Rimming flows within a rotating horizontal cylinder: asymptotic analysis of the thin-film lubrication equations and stability of their solutions

It is well-known that a standard lubrication analysis of the equations of motion in thin liquid films coating the inside surface of a rotating horizontal cylinder leads, under creeping-flow conditions, to a cubic equation for the film thickness profile which, depending on the fluid properties of the liquid, the speed of rotation and the fill fraction F, has either (a) it continuous, symmetric (homogeneous) solution; (b) a solution containing a shock; or (c) no solution below a certain speed. By means of an asymptotic analysis of the recently proposed "modified lubrication equation" (MLE) [M. Tirumkudulu and A. Acrivos, Phys. Fluid 13 (2000) 14-19], it is shown that the solutions of the cubic equation referred to above correctly describe the film-thickness profiles although, when shocks are involved, under exceedingly restrictive conditions, typically Fsimilar to10(-3) or less. In addition, using the MLE, the linear stability of these film profiles is investigated and it is shown that: the "homogeneous" profiles are neutrally stable if surface-tension effects are neglected but, if the latter are retained, the films are asymptotically stable to two-dimensional disturbances and unstable to axial disturbances; on the other hand, the non-homogeneous profiles are always asymptotically stable, thus confirming results given earlier [T.B. Benjamin, WG. Pritchard, and S.J. Tavener (preprint, 1993)] on the basis of the standard lubrication analysis.