Lamb-type solution and properties of unsteady Stokes equations

Abstract

We derive the general solution of the unsteady Stokes equations for an unbounded fluid in spherical polar coordinates, in both time and frequency domains. The solution is an expansion in vector spherical harmonics and given as a sum of a particular solution, proportional to pressure gradient exhibiting power-law spatial dependence, and a solution of vector Helmholtz equation decaying exponentially in far field, the decomposition originally introduced by Lamb. The solution can be applied to construct the transient exterior flow induced by an arbitrary velocity distribution at the spherical boundary, such as arising in the squirmer model of a microswimmer. It can be used to construct solutions for transient flows driven by initial conditions, unbounded flows driven by volume forces or disturbance to the unsteady flow due to a stationary spherical particle. The long-time behavior of solution is controlled by the flow component corresponding to average (or collective) motion of the boundary. This conclusion is illustrated by the study of decay of transversal wave in the presence of a fixed sphere. We further show that the general representation reduces to the well-known solutions for unsteady flow around a sphere undergoing oscillatory rigid-body (translation and rotation) motion. The proposed solution representation provides an explicit form of the velocity potential far from an oscillating body ("generalized" Darcy's law) and high- and low-frequency expansions. The leading-order high-frequency expansion yields the well-known ideal (inviscid) flow approximation, and the leading-order low-frequency expansion yields the steady Stokes equations. We derive the higher-order corrections to these approximations and discuss d'Alembert paradox. Continuation of the general solution to imaginary frequency yields the general solution of the Brinkman equations describing viscous flow in porous medium.