Analytical representations of the flows associated with doubly periodic arrangements of point singularities of two-dimensional Stokes flow are derived. The analysis makes use of a conformal mapping from a concentric annulus to a rectangle. A natural mathematical object known as the Schottky–Klein prime function is used to derive the solutions avoiding the theory of elliptic functions. The new expressions have the advantage of being immediately amenable to fast evaluation without the need for Ewald or other fast summation methods or acceleration techniques for lattice sums.