Leading-order Stokes flows near a corner

TL;DR

This paper derives the asymptotic behavior of fundamental Stokes flow solutions near corners, providing insights useful for biological physics, colloidal science, and microfluidics.

Contribution

It introduces new asymptotic solutions for Stokes flows near corners, extending classical results to complex geometries and orientations.

Findings

Derived asymptotic behavior of Stokeslet near corners.

Analyzed all orientations and geometries from acute to obtuse.

Validated results with bead sedimentation experiments.

Abstract

Singular solutions of the Stokes equations play important roles in a variety of fluid dynamics problems. They allow the calculation of exact flows, are the basis of the boundary integral methods used in numerical computations, and can be exploited to derive asymptotic flows in a wide range of physical problems. The most fundamental singular solution is the flow's Green function due to a point force, termed the Stokeslet. Its expression is classical both in free space and near a flat surface. Motivated by problems in biological physics occurring near corners, we derive in this paper the asymptotic behaviour for the Stokeslet both near and far from a corner geometry by using complex analysis on a known double integral solution for corner flows. We investigate all possible orientations of the point force relative to the corner and all corner geometries from acute to obtuse. The case of salient corners is also addressed for point forces aligned with both walls. We use experiments on beads sedimenting in corn syrup to qualitatively test the applicability of our results. The final results and scaling laws will allow to address the role of hydrodynamic interactions in problems from colloidal science to microfluidics and biological physics.