The free surface on a simple fluid between eccentric cylinders rotating at different speeds

The free surface flow of a simple fluid between eccentric cylinders is considered. Flow between rotating cylinders is reviewed. The rod climbing problem between eccentric cylinders is defined and set in a mathematical context. The analytical problem is solved using domain perturbations. At the 1st order a plane flow is obtained with an asymmetrical free surface. At the 2nd order the problem is reduced to three subproblems: (1) a plane Newtonian flow which collapses, as the eccentricity goes to zero, on to the axially symmetrical solution for the case of two concentric cylinders; (2) a top driven, three dimensional, Newtonian flow which tends to zero as the eccentricity tends to zero; and (3) a non-Newtonian pressure gradient which is balanced by a gradient with zero flow.