A solution technique for longitudinal Stokes flow around multiple aligned cylinders

Abstract

This technique solves the two-dimensional Poisson equations in geometries involving cylindrical objects. The method uses three fundamental solutions, corresponding to a line force, a line couple and a pressure gradient, on each cylinder. Superposition of the fundamental solutions due to all the cylinders involved, while approximately satisfying the no-slip condition on each cylinder, yields a mobility matrix relating the various forces and motions of all the cylinders. Any specific problem can be solved by prescribing the motions of the cylinders and solving the matrix. For problems involving few cylinders or with a sufficient degree of symmetry this can be done analytically.

Once constructed, the general method is applied analytically to a series of specific problems. The permeability of an eccentric annulus is derived. The result is numerically indistinguishable from the exact solution to the problem, but unlike the exact solution the present one is obtained in closed form. The drag on two parallel rods moving past one another is also derived and compared to the exact solution. In this case the result is accurate for rod separations down to about 0.2 times the rod diameter. Finally the drag on a rod moving in a triangular array of identical rods is derived. Here it is shown that due to screening it is sufficient to include the six nearest neighbours, regardless of the rod separation. Although the present examples are all worked out analytically, the matrix can also be solved numerically, in which case any two-dimensional arrangement of cylindrical objects can be studied.