Computation of 2D Stokes flows via lightning and AAA rational approximation

TL;DR

The paper develops LARS (Lightning-AAA Rational Stokes), a fast, accurate solver for 2D Stokes flows in general bounded domains by integrating lightning-based corner handling, AAA rational approximation for curved boundaries, and a series method for multiply connected regions. It leverages Goursat functions to represent the biharmonic Stokes problem, constructs well-conditioned rational bases via Vandermonde–Arnoldi, and solves boundary-value problems with a linear least-squares formulation to obtain high-precision velocity, pressure, and vorticity fields. Key contributions include enabling sub-second solutions with 6+ digit accuracy across smooth and multiply connected geometries, demonstrated on constricted channels and cylinder configurations, and providing a practical framework for rapid quasi-steady simulations in microfluidics and related applications. The approach broadens the use of rational approximation in 2D Stokes problems, offering a general, implementable tool for complex geometries and potentially extensible to time-dependent and unbounded-domain settings.

Abstract

Low Reynolds number fluid flows are governed by the Stokes equations. In two dimensions, Stokes flows can be described by two analytic functions, known as Goursat functions. Brubeck and Trefethen (2022) recently introduced a lightning Stokes solver that uses rational functions to approximate the Goursat functions in polygonal domains. In this paper, we present the "LARS" algorithm (Lightning-AAA Rational Stokes) for computing 2D Stokes flows in domains with smooth boundaries and multiply-connected domains using lightning and AAA rational approximation (Nakatsukasa et al., 2018). After validating our solver against known analytical solutions, we solve a variety of 2D Stokes flow problems with physical and engineering applications. Using these examples, we show rational approximation can now be used to compute 2D Stokes flows in general domains. The computations take less than a second and give solutions with at least 6-digit accuracy.

Figures (11)

• Figure 1: Schematic of Stokes flow through a smoothly constricted channel, after Tavakol2017. (a) Geometry and boundary conditions. (b) Shape function $H(X)$ of the upper boundary for different $\lambda$.
• Figure 1: Schematic of a translating and rotating cylinder in a rotating cylinder, after Finn2001 (dimensionless quantities).
• Figure 1: Stokes flow around a translating and rotating elliptical cylinder inside a fixed elliptical cylinder. The same parameter values as for Case 'c' in \ref{['tab:parameter_value']} are used here. The outer ellipse has eccentricity $0.6$ and the inner ellipse has eccentricity $0.8$. The translation and rotation of the inner ellipse are indicated by white arrows.
• Figure 2: Pressure drop as a function of constriction parameter $\lambda$ when $\delta=1$ computed using polynomials with degrees $200$, $300$ and $400$. The simulation results are compared with the solutions derived using a $4$th-order extended lubrication theory Tavakol2017. The numbers of sample points for the polynomial approximations are $4200$, $6300$ and $8400$.
• Figure 2: Streamlines for Stokes flow between two cylinders for nine different boundary conditions, following Finn2001. The parameter values are listed in \ref{['tab:parameter_value']}. The stream function is $0$ on the outer cylinder.