(242i) Stokesian Image Systems and Lorentz’ Reflection Theorem

Lorentz’ reflection theorem, a lemma of the classic Lorentz reciprocal theorem (1896), allows one to compute the creeping-flow motion of a fluid near a no-slip plane wall given an arbitrary known solution to the Stokes equations that generally does not satisfy the condition at the wall. This theorem may be applied, for example, to obtain the drag on colloidal particles or droplets near a wall. A related concept is that of an “image system”, which is an auxiliary fluid flow that, when added to a Stokeslet—the singular point-force solution of the Stokes equations in an infinite medium—produces a new point-force solution subject to a desired boundary condition. The image system for a no-slip plane wall was first derived by Blake (1971). Image systems have been used extensively to improve the efficiency of numerical boundary integral methods.

We show the existence of a close mathematical connection between Lorentz’ theorem and the Stokeslet image system which appears not to be discussed in previous literature. In particular, if one possesses a known image system for a given geometry with appropriate boundary conditions, then a formula analogous to Lorentz’ reflection theorem may be immediately produced via a simple rule. Our work connects and extends several theoretical results in the microhydrodynamics literature derived via different methods and leads to some new results. For example, a Lorentz reflection theorem for a spherical cavity or droplet may be determined from known, point-force image systems. Finally, we present an application of our results to the method of regularized Stokeslets to compute the motion of colloidal particles in the presence of planar or spherical boundaries.