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A novel regularization for higher accuracy in the solution of the 3-dimensional Stokes flow

J. Thomas Beale, Christina Jones, Jillian Reale and Svetlana Tlupova

Vol. 15 (2022), No. 3, 515–524
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 by Tlupova and Beale (J. Comput. Phys. 386 (2019), 568–584). 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.

Keywords
Stokes flow, boundary integral equations, regularization
Mathematical Subject Classification
Primary: 65B99
Milestones
Received: 30 August 2021
Revised: 16 November 2021
Accepted: 18 November 2021
Published: 2 December 2022

Communicated by Michael Dorff
Authors
J. Thomas Beale
Department of Mathematics
Duke University
Durham, NC
United States
Christina Jones
Department of Mathematics
Farmingdale State College
Farmingdale, NY
United States
Jillian Reale
Department of Mathematics
Farmingdale State College
Farmingdale, NY
United States
Svetlana Tlupova
Department of Mathematics
Farmingdale State College
Farmingdale, NY
United States