Motions of a sphere in a time-dependent stokes flow: A generalization of Faxen’s law

A general solution of the unsteady Stokes equation in spherical coordinates is derived for flow in the exterior of a sphere, and then applied to study the arbitrary unsteady motion of a rigid sphere in an unbounded single fluid domain which is undergoing a time-dependent mean flow. Calculation of the hydrodynamic force and torque on the sphere leads to a generalization of the Faxen’s law to time-dependent flow fields which satisfy the unsteady Stokes equation. For illustrative purposes, we consider the relative motion of gas bubbles which undergo very rapid oscillations so that the generalized Faxen’s law derived for a solid sphere can be applied. We also demonstrate that our results reduce to those of Faxen for the steady flow limit.