Long waves on inclined films at high Reynolds number

Abstract

At large Reynolds number (Re > 10), waves on inclined films grow rapidly downstream in both amplitude and wavelength to the extent that linear stability theory cannot adequately describe their velocity–wavenumber relationship. The wavelength increases indefinitely until solitary waves are formed very far downstream. In a recent experiment of Brauner & Maron (1982), this evolution to long waves is observed to occur by successive wavelength doubling. In this analysis, we develop a second-order integral boundary-layer approximation for long waves at intermediate Re of O(ε−1), where ε is the dimensionless wavenumber scaled with respect to the film thickness. (A second-order theory is needed because it introduces important dissipation terms which allow periodic and solitary waveforms to exist when surface tension is negligible.) After showing that this model can adequately describe infinitesimal waves at inception, we verify the existence of solitary waves and long-wavelength periodic waves near the critical Reynolds number with a weakly nonlinear analysis. These finite-amplitude waves are then