The squirmer model is used to model the swimming of microorganisms with hair-like structures known as cilia on their surface. The model uses a slip velocity defined on the swimmer surface to represent the collective motion of the cilia, resulting in a Dirichlet boundary value problem for the Stokes equations. We use the Helmholtz decomposition to define the slip velocity in terms of tangential basis functions on a surface of arbitrary shape. The reciprocal theorem (Green’s second identity for the Stokes equations) can be used to determine the translational and rotational velocities of the swimmer from a prescribed slip velocity – closed-form expressions for these quantities can be obtained for a spheroidal squirmer.

We then explore an inverse problem – given an arbitrary swimmer shape of spherical topology, which slip profile minimizes power loss? While the space of tangential slip velocity fields is infinite-dimensional, this optimization problem can be reduced to six dimensions. The problem consists of two nested optimizations – a partial minimization in which the direction of net motion of the swimmer is prescribed, followed by a global optimization procedure in which the best net motion direction is determined. We develop a solution algorithm for this optimization problem that involves only 12 auxiliary flow problems which can be solved numerically using boundary integral methods. The optimization results suggest that the competition between linear and rotational optimal motion is linked to symmetries in the swimmer shape.