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Hydrodynamic bound states of a low-Reynolds-number swimmer near a gap in a wall

Published online by Cambridge University Press:  16 November 2010

DARREN CROWDY  and
OPHIR SAMSON
DARREN CROWDY*
Affiliation:
Department of Mathematics, Imperial College, 180 Queen's Gate, London SW7 2AZ, UK
OPHIR SAMSON
Affiliation:
Department of Mathematics, Imperial College, 180 Queen's Gate, London SW7 2AZ, UK
*
†Email address for correspondence: d.crowdy@imperial.ac.uk
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Abstract

The motion of an organism swimming at low Reynolds number near an infinite straight wall with a finite-length gap is studied theoretically within the framework of a two-dimensional model. The swimmer is modelled as a point singularity of the Stokes equations dependent on a single real parameter. A dynamical system governing the position and orientation of the model swimmer is derived in analytical form. The dynamical system is studied in detail and a bifurcation analysis performed. The analysis reveals, inter alia, the presence of stable equilibrium points in the gap region as well as Hopf bifurcations to periodic bound states. The reduced-model system also exhibits a global gluing bifurcation in which two symmetric periodic orbits merge at a saddle point into symmetric ‘figure-of-eight’ bound states having more complex spatiotemporal structure. The additional effect of a background shear is also studied and is found to introduce new types of bound state. The analysis allows us to make theoretical predictions as to the possible behaviour of a low-Reynolds-number swimmer near a gap in a wall. It offers insights into the use of gaps or orifices as possible control devices for such swimmers in confined environments.

JFM classification

Low-Reynolds-number flows: Low-Reynolds-number flows Nonlinear Dynamical Systems: Nonlinear Dynamical Systems
Type
Papers
Information
Journal of Fluid Mechanics , Volume 667 , 25 January 2011 , pp. 309 - 335
Copyright
Copyright © Cambridge University Press 2010

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