ON EVOLUTION-EQUATIONS FOR THIN-FILMS FLOWING DOWN SOLID-SURFACES

A wavy free-surface flow of a viscous film down a cylinder is considered. It is shown that if the cylinder radius is large, as compared to the film thickness, the long-wave perturbation approach yields a rather simple evolution equation. This nonlinear equation is similar to the well-known Benney equation of planar films, and becomes exactly the latter in the limit of infinite radius. Thus it is the annular-case analog-which was missing in the literature-of the Benney equation. It is argued that under conditions implicitly implied in their derivation, the Benney-type equations are not uniformly valid for large times, However, both the new and Benney equations are important heuristically-as sources of other, simpler, equations which, in certain domains of system parameters, are valid for all time. Also, the new equation of annular films is important as a qualitative model incorporating all significant physical factors.