The flow of a liquid film on the inside of a rotating cylinder, and some related problems

Abstract

A liquid film flows on the inner surface of a rotating horizontal cylinder. The simplest lubrication model assumes a force balance between viscosity and the streamwise component of gravity, and the equation for the film thickness admits a continuous solution only when the average thickness is less than a certain critical value. Above this value, a discontinuous solution is possible but the details are not accessible by means of the simple theory. This behavior can be traced to the gradual periodic variation of the streamwise component of gravity in the streamwise direction. We consider also two related problems in which this variation occurs more or less abruptly: (i) when the moving wall comprises two straight segments inclined at different angles, and (ii) when the dragged film emerges through the free surface of a second, overlying liquid. These problems are approached by introducing a smoothing parameter, namely surface tension, and solving a suitable initial value problem. We use the method of lines for this purpose because of the availability of robust ODE software which can exploit the structure of the problem; however, the periodic conditions of the cylinder problem necessitate a special approach to the discretization.