# Boundary regularized integral equation formulation (BRIEF) of Stokes flow

## TL;DR

This paper introduces a boundary regularized integral equation method for Stokes flow that eliminates singularities, leading to more accurate and easier-to-implement solutions, especially in complex boundary contact scenarios.

## Contribution

It presents an exact, singularity-free boundary integral formulation for Stokes flow using auxiliary fields, simplifying implementation and enhancing numerical accuracy.

## Key findings

- Eliminates mathematical singularities in boundary integral equations.
- Provides significant coding and computational savings.
- Maintains high accuracy even with nearly contacting boundaries.

## Abstract

Single-phase Stokes flow problems with prescribed boundary conditions can be formulated in terms of a boundary regularized integral equation that is completely free of singularities that exist in the traditional formulation. The usual mathematical singularities that arise from using the fundamental solution in the conventional boundary integral method are removed by subtracting a related auxiliary flow field, $\boldsymbol{w}$, that can be constructed from one of many known fundamental solutions of the Stokes equation. This approach is exact and does not require the introduction of additional cutoff parameters. The numerical implementation of this boundary regularized integral equation formulation affords considerable savings in coding effort with improved numerical accuracy. The high accuracy of this formulation is retained even in problems where parts of the boundaries may almost be in contact.