Fast evaluation of the fundamental singularities of two-dimensional doubly periodic Stokes flow

Abstract

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 methods for lattice sums.

Method of fundamental solutions for the problem of doubly-periodic potential flow

“…In the work on periodic Stokes flow [21], the author et al proposed to use the periodic fundamental solutions, which was presented by Hasimoto [9] and is given by a Fourier series. It is expected to construct a method of fundamental solutions using periodic fundamental solutions given by the theta functions or elliptic functions [11,6] as in this paper.…”

The slow viscous flow around doubly-periodic arrays of infinite slender cylinders

“…Though conformal maps do not preserve the boundary conditions for Stokes flow, this complex representation can be useful in determining the solutions to various problems [31][32][33][34].…”

“…The two-dimensional flow from a point force per unit length in a doubly-periodic domain was first determined by Hasimoto [35]. Luca and Crowdy [32] later revisited this problem to determine higher-order singularities and express it as a rapidly converging series. They showed that the flow from a two-dimensional point force per unit length of strength −8πF = −8π(F x + iF y ) located at z 0 = x 0 + iy 0 in a doubly-periodic cell with dimensions x ∈ [0, 1) and y ∈ [0, h) [32] can be written as…”

Analytical solutions for two-dimensional singly periodic Stokes flow singularity arrays near walls

“…The present authors [2] have recently given analytical representations that are amenable to fast numerical evaluation of the flows associated with doubly periodic arrangements of point singularities of two-dimensional Stokes flow. They used analytic function theory, a conformal mapping from a concentric annulus and the so-called Schottky-Klein prime function associated with that annulus to derive their new form of the solutions.…”

“…The present paper makes a basic theoretical contribution to the study of singly periodic two-dimensional Stokes flows in a no-slip channel and in the half-plane near a single no-slip wall. This paper can be viewed as a sequel to [2] where doubly periodic arrays of Stokes flow singularities are considered. Here we focus on singly periodic flows and, specifically, those generated by arrays of fundamental singularities, i.e.…”