Green function and singularities in Stokes flow confined by cylindrical walls

TL;DR

This paper derives the Green function for Stokes flow in cylindrical regions, enabling analysis of singularities and hydrodynamic interactions near cylindrical boundaries, with applications to colloids and microswimmers.

Contribution

It introduces a cylindrical harmonic expansion approach to derive Green functions and higher-order singularities for Stokes flow in cylindrical geometries, including a new method for confined Sourcelets.

Findings

Derived Green functions for various cylindrical regions.

Obtained higher-order singularities like Couplet and Stresslet.

Applied results to colloids and microswimmer hydrodynamics.

Abstract

In this article, the Green function for the Stokes flow in the interior, exterior, and annular regions bounded by cylindrical walls is derived as a function of the pole position and expressed invariantly both at the field and pole points. Specifically, the Green function is obtained using a cylindrical harmonic expansion of the Stokes flow within the bitensorial formulation. This formulation allows us to obtain higher-order singularities within the same domains, such as the confined Couplet and Stresslet, by simply differentiating the Green function at its pole. Moreover, the confined Sourcelet and its associated multipoles are derived from the Green function through a new method that enforces the reciprocal properties of the Stokes flow. The resulting singularities are then employed to address hydrodynamic problems involving active and passive colloids interacting with cylindrical walls, such as sedimenting particles in the annular cylindrical region and the attractive or repulsive hydrodynamic forces exerted by the cylindrical boundaries on a microswimmer.