Skip to main content Accessibility help

Exact solutions for hydrodynamic interactions of two squirming spheres

  • Dario Papavassiliou (a1) and Gareth P. Alexander (a1)


We provide exact solutions of the Stokes equations for a squirming sphere close to a no-slip surface, both planar and spherical, and for the interactions between two squirmers, in three dimensions. These allow the hydrodynamic interactions of swimming microscopic organisms with confining boundaries, or with each other, to be determined for arbitrary separation and, in particular, in the close proximity regime where approximate methods based on point-singularity descriptions cease to be valid. We give a detailed description of the circular motion of an arbitrary squirmer moving parallel to a no-slip spherical boundary or flat free surface at close separation, finding that the circling generically has opposite sense at free surfaces and at solid boundaries. While the asymptotic interaction is symmetric under head–tail reversal of the swimmer, in the near field, microscopic structure can result in significant asymmetry. We also find the translational velocity towards the surface for a simple model with only the lowest two squirming modes. By comparing these to asymptotic approximations of the interaction we find that the transition from near- to far-field behaviour occurs at a separation of approximately two swimmer diameters. These solutions are for the rotational velocity about the wall normal, or common diameter of two spheres, and the translational speed along that same direction, and are obtained using the Lorentz reciprocal theorem for Stokes flows in conjunction with known solutions for the conjugate Stokes drag problems, the derivations of which are demonstrated here for completeness. The analogous motions in the perpendicular directions, i.e. parallel to the wall, currently cannot be calculated exactly since the relevant Stokes drag solutions needed for the reciprocal theorem are not available.

  • View HTML
    • Send article to Kindle

      To send this article to your Kindle, first ensure is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

      Note you can select to send to either the or variations. ‘’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

      Find out more about the Kindle Personal Document Service.

      Exact solutions for hydrodynamic interactions of two squirming spheres
      Available formats

      Send article to Dropbox

      To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

      Exact solutions for hydrodynamic interactions of two squirming spheres
      Available formats

      Send article to Google Drive

      To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

      Exact solutions for hydrodynamic interactions of two squirming spheres
      Available formats


This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (, which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.

Corresponding author

Email address for correspondence:


