Stokesian image systems and Lorentz' reflection formula

Lorentz' reflection formula, a corollary of the Lorentz reciprocal theorem (1896), allows one to compute the creeping-flow motion of a fluid near a no-slip plane wall given a known solution to the Stokes equations in an unbounded fluid. This theorem may be applied to obtain the drag on colloidal particles or droplets near the wall. A related concept is that of an "image system", which is an auxiliary fluid flow that, when added to a Stokeslet---the 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), who noted that, while Lorentz' formula could be used to obtain a similar result, it did not produce his image system in a convenient form. Image systems have been used extensively to improve the efficiency of numerical boundary integral methods for Stokes flow.

We present a close 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 at which appropriate boundary conditions are applied, then a formula analogous to Lorentz' reflection formula may be immediately produced (and vice versa) via a simple rule. Our work connects and extends several results in the microhydrodynamics literature derived via different methods and leads to some new results. 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.