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.

翻译：有理逼近已被证明是求解二维流体问题的有效方法。在低雷诺数条件下，二维斯托克斯流可由两个解析函数表示，即古尔萨函数。Xue、Waters与Trefethen近期在[SIAM J. Sci. Comput., 46 (2024), pp. A1214-A1234]中提出LARS算法（闪电AAA有理斯托克斯法），通过有理函数逼近古尔萨函数来计算一般区域中的二维斯托克斯流。本文提出一种新算法，利用三角有理函数计算周期通道中的二维斯托克斯流，其极点通过AAA-LS算法[Costa与Trefethen, European Congr. Math., 2023]配置于区域边界的共形映射中。我们将该算法应用于不同周期通道几何构型下的泊肃叶与库埃特问题，在1秒内获得至少6位有效数字的精确解。该算法在非定常库埃特流中流体粒子动力学计算方面的适用性得到了重点展示。