Deformation of Supercritically Stable Waves on a Viscous Liquid Film down an Inclined Plane Wall with the Decrease of Wave Number

Abstract

The deformation of supercritically stable monochromatic waves on a viscous liquid film down an inclined plane wall with the decrease of wave number from that for the neutral stability is studied on the basis of a weakly nonlinear surface height equation. The equation is solved by using a two-amplitudes expansion technique to obtain two types of supercritically stable wave solutions and to examine their stability. It is found that the well-known first type of solution becomes unstable for the wave number smaller than a certain critical value, and the second type of solution becomes stable to be realized in place of the first type. With the further decrease of wave number, the second type of supercritically stable wave is transformed again into the first type with the wave number twice that of the sinusoidal disturbance imparted initially.