 Biofilm formation as microbial development. Annu. Rev. Microbiol. 54, 4979.

67.O. S. Pak , W. Gao , J. Wang & E. Lauga 2011 High-speed propulsion of flexible nanowire motors: theory and experiments. Soft Matt. 7, 81698181.

68.W. F. Paxton , K. C. Kistler , C. C. Olmeda , A. Sen , S. K. St. Angelo , Y. Cao , T. E. Mallouk , P. E. Lammert & V. H. Crespi 2004 Catalytic nanomotors: autonomous movement of striped nanorods. J. Am. Chem. Soc. 126, 13424.

69.A. T. Poortinga , R. Bos , W. Norde & H. J. Busscher 2002 Electric double layer interactions in bacterial adhesion to surfaces. Surf. Sci. Rep. 47, 132.

70.H. Power & G. Miranda 1987 Second kind integral equation formulation of Stokes’ flows past a particle of arbitrary shape. SIAM J. Appl. Maths 47, 689698.

72.E. M. Purcell 1977 Life at Low Reynolds number. Am. J. Phys. 45, 311.

73.L. J. Rothschild 1963 Non-random distribution of bull spermatazoa in a drop of sperm suspension. Nature (London) 198, 1221222.

74.G. Rückner & R. Kapral 2007 Chemically powered nanodimers. Phys. Rev. Lett. 98, 150603.

75.D. Saintillan & M. J. Shelley 2008 Instabilities and pattern formation in active particle suspensions: kinetic theory and continuum simulation. Phys. Rev. Lett. 100, 178103.

76.H. Shum , E. A. Gaffney & D. J. Smith 2010 Modelling bacterial behaviour close to a no-slip plane boundary: the influence of bacterial geometry. Proc. R. Soc. Lond. A 466, 17251748.

79.S. E. Spagnolie & E. Lauga 2010 Jet propulsion without inertia. Phys. Fluids 22, 081902.

81.G. I. Taylor 1951 Analysis of the swimming of microscopic organisms. Proc. R. Soc. Lond. A 209, 447461.

82.I. Tuval , L. Cisneros , C. Dombrowski , C. W. Wolgemuth , J. O. Kessler & R. E. Goldstein 2005 Bacterial swimming and oxygen transport near contact lines. Proc. Natl. Acad. Sci. USA 102, 22772282.

84.J. Wang 2009 Can man-made nanomachines compete with nature biomotors? ACS Nano 3, 49.

85.R. Zargar , A. Najafi & M. Miri 2009 Three-sphere low-Reynolds-number swimmer near a wall. Phys. Rev. E 80, 026308.

86.L. Zhu , M. Do-Quang , E. Lauga & L. Brandt 2011 Locomotion by tangential deformation in a polymeric fluid. Phys. Rev. E 83, 011901.

Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Journal of Fluid Mechanics
  • ISSN: 0022-1120
  • EISSN: 1469-7645
  • URL: /core/journals/journal-of-fluid-mechanics
Please enter your name
Please enter a valid email address
Who would you like to send this to? *



Altmetric attention score