On the solution of Stokes equation on regions with corners

TL;DR

This paper provides a new analytical series representation for solutions to the biharmonic equation in Stokes flow near corners, enabling more accurate and efficient numerical solutions for such geometries.

Contribution

It introduces a rapidly convergent series solution for the biharmonic equation on corner regions, improving numerical discretization and understanding of flow behavior.

Findings

Series representation accurately models solutions near corners.

Reduces degrees of freedom needed for numerical solutions.

Numerical examples demonstrate method effectiveness.

Abstract

In Stokes flow, the stream function associated with the velocity of the fluid satisfies the biharmonic equation. The detailed behavior of solutions to the biharmonic equation on regions with corners has been historically difficult to characterize. The problem was first examined by Lord Rayleigh in 1920; in 1973, the existence of infinite oscillations in the domain Green's function was proven in the case of the right angle by S.~Osher. In this paper, we observe that, when the biharmonic equation is formulated as a boundary integral equation, the solutions are representable by rapidly convergent series of the form $\sum_{j} ( c_{j} t^{\mu_{j}} \sin{(\beta_{j} \log{(t)})} + d_{j} t^{\mu_{j}} \cos{(\beta_{j} \log{(t)})} )$, where $t$ is the distance from the corner and the parameters $\mu_{j},\beta_{j}$ are real, and are determined via an explicit formula depending on the angle at the corner. In addition to being analytically perspicuous, these representations lend themselves to the construction of highly accurate and efficient numerical discretizations, significantly reducing the number of degrees of freedom required for the solution of the corresponding integral equations. The results are illustrated by several numerical examples.