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
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.