Behaviour of a viscous film on the outer surface of a rotating cylinder

The problem of the dynamics of a thin film of viscous fluid on the outer surface of a horizontal rotating roller is considered by the techniques of lubrication theory. The condition for the existence of a steady solution is first obtained, and the maximum load that can be supported for given rotation rate is found on the basis of the steady solution. Kinematic wave theory shows that the steady state is not approached in general from arbitrary initial conditions, but that discontinuities in depth appear, like backward breaking waves relative to the direction of rotation. Weak diffusion can eliminate this tendency and can restore an approach to the steady state. A qualitative experiment is described showing that the steady state can be realized, but that it is in general unstable as the rotation rate is increased. The instability evolves into a sequence of rings of liquid around the cylinder, each ring carrying one or more 'lobes' exhibiting depth discontinuities of the kind suggested by kinematic wave theory. The observed behavior of the rings is described and discussed.