  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.
