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Green function and singularities in Stokes flow confined by cylindrical walls

Giuseppe Procopio*

  • *Contact author: giuseppe.procopio@uniroma1.it
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Phys. Rev. Fluids 11, 034201 – Published 13 March, 2026

DOI: https://doi.org/10.1103/jt65-63vb

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 in an invariant form at both the field and pole points. Specifically, the Green function, assuming no-slip boundary conditions, is obtained using a cylindrical harmonic expansion of the Stokes flow within the bitensorial formulation introduced in Procopio and Giona [Math. Eng. 5, 1 (2022)]. This formulation allows us to obtain higher-order singularities within the same domains and under the same boundary conditions, such as the confined couplet and stresslet, by simply differentiating the Green function at its pole. Moreover, the confined sourcelet (or point source) 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 and planar walls, such as sedimenting particles in an annular cylindrical region and between two parallel plane walls, and the attractive or repulsive hydrodynamic forces exerted by cylindrical boundaries on microswimmers.

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Phys. Rev. Fluids 11, 034201– Published 13 March, 2026
  • Received 7 July 2025
  • Accepted 3 February 2026
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