Stokes flow due to point torques and sources in a spherical geometry

TL;DR

This paper derives new solutions for Stokes flow caused by point torques and sources near a sphere, enhancing theoretical understanding and computational modeling of microswimmer propulsion and hydrodynamic interactions.

Contribution

It extends the catalog of known Stokes flow solutions by deriving flow expressions for point torques and sources around a sphere with various boundary conditions, including simplified cases.

Findings

Single image point torque for axisymmetric cases with no-slip boundary

Explicit solutions involving point and line singularities for complex cases

Useful for modeling microswimmer propulsion and numerical simulations

Abstract

Solutions to the Stokes equations written in terms of a small number of hydrodynamic image singularities have been a useful tool in theoretical and numerical computations for nearly fifty years. In this article, we extend the catalogue of known solutions by deriving the flow expressions due to a general point torque and point source in the presence of a stationary sphere with either a no-slip or a stress-free (no shear) boundary condition. For an axisymmetric point torque and a no-slip sphere the image system simplifies to a single image point torque, reminiscent of the solution for a point charge outside an equipotential sphere in electrostatics. By symmetry, this also gives a simple representation of the solution due to an axisymmetric point torque inside a rigid spherical shell. In all remaining cases, the solution can be described by a collection of physically intuitive point and line singularities. Our results will be useful for the theoretical modelling of the propulsion of microswimmers and efficient numerical implementation of far-field hydrodynamic interactions in this geometry.