There's more than one way to cancel a regularized Stokeslet

TL;DR

This paper explores different methods for canceling regularized Stokeslets in microscale flow simulations, revealing discrepancies between solutions and suggesting potential advantages of the Lorentz-based approach.

Contribution

It adapts Lorentz's method of images to regularized Stokes flow, producing solutions that differ from previous reports and analyzing their practical implications.

Findings

Different solutions arise from variations in forcing terms.

Lorentz-based solutions may be advantageous in certain applications.

Discrepancies challenge the assumed uniqueness of solutions.

Abstract

The Green's functions of Stokes flow are widely used analytical and computational tools for microscale flows. We adapt a procedure from H.A. Lorentz for the method of images in Stokes flow to the regularized setting. Our solutions differ from those previously reported, a surprising result given the uniqueness theory for elliptic partial differential equations. The discrepancy originates in the fact that the two version are exact solutions of inhomogeneous Stokes systems with slightly different forcing on the right-hand sides. We compare the fluid flows produced by the two methods and conclude that the Lorentz versions may be advantageous in some settings.