Elliptic Function Representation of Doubly Periodic Two-Dimensional Stokes Flows

We construct doubly periodic Stokes flows in two dimensions using elliptic functions. This method has advantages when the doubly periodic lattice of obstacles has less than maximal symmetry. We find the mean flow through an arbitrary lattice in response to a pressure gradient in an arbitrary direction, and show in a typical example that the shorter of the two period lattice vectors is an “easy direction” for the flow, an eigenvector of the conductance tensor corresponding to maximal conductance. It is known, and we rederive it below, that two-dimensional (2D) Stokes flows can be represented in terms of two complex analytic functions [1]. It is plausible, then, that doubly periodic 2D Stokes flows should have a representation in terms of doubly periodic complex analytic functions, that is, elliptic functions [2, 3]. Such a representation was promised in 1959 by H. Hasimoto [4], but it did not appear. Other authors in the succeeding decades alluded to such a representation [5], and even, like Hasimoto, quoted results following from it [6]. The Hasimoto article may have appeared much later as lecture notes in Japanese [7]. Meanwhile, there are other, perhaps more straightforward, ways to represent doubly periodic 2D Stokes flows. These include matching of flows around a single obstacle across periodic cell boundaries [6, 8, 9, 10, 11], integral equation methods [12], biharmonic solvers on a grid [5], and finite element methods. In spite of this long history, we have thought it useful to present the elliptic function method, because there is still, apparently, no readily available description of it. Furthermore, as we shall show, this approach solves one aspect of the problem which is not at all simple in the most common cell matching approach, namely the appropriate boundary condition for flow through a general periodic lattice in a general direction. With this method we describe the typical flow through a generic lattice.