Computing Stokes flows in periodic channels via rational approximation

TL;DR

The paper develops a fast, accurate framework for 2D Stokes flows in periodic channels by representing Goursat functions with trigonometric rational bases and placing poles via AAA-LS in a conformal boundary map. The method enforces 2π-periodicity in the basis, solves a well-conditioned least-squares system to obtain the Goursat coefficients, and uses Vandermonde–Arnoldi orthogonalization to stabilize the basis. It demonstrates Poiseuille and Couette problems with analytic and polygonal boundaries, achieving 6-digit accuracy in under a second and enabling efficient computation of unsteady particle trajectories and mixing via Poincaré maps. The work highlights the practical impact for rapid, high-fidelity simulations of periodic Stokes flows and discusses extensions to related periodic Laplace/Stokes problems and multiply connected domains.

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.

Figures (9)

• Figure 1: Schematic of 2D Stokes flow through a periodic channel.
• Figure 1: AAA poles outside the domain boundaries in the $\zeta=e^{iz}$ and $z$ planes. The poles are marked by red dots. In the top row, the poles are placed outside sinusoidal boundaries, which are periodic analytic curves. In the bottom row, the poles are placed outside polygonal boundaries, which are periodic piecewise functions with sharp corners. The AAA approximations are computed for the domain boundary in the blue region, where $\mathrm{Re}(z)\in[0,2\pi)$.
• Figure 1: Stokes flows in three periodic channels with analytic boundaries. The streamlines, poles and velocity magnitude are represented by solid black lines, red dots and a colour scale. For each example, we present solutions from $x=-\pi$ to $x=3\pi$, where the period from $x=0$ to $x=2\pi$ is marked by two dashed black lines.
• Figure 2: Stokes flows in two periodic channels with sharp corners. In (b), the eddies near sharp corners are shown using yellow contours of the stream function.
• Figure 3: Stokes flows in periodic channels constricted by a moving flat wall and a steady sinusoidal wall. The first and second Moffatt eddies are indicated by yellow and white contours, respectively. The same problem has been computed in Figure $3$b of Pozrikidis1987 using a boundary integral method.