Nonlinear stability of a viscous film with respect to three-dimensional side-band disturbances

Abstract

The analysis of Lin is extended to investigate the nonlinear stability of a liquid film with respect to three‐dimensional side‐band disturbances. Near the upper branch of the linear‐stability curve where the amplification c_i is O (ε²), ε being proportional to the ratio of the amplitude to the film thickness, the nonlinear evolution of initially infinitestimal three‐dimensional disturbances of a finite band width is shown to obey the nonlinear Schrödinger equation. Near the lower branch of the neutral curve, the nonlinear evolution is stronger. An equation is derived which describes this strong nonlinear development of relatively long three‐dimensional waves. It is shown that the supercritically stable, finite amplitude, long monochromatic wave is stable to three‐dimensional side‐band disturbances under modal interaction if the bandwidth is less in magnitude than ε.