Title:

The reciprocal theorem and swimmer interactions

We present a number of solutions for the hydrodynamic interaction between microscopic swimmers in a viscous fluid and confining geometries. The reciprocal theorem is adapted for this use, allowing existing solutions for Stokes drag problems to be used to calculate the motion and rotation of forcefree swimmers as well as other aspects of the hydrodynamics, such as flow fields. We outline the general procedure for approximating the reciprocal theorem to calculate motion for an arbitrary slip velocity by exploiting existing solutions for point forces and point torques in Stokes flows. This is demonstrated with two examples: firstly, the commonly studied case of a swimmer in the presence of an infinite wall, where we find the reported circling of certain bacteria near a surface, and reproduce the equations of motion for a swimmer in the presence of a wall found by other means; and secondly, a calculation giving the leading contributions to the motion of a swimmer between two infinite parallel plates, representing a strongly confining geometry, and relying upon the derivation of the flow due to a point torque in this geometry, a new result. We then derive exact solutions in two and three dimensions. In two dimensions we find the equations of motion for a circular squirmer with arbitrary axisymmetric slip velocity near a plane wall or inside a circular cavity, and discuss the extension to the case of two squirmers interacting with each other, which presents some additional mathematical difficulties. In three dimensions we provide exact solutions for the axisymmetric motion of a squirming sphere close to a noslip surface, both planar and spherical. These allow the hydrodynamic interactions of swimming microscopic organisms with confining boundaries, or 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 find that the circling motion of flagellated bacteria generically has opposite sense at free surfaces and at solid boundaries, as seen experimentally. By comparing these to asymptotic approximations of the interaction we find that the transition from nearto farfield behaviour occurs at a separation of about two swimmer diameters. Finally we discuss possible extensions to this work, and limitations of the approach used.
