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.