Matched asymptotics for a treadmilling low-Reynolds-number swimmer near a wall

The asymptotic equations governing the evolution of a small circular treadmilling swimmer in a two-dimensional fluid confined by no-slip walls in the Stokes régime are derived correct to third order in the swimmer radius. The analysis provides the appropriate asymptotic corrections to a point singularity model of the swimmer recently proposed by Crowdy and Or [Phys. Rev. E.81 (2010), p. 036313]. The new equations are used to study the motion of a treadmiller in a circular tank including the case where an internal circular wall is present.