Hide All
Berg, H. C. 2000 Motile behavior of bacteria. Phys. Today 53 (1), 2430.
Berke, A. P., Turner, L., Berg, H. C. & Lauga, E. 2008 Hydrodynamic attraction of swimming microorganisms by surfaces. Phys. Rev. Lett. 101, 038102.
Blake, J. R. 1971a Self propulsion due to oscillations on the surface of a cylinder at low Reynolds number. Bull. Austral. Math. Soc. 5 (02), 255264.
Blake, J. R. 1971b A spherical envelope approach to ciliary propulsion. J. Fluid Mech. 46 (1), 199208.
Blake, J. R. & Chwang, A. T. 1974 Fundamental singularities of viscous flow. J. Engng Maths 8 (1), 2329.
Brenner, H. 1961 The slow motion of a sphere through a viscous fluid towards a plane surface. Chem. Engng Sci. 16 (3), 242251.
Brenner, H. 1964 Slow viscous rotation of an axisymmetric body within a circular cylinder of finite length. Appl. Sci. Res. 13 (1), 81120.
Brumley, D. R., Wan, K. Y., Polin, M. & Goldstein, R. E. 2014 Flagellar synchronization through direct hydrodynamic interactions. eLife 3, e02750.
Cates, M. E., Marenduzzo, D., Pagonabarraga, I. & Tailleur, J. 2010 Arrested phase separation in reproducing bacteria creates a generic route to pattern formation. Proc. Natl Acad. Sci. USA 107 (26), 1171511720.
Cisneros, L. H., Kessler, J. O., Ganguly, S. & Goldstein, R. E. 2011 Dynamics of swimming bacteria: transition to directional order at high concentration. Phys. Rev. E 83 (6), 061907.
Contino, M., Lushi, E., Tuval, I., Kantsler, V. & Polin, M. 2015 Microalgae scatter off solid surfaces by hydrodynamic and contact forces. Phys. Rev. Lett. 115 (25), 258102.
Cox, R. G. & Brenner, H. 1967 The slow motion of a sphere through a viscous fluid towards a plane surface – II. Small gap widths, including inertial effects. Chem. Engng Sci. 22 (12), 17531777.
Crowdy, D. G. 2011 Treadmilling swimmers near a no-slip wall at low Reynolds number. Intl J. Non-linear Mech. 46 (4), 577585.
Crowdy, D. G. 2013 Wall effects on self-diffusiophoretic Janus particles: a theoretical study. J. Fluid Mech. 735, 473498.
Crowdy, D. G., Lee, S., Samson, O., Lauga, E. & Hosoi, A. E. 2011 A two-dimensional model of low-Reynolds number swimming beneath a free surface. J. Fluid Mech. 681, 2447.
Davis, A. M. J. & Crowdy, D. G. 2015 Matched asymptotics for a spherical low-Reynolds-number treadmilling swimmer near a rigid wall. IMA J. Appl. Maths 80 (3), 634650.
Dean, W. R. & O’Neill, M. E. 1963 A slow motion of viscous liquid caused by the rotation of a solid sphere. Mathematika 10 (01), 1324.
Denissenko, P., Kantsler, V., Smith, D. J. & Kirkman-Brown, J. 2012 Human spermatozoa migration in microchannels reveals boundary-following navigation. Proc. Natl Acad. Sci. USA 109 (21), 80078010.
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 (21), 95419545.
Di Leonardo, R., Dell’Arciprete, D., Angelani, L. & Iebba, V. 2011 Swimming with an image. Phys. Rev. Lett. 106 (3), 038101.
DiLuzio, W. R., Turner, L., Mayer, M., Garstecki, P., Weibel, D. B., Berg, H. C. & Whitesides, G. M. 2005 Escherichia coli swim on the right-hand side. Nature 435 (7046), 12711274.
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 (9), 098103.
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), 1094010945.
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.
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.
Dreyfus, R., Baudry, J., Roper, M. L., Fermigier, M., Stone, H. A. & Bibette, J. 2005 Microscopic artificial swimmers. Nature 437 (7060), 862865.
Dunkel, J., Heidenreich, S., Drescher, K., Wensink, H. H., Bär, M. & Goldstein, R. E. 2013 Fluid dynamics of bacterial turbulence. Phys. Rev. Lett. 110 (22), 228102.
Ehlers, K. M., Samuel, A. D., Berg, H. C. & Montgomery, R. 1996 Do cyanobacteria swim using traveling surface waves? Proc. Natl Acad. Sci. USA 93 (16), 83408343.
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), 61956199.
Goldman, A. J., Cox, R. G. & Brenner, H. 1966 The slow motion of two identical arbitrarily oriented spheres through a viscous fluid. Chem. Engng Sci. 21, 11511170.
Goldman, A. J., Cox, R. G. & Brenner, H. 1967a Slow viscous motion of a sphere parallel to a plane wall – I. Motion through a quiescent fluid. Chem. Engng Sci. 22 (4), 637651.
Goldman, A. J., Cox, R. G. & Brenner, H. 1967b Slow viscous motion of a sphere parallel to a plane wall – II. Couette flow. Chem. Engng Sci. 22 (4), 653660.
Golestanian, R., Liverpool, T. B. & Adjari, A. 2005 Propulsion of a molecular machine by asymmetric distribution of reaction products. Phys. Rev. Lett. 94, 220801.
