Stokes flow due to point torques and sources in a spherical geometry

Abstract

Solutions to the Stokes equations written in terms of a small number of hydrodynamic image singularities have been a useful tool in theoretical and numerical computations for nearly fifty years. In this article, we extend the catalogue of known solutions by deriving the flow expressions due to a general point torque and point source in the presence of a stationary sphere with either a no-slip or a stress-free (no shear) boundary condition. For an axisymmetric point torque and a no-slip sphere the image system simplifies to a single image point torque, reminiscent of the solution for a point charge outside an equipotential sphere in electrostatics. By symmetry, this also gives a simple representation of the solution due to an axisymmetric point torque inside a rigid spherical shell. In all remaining cases, the solution can be described by a collection of physically intuitive point and line singularities. Our results will be useful for the theoretical modelling of the propulsion of microswimmers and efficient numerical implementation of far-field hydrodynamic interactions in this geometry.