  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
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 about 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.
