Three-Dimensional Motion of a Liquid Film Induced by Surface-Tension Variation or Gravity

Abstract

Steady flows of a thin layer of viscous liquid on a horizontal plane induced by the nonuniformity of surface tension at its free surface are treated. If the film is very thin, surface-tension effects dominate gravity effects. Under that circumstance and away from vertical boundaries, a binomial of depth h of the liquid layer is a harmonic function of the Cartesian coordinates x and y in a horizontal plane, and the surface tension is a function of h. Near any vertical boundary there is a velocity boundary layer whose thickness is of the order of h. The velocity distribution in this boundary layer is given explicitly. The diffusion of the surface material affecting the surface tension is considered. Steady flows of a liquid film induced by gravity are also discussed. Simple solutions are possible if the film flows over a horizontal plane.