Computing Stokes flows in periodic channels via rational approximation

TL;DR

This paper introduces a novel algorithm using rational approximation and conformal mapping to efficiently compute highly accurate 2D Stokes flows in periodic channels, applicable to various geometries and unsteady flows.

Contribution

A new trigonometric rational approximation algorithm for 2D Stokes flows in periodic channels using AAA-LS pole placement in conformal maps.

Findings

Achieves at least 6-digit accuracy in less than 1 second.

Successfully applied to Poiseuille and Couette flow problems.

Demonstrates effectiveness in unsteady flow particle dynamics.

Abstract

Rational approximation has proven to be a powerful method for solving two-dimensional (2D) fluid problems. At small Reynolds numbers, 2D Stokes flows can be represented by two analytic functions, known as Goursat functions. Xue, Waters and Trefethen [SIAM J. Sci. Comput., 46 (2024), pp. A1214-A1234] recently introduced the LARS algorithm (Lightning-AAA Rational Stokes) for computing 2D Stokes flows in general domains by approximating the Goursat functions using rational functions. In this paper, we introduce a new algorithm for computing 2D Stokes flows in periodic channels using trigonometric rational functions, with poles placed via the AAA-LS algorithm [Costa and Trefethen, European Congr. Math., 2023] in a conformal map of the domain boundary. We apply the algorithm to Poiseuille and Couette problems between various periodic channel geometries, where solutions are computed to at least 6-digit accuracy in less than 1 second. The applicability of the algorithm is highlighted in the computation of the dynamics of fluid particles in unsteady Couette flows.