The rotation of two circular cylinders in a viscous fluid

In a previous communication we employed the solution of the equation ∇4ψ = 0 in bipolar co-ordinates defined by α + iβ = log x + i(y + a)/x + i(y - a) (1) to discuss the problem of the elastic equilibrium of a plate bounded by any two non-concentric circles. There is a well-known analogy between plain elastic stress and two-dimensional steady motion of a viscous fluid, for which the stream-function satisfies ∇4ψ = 0. The boundary conditions are, however, different in the two cases, and the hydrodynamical problem has its own special difficulties.