Golestanian, R., Liverpool, T. B. & Adjari, A. 2007 Designing phoretic micro-and nano-swimmers. New J. Phys. 9, 126.
Happel, J. & Brenner, H. 1983 Low Reynolds Number Hydrodynamics. Martinus Nijhoff.
Happel, J. & Pfeffer, R. 1960 The motion of two spheres following each other in a viscous fluid. AIChE J. 6, 129133.
Higdon, J. J. L. 1979 The generation of feeding currents by flagellar motions. J. Fluid Mech. 94 (02), 305330.
Howse, J. R., Jones, R. A. L., Ryan, A. J., Gough, T., Vafabakhsh, R. & Golestanian, R. 2007 Self-motile colloidal particles: from directed propulsion to random walk. Phys. Rev. Lett. 99 (4), 048102.
Ishikawa, T., Simmonds, M. P. & Pedley, T. J. 2006 Hydrodynamic interaction of two swimming model micro-organisms. J. Fluid Mech. 568, 119160.
Jeffery, G. B. 1912 On a form of the solution of Laplace’s equation suitable for problems relating to two spheres. Proc. R. Soc. Lond. A 109120.
Jeffery, G. B. 1915 On the steady rotation of a solid of revolution in a viscous fluid. Proc. Lond. Math. Soc. 2 (1), 327338.
Jeffery, G. B. 1922 The rotation of two circular cylinders in a viscous fluid. Proc. R. Soc. Lond. A 101 (709), 169174.
Jeffrey, D. J. & Onishi, Y. 1981 The slow motion of a cylinder next to a plane wall. Q. J. Mech. Appl. Maths 34 (2), 129137.
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), 11871192.
Kanwal, R. P. 1961 Slow steady rotation of axially symmetric bodies in a viscous fluid. J. Fluid Mech. 10 (01), 1724.
Kim, S. & Karrila, S. J. 2013 Microhydrodynamics: Principles and Selected Applications. Courier Corporation.
Koumakis, N., Lepore, A., Maggi, C. & Di Leonardo, R. 2013 Targeted delivery of colloids by swimming bacteria. Nat. Commun. 4, 2588.
Lauga, E. 2014 Locomotion in complex fluids: integral theorems. Phys. Fluids 26 (8), 081902.
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), 400412.
Lauga, E. & Powers, T. R. 2009 The hydrodynamics of swimming microorganisms. Rep. Prog. Phys. 72 (9), 096601.
Leshansky, A. M. & Kenneth, O. 2008 Surface tank treading: propulsion of Purcell’s toroidal swimmer. Phys. Fluids 20 (6), 063104.
Leshansky, A. M., Kenneth, O., Gat, O. & Avron, J. E. 2007 A frictionless microswimmer. New J. Phys. 9 (5), 145.
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.
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 (1052), 109118.
Liron, N. & Mochon, S. 1976 Stokes flow for a Stokeslet between two parallel flat plates. J. Engng Maths 10 (4), 287303.
Liron, N. & Shahar, R. 1978 Stokes flow due to a Stokeslet in a pipe. J. Fluid Mech. 86 (04), 727744.
Lopez, D. & Lauga, E. 2014 Dynamics of swimming bacteria at complex interfaces. Phys. Fluids 26 (7), 071902.
Magariyama, Y., Ichiba, M., Nakata, K., Baba, K., Ohtani, T., Kudo, S. & Goto, T. 2005 Difference in bacterial motion between forward and backward swimming caused by the wall effect. Biophys. J 88 (5), 36483658.
Magariyama, Y., Sugiyama, S. & Kudo, S. 2001 Bacterial swimming speed and rotation rate of bundled flagella. FEMS Microbiol. Lett. 199 (1), 125129.
Majumdar, S. R. & O’Neill, M. E. 1977 On axisymmetric Stokes flow past a torus. Z. Angew. Math. Phys. 28 (4), 541550.
Maleček, K. & Nádeník, Z. 2001 On the inductive proof of Legendre addition theorem. Stud. Geophys. Geod. 45 (1), 111.
Marchetti, M. C., Joanny, J. F., Ramaswamy, S., Liverpool, T. B., Prost, J., Rao, M. & Simha, R. A. 2013 Hydrodynamics of soft active matter. Rev. Mod. Phys. 85 (3), 11431189.
Masoud, H. & Stone, H. A. 2014 A reciprocal theorem for Marangoni propulsion. J. Fluid Mech. 741, R4.
Michelin, S. & Lauga, E. 2015a Autophoretic locomotion from geometric asymmetry. Eur. Phys. J. E 38 (2), 116.
Michelin, S. & Lauga, E. 2015b A reciprocal theorem for boundary-driven channel flows. Phys. Fluids 27 (11), 111701.
Mozaffari, A., Sharifi-Mood, N., Koplik, J. & Maldarelli, C. 2016 Self-diffusiophoretic colloidal propulsion near a solid boundary. Phys. Fluids 28 (5), 053107.
O’Neill, M. E. 1964 A slow motion of viscous liquid caused by a slowly moving solid sphere. Mathematika 11 (01), 6774.
O’Neill, M. E. & Majumdar, R. 1970a Asymmetrical slow viscous fluid motions caused by the translation or rotation of two spheres. Part I. The determination of exact solutions for any values of the ratio of radii and separation parameters. Z. Angew. Math. Phys. 21 (2), 164179.
