Stability of a liquid film with respect to initially finite three-dimensional disturbances

Abstract

The nonlinear stability of a viscous liquid film flowing steadily down an inclined plane is studied in the phase plane. Explicit expressions of the critical points of the governing differential system are obtained. The nature of the critical points and the integral curves corresponding to various flow parameters, initial conditions, and disturbance characteristics are determined numerically. Based on the phase plane analysis and the numerical results, the following conclusions are reached: The film which is unstable according to linear theory may be stable with respect to a finite three-dimensional disturbance if the initial amplitude and the side-band width are sufficiently small. The particular values of the initial amplitude and the side-band width beyond which the film becomes unstable depend on the relevant flow parameters. The film which is stable according to linear theory is also shown to be stable with respect to three-dimensional small finite amplitude disturbances with finite bandwidths.