Purcell swimmer near a wall

We study the effects of hydrodynamic interactions between a wall and a Purcell three-link swimmer in the case of motion in a plane. We extend previous theoretical studies of the three-link swimmer based on resistive-force theory by taking into account drag coefficients that are modified by the presence of a no-slip wall, considering the case where the distance to the wall is much less than the length of the swimmer and much larger than the thickness of the links. After deriving the equations of motion, we show, by means of criteria from Geometric Control Theory, that the system is locally controllable at configurations that are nearly parallel to the wall. Our result is obtained for a three-link swimmer in which the outer links have equal length and the central one differs by a factor λ> 0: controllability is achieved for all values of λ. For initially horizontal swimmers, the analytical displacements obtained from Lie brackets agree with the ones computed numerically with corresponding control gaits; moreover, we find values of λ that maximize the horizontal components of displacement. Numerical calculations are used to assess the effects of an approximation to the wall-induced drag correction used in the controllability proof. Furthermore, we numerically study first- and second-order controls applied to swimmers that are slightly tilted with respect to the wall and determine which components of net displacement depend on the orientation of the swimmer relative to the wall.