Exact boundary integral solution for the Stokes traction on an active particle

Active particles produce fluid flow around them even when stationary and often this can lead to self-propulsion. Examples include microorganisms and autophoretic colloids. Activity, consisting of non-equilibrium processes at the surface of the particle (e.g., ciliary motion in microorganisms), can be modelled by a velocity boundary condition. The boundary integral formulation of Stokes flow has been used extensively in the dynamics of passive colloidal particles and, more recently, for active particles. It provides the traction on an active particle directly, obviating the need to solve for the fluid flow in the bulk. Using spectral expansions of the traction and the velocity boundary condition in a basis of tensorial spherical harmonics and Ritz-Galerkin discretisation the direct boundary integral equation can be reduced to an infinite-dimensional linear system. We have diagonalised this linear system exactly and obtained the solution for the traction in terms of the velocity boundary condition. We call these relations the generalised Stokes laws­_­. Apart from intrinsic theoretical interest, such a solution is of use in numerical solutions of the boundary integral equation for many particles, where numerical iterations can be initialized with the exact one-particle solution.