Stokes flow in a trench between concentric cylinders

In this paper a separation of variables theory is developed for solving problems of Stokes flow in annular trenches bounded by horizontal parallel planes and concentric vertical cylinders. The theory leads to a new set of Stokes flow eigenfunctions, adjoint eigenfunctions, biorthogonality conditions and an algorithm for the computation of the coefficients in an eigenfunction expansion of edge data prescribed on the horizontal boundaries. To illustrate the algorithm, the motion and the shape of the free surface on a liquid in the space between two cylinders are computed; the cylinders are maintained at different temperatures. The solution is constructed as a domain perturbation of the rest state in powers of the temperature difference. At the lowest significant order the problem is reduced to a Stokes flow problem of the desired type with edge data prescribed on the horizontal boundaries.