Linear stability of rimming flow

We consider the case of rimming flow where a thin film of viscous liquid coats the walls of a cylinder whose axis is horizontal and which is rotating with constant angular velocity. It has been experimentally established that such flows are often unstable and that the liquid often segregates into "rings" along the length of the tube. Using a leading-order lubrication theory, we utilise recently established steady solutions, which in some instances contain shocks, to examine the linear stability of the flow when subjected to two-dimensional disturbances. All solutions are shown to be at least neutrally stable. We suggest that further investigations should include higher-order (small) effects and that the origin of the observed instabilities lies in these terms.