Rising and falling film flows: Viewed from a first-order approximation

Steady flow of liquid film on a vertical plane solid surface that is either fixed, or rising in its own plane out of a bath of liquid, occurs in coating and various other applications. At low capillary number the free surface flow is nearly rectilinear and invites approximate analysis which reveals two families of solutions. In the first, the configuration of the bath meters rising film flow and the film thickness above. In the other, the thickness of a rising or falling film is controlled by conditions far above the bath; bath configuration affects film thickness only in a boundary layer-like region adjoining the bath. Between the families, a range of film thickness is found that cannot be realized in steady flow.