Stability of wavy conditions in a film flowing down an inclined plane

The nonlinear theory of motion in a film of liquid flowing down an inclined plane predicts the existence of an interval k₀m, inside of which the wave number of periodic wave motion may lie 1]. The condition of the stability of experimentally attained motions imposes a limitation on their wave numbers. In 2] a numerical investigation of the stability of wavy motions was made; in the investigated range of change in the Galileo number and the wave number all the motions were found to be unstable; however, the fastest growing were perturbations imposed on a motion with a determined wave number (“optimal” conditions). In 3] the instability of motions with a wavelength exceeding some limiting value was established in a long-wave approximation. In the present work, within the framework of the two-dimensional problem, an investigation was made of the stability of periodic wavy motions, based on expansion in terms of the small parameter k_m. It is established that, within the interval k₀m, there lies a finite subinterval of wave numbers for which wavy motions are stable. The narrowness of this interval (δk≈0.07 k_m) may be the reason why, in the experiment, with not too great Galileo numbers for fully established periodic wavy motions, no substantial differences in the wave-length are observed 4].