A Novel Regularization for Higher Accuracy in the Solution of 3D Stokes Flow

TL;DR

This paper introduces a simplified and more accurate regularization method for boundary integral equations in 3D Stokes flow, improving numerical precision while maintaining convergence order.

Contribution

It presents a new stresslet regularization that simplifies the polynomial formulation and enhances accuracy over previous methods.

Findings

Retains the same order of convergence as previous methods

Reduces the magnitude of numerical error

Simplifies the polynomial formulation for regularization

Abstract

Many problems in fluid dynamics are effectively modeled as Stokes flows - slow, viscous flows where the Reynolds number is small. Boundary integral equations are often used to solve these problems, where the fundamental solutions for the fluid velocity are the Stokeslet and stresslet. One of the main challenges in evaluating the boundary integrals is that the kernels become singular on the surface. A regularization method that eliminates the singularities and reduces the numerical error through correction terms for both the Stokeslet and stresslet integrals was developed in Tlupova and Beale, JCP (2019). In this work we build on the previously developed method to introduce a new stresslet regularization that is simpler and results in higher accuracy when evaluated on the surface. Our regularization replaces a seventh-degree polynomial that results from an equation with two conditions and two unknowns with a fifth-degree polynomial that results from an equation with one condition and one unknown. Numerical experiments demonstrate that the new regularization retains the same order of convergence as the regularization developed by Tlupova and Beale but shows a decreased magnitude of the error.