An algorithm to compute 2D Stokes flows with great speed and accuracy
We developed a 2D Stokes flow solver LARS (Lightning-AAA Rational Stokes) using rational approximation (Xue et al., 2024). It uses the lightning algorithm for sharp corners (Gopal and Trefethen, 2019; Brubeck and Trefethen, 2022), the AAA algorithm for curved boundaries (Nakatsukasa et al., 2018), and a series method (Trefethen, 2018) for multiply connected domains. The solver works for most 2D Stokes problems with different geometries and boundary conditions. In most cases, a solution can be obtained to 6-digit accuracy in less than 1 second on a laptop. Example MATLAB codes can be found on GitHub. Further application of the algorithm to Stokes flows in a 2D bifurcation can be found in (Xue et al., 2025). An extension of rational approximation methods for periodic Stokes flows can be found in (Xue, 2025).
The flow network model is an established approach to approximate pressure-flow relationships in a bifurcating network, and has been widely used in many contexts. Existing models typically assume unidirectional flow and exploit Poiseuille’s law, and thus neglect the impact of bifurcation geometry and finite-sized objects on the flow. We determine the impact of bifurcation geometry and objects by computing Stokes flows in a two-dimensional (2D) bifurcation using the Lightning-AAA Rational Stokes algorithm, a novel mesh-free algorithm for solving 2D Stokes flow problems utilizing an applied complex analysis approach based on rational approximation of the Goursat functions. We compute the flow conductances of bifurcations with different channel widths, bifurcation angles, curved boundary geometries and fixed circular objects. We quantify the difference between the computed conductances and their Poiseuille law approximations to demonstrate the importance of incorporating detailed bifurcation geometry into existing flow network models. We parametrize the flow conductances of 2D bifurcation as functions of the dimensionless parameters of bifurcation geometry and a fixed object using a machine learning approach, which is simple to use and provides more accurate approximations than Poiseuille’s law. Finally, the details of the 2D Stokes flows in bifurcations are presented.
Computing Stokes flows in periodic channels via rational approximation
Rational approximation has proven to be a powerful method for solving two-dimensional fluid problems. At small Reynolds numbers, two-dimensional Stokes flows can be represented by two analytic functions, known as Goursat functions. (Xue et al. 2024 SIAM J. Sci. Comput. 46, A1214-A1234 (doi:10.1137/23m1576876)) recently introduced the lightning-adaptive Antoulas-Anderson (AAA) rational Stokes algorithm for computing two-dimensional Stokes flows in general domains by approximating the Goursat functions using rational functions. In this paper, we introduce a new algorithm for computing two-dimensional Stokes flows in periodic channels using trigonometric rational functions, with poles placed via the AAA-least squares algorithm (Costa & Trefethen 2023 In European Congress of Mathematics (ed. A Hujdurović), pp. 511-534 (doi:10.4171/8ecm/16)) 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 six-digit accuracy in less than 1 s. The applicability of the algorithm is highlighted in the computation of the dynamics of fluid particles in unsteady Couette flows.
2024
Computation of two-dimensional Stokes flows via lightning and AAA rational approximation
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 [SIAM J. Sci. Comput., 44 (2022), pp. A1205-A1226] 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 two-dimensional (2D) Stokes flows in domains with smooth boundaries and multiply connected domains using lightning and AAA rational approximation [Y. Nakatsukasa, O. Sète, and L. N. Trefethen, SIAM J. Sci. Comput., 40 (2018), pp. A1494-A1522]. 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.
