Treadmilling swimmers near a no-slip wall at low Reynolds number

The motion of a circular treadmilling low Reynolds number swimmer near a no-slip wall is studied analytically. First, the exact solution of Jeffrey and Onishi [Q. J. Mech. Appl. Math., 34 (1981)] for a translating and rotating solid cylinder near a no-slip wall is rederived using a novel conformal mapping approach that differs from the original derivation which employed bipolar coordinates. Then it is shown that this solution can be combined with the reciprocal theorem, and the calculus of residues, to produce an explicit non-linear dynamical system for the treadmilling swimmer's velocity and angular velocity. The resulting non-linear dynamical system governing the swimmer motion is used to corroborate the qualitative results obtained by an approximate model of the same swimmer recently presented in Crowdy and Or [Phys. Rev. E., 81 (2010)].