Simple and efficient representations for the fundamental solutions of Stokes flow in a half-space

TL;DR

This paper introduces simplified, efficient formulas for fundamental solutions of Stokes flow in a half-space, enabling easier implementation and physical interpretation compared to classical methods.

Contribution

It presents new, simpler formulas for Stokes flow solutions in a half-space that are easier to implement and interpret than classical representations.

Findings

New formulas are simpler than classical ones.

Efficient implementation using existing fast solvers.

Velocity field can be controlled with a single reflected Stokeslet and scalar harmonic potential.

Abstract

We derive new formulas for the fundamental solutions of slow, viscous flow, governed by the Stokes equations, in a half-space. They are simpler than the classical representations obtained by Blake and collaborators, and can be efficiently implemented using existing fast solvers libraries. We show, for example, that the velocity field induced by a Stokeslet can be annihilated on the boundary (to establish a zero slip condition) using a single reflected Stokeslet combined with a single Papkovich-Neuber potential that involves only a scalar harmonic function. The new representation has a physically intuitive interpretation.