O’Neill, M. E. & Majumdar, S. R. 1970b Asymmetrical slow viscous fluid motions caused by the translation or rotation of two spheres. Part II. Asymptotic forms of the solutions when the minimum clearance between the spheres approaches zero. Z. Angew. Math. Phys. 21 (2), 180187.
O’Neill, M. E. & Ranger, K. B. 1979 On the rotation of a rotlet or sphere in the presence of an interface. Intl J. Multiphase Flow 5 (2), 143148.
O’Neill, M. E. & Stewartson, K. 1967 On the slow motion of a sphere parallel to a nearby plane wall. J. Fluid Mech. 27 (04), 705724.
O’Toole, G., Kaplan, H. B. & Kolter, R. 2000 Biofilm formation as microbial development. Ann. Rev. Microbiol. 54 (1), 4979.
Pak, O. S. & Lauga, E. 2014 Generalized squirming motion of a sphere. J. Engng Maths 88 (1), 128.
Papavassiliou, D. & Alexander, G. P. 2015 The many-body reciprocal theorem and swimmer hydrodynamics. Europhys. Lett. 110 (4), 44001.
Paxton, W. F., Sen, A. & Mallouk, T. E. 2005 Motility of catalytic nanoparticles through self-generated forces. Chem. Eur. J. 11 (22), 64626470.
Payne, L. E. & Pell, W. H. 1960 The Stokes flow problem for a class of axially symmetric bodies. J. Fluid Mech. 7 (04), 529549.
Pedley, T. J. & Kessler, J. O. 1992 Hydrodynamic phenomena in suspensions of swimming microorganisms. Annu. Rev. Fluid Mech. 24, 313358.
Popescu, M. N., Tasinkevych, M. & Dietrich, S. 2011 Pulling and pushing a cargo with a catalytically active carrier. Europhys. Lett. 95 (2), 28004.
Pozrikidis, C. 1992 Boundary Integral and Singularity Methods for Linearized Viscous Flow. Cambridge University Press.
Purcell, E. M. 1977 Life at low Reynolds number. Am. J. Phys. 45 (1), 311.
Pushkin, D. O., Shum, H. & Yeomans, J. M. 2013 Fluid transport by individual microswimmers. J. Fluid Mech. 726, 525.
Ramaswamy, S. 2010 The mechanics and statistics of active matter. Annu. Rev. Condens. Matter Phys. 1, 323345.
Rothschild, L. 1963 Non-random distribution of bull spermatozoa in a drop of sperm suspension. Nature 198 (488), 12211222.
Rückner, G. & Kapral, R. 2007 Chemically powered nanodimers. Phys. Rev. Lett. 98 (15), 150603.
Sharifi-Mood, N., Mozaffari, A. & Córdova-Figueroa, U. M. 2016 Pair interaction of catalytically active colloids: from assembly to escape. J. Fluid Mech. 798, 910954.
Sneddon, I. N. 1956 Special Functions of Mathematical Physics and Chemistry. Oliver and Boyd.
Spagnolie, S. E. & Lauga, E. 2012 Hydrodynamics of self-propulsion near a boundary: predictions and accuracy of far-field approximations. J. Fluid Mech. 700, 105147.
Squires, T. M. & Bazant, M. Z. 2004 Induced-charge electro-osmosis. J. Fluid Mech. 509, 217252.
Stimson, M. & Jeffery, G. B. 1926 The motion of two spheres in a viscous fluid. Proc. R. Soc. Lond. A 111, 110116.
Stone, H. A. & Samuel, A. D. T. 1996 Propulsion of microorganisms by surface distortions. Phys. Rev. Lett. 77 (19), 41024104.
Takagi, D., Palacci, J., Braunschweig, A. B., Shelley, M. J. & Zhang, J. 2014 Hydrodynamic capture of microswimmers into sphere-bound orbits. Soft Matt. 10, 17841789.
Taylor, G. I. 1951 Analysis of the swimming of microscopic organisms. Proc. R. Soc. Lond. A 209 (1099), 447461.
Taylor, G. I. 1952 The action of waving cylindrical tails in propelling microscopic organisms. Proc. R. Soc. Lond. A 225239.
Valadares, L. F., Tao, Y.-G., Zacharia, N. S., Kitaev, V., Galembeck, F., Kapral, R. & Ozin, G. A. 2010 Catalytic nanomotors: self-propelled sphere dimers. Small 6 (4), 565572.
Weibel, D. B., Garstecki, P., Ryan, D., Diluzio, W. R., Mayer, M., Seto, J. E. & Whitesides, G. M. 2005 Microoxen: microorganisms to move microscale loads. Proc. Natl Acad. Sci. USA 102 (34), 1196311967.
Whittaker, E. T. & Watson, G. N. 1996 A Course of Modern Analysis. Cambridge University Press.
Wioland, H., Woodhouse, F. G., Dunkel, J., Kessler, J. O. & Goldstein, R. E. 2013 Confinement stabilizes a bacterial suspension into a spiral vortex. Phys. Rev. Lett. 110 (26), 268102.
Wu, X.-L. & Libchaber, A. 2000 Particle diffusion in a quasi-two-dimensional bacterial bath. Phys. Rev. Lett. 84 (13), 30173020.
Zargar, R., Najafi, A. & Miri, M. 2009 Three-sphere low-Reynolds-number swimmer near a wall. Phys. Rev. E 80 (2), 026308.
MathJax is a JavaScript display engine for mathematics. For more information see

JFM classification

Exact solutions for hydrodynamic interactions of two squirming spheres

  • Dario Papavassiliou (a1) and Gareth P. Alexander (a1)


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed