Optimal slip velocities of micro-swimmers with arbitrary axisymmetric shapes

This article presents a computational approach for determining the optimal slip velocities on any given shape of an axisymmetric micro-swimmer suspended in a viscous fluid. The objective is to minimize the power loss to maintain a target swimming speed, or equivalently to maximize the efficiency of the micro-swimmer. Owing to the linearity of the Stokes equations governing the fluid motion, we show that this PDE-constrained optimization problem can be reduced to a simpler quadratic optimization problem, which we solve using a high-order accurate boundary integral method. We consider various families of shapes parameterized by the reduced volume and compute their swimming efficiency. We found that for a given reduced volume, prolate ellipsoids are the most efficient micro-swimmer shapes and that, irrespective of the shape, the optimal slip always corresponds to a neutral swimmer (as opposed to a pusher or a puller).