Green function and singularities in Stokes flow confined by cylindrical walls

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.

💡 Research Summary

This paper presents a comprehensive analytical framework for Stokes flow confined by infinitely long cylindrical walls, covering the interior, exterior, and annular domains. The authors adopt a bitensorial formulation, treating the field point x and the pole point ξ in separate coordinate systems and introducing a parallel propagator to correctly handle the Dirac delta distribution. The Stokes Green function Gᵇ_β(x,ξ) and the associated pressure tensor P_β(x,ξ) are expressed as a sum of the free‑space Stokeslet Sᵇ_β(x,ξ) and a regular part Wᵇ_β(x,ξ) that enforces the no‑slip boundary condition on the cylindrical surfaces.

Using a cylindrical harmonic expansion, the authors solve for the coefficients of the regular part in each geometry. Bessel functions (Jₙ, Yₙ) and modified Bessel functions (Iₙ, Kₙ) appear naturally in the series, and the method accommodates arbitrary inner radius a and outer radius b. Special limits (a→0 for a single interior cylinder, b→∞ for a single exterior cylinder) are obtained smoothly, correcting inconsistencies found in earlier literature.

Higher‑order singularities are generated simply by differentiating the Green function with respect to the pole coordinates. The first derivative yields the Stokeslet dipole, whose antisymmetric component is identified as the Couplet and the symmetric component as the Stresslet. Second‑order derivatives give rise to a new “Sourcelet” – a point source/sink solution that satisfies ∇·M = –4πδ without violating incompressibility because the cylinders are infinitely long and the fluid does not exchange mass with the exterior. The Sourcelet dipole is also derived, completing the set of fundamental singularities needed for confined Stokes flow.

The paper also reformulates the unbounded Stokeslet in cylindrical coordinates within the bitensorial language, providing a clear basis for constructing all confined singularities.

Applications are demonstrated in three contexts: (i) sedimenting particles in the annular region, where the drag is obtained from the regular part of the Green function evaluated at the particle’s location; (ii) comparison of flow fields for a particle near a cylinder versus near a sphere, showing the expected convergence to the planar‑wall limit as the cylinder curvature vanishes; (iii) hydrodynamic forces on microswimmers near cylindrical walls, where the combined effect of Couplet, Stresslet, and Sourcelet produces orientation‑dependent attractive or repulsive forces. The authors verify that taking the curvature to zero recovers known results for two parallel plates, confirming the consistency of their expressions.

In summary, the bitensorial approach yields a fully differentiable Green function that simultaneously provides the free‑space singularities and their regular corrections for any cylindrical confinement. This unified treatment overcomes the fragmented and sometimes erroneous solutions previously available, and it offers a versatile toolbox for analytical studies of colloidal, sedimentation, and microswimmer dynamics in cylindrical microfluidic devices. The methodology is readily extensible to other geometries (e.g., conical or spherical boundaries) where similar confinement effects are important.