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Exact solutions for hydrodynamic interactions of two squirming spheres

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Papavassiliou, Dario and Alexander, Gareth P. (2017) Exact solutions for hydrodynamic interactions of two squirming spheres. Journal of Fluid Mechanics, 813 . pp. 618-646. doi:10.1017/jfm.2016.837

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Official URL: http://dx.doi.org/10.1017/jfm.2016.837

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Abstract

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.

Item Type: Journal Article
Subjects: Q Science > QC Physics
Divisions: Faculty of Science > Centre for Complexity Science
Faculty of Science > Physics
Library of Congress Subject Headings (LCSH): Fluid dynamics., Reynolds number., Fluid mechanics.
Journal or Publication Title: Journal of Fluid Mechanics
Publisher: Cambridge University Press
ISSN: 0022-1120
Official Date: February 2017
Dates:
DateEvent
February 2017Published
20 January 2017Available
7 December 2016Accepted
Volume: 813
Page Range: pp. 618-646
DOI: 10.1017/jfm.2016.837
Status: Peer Reviewed
Publication Status: Published
Access rights to Published version: Open Access
Funder: Engineering and Physical Sciences Research Council (EPSRC)
Grant number: A.MACX.0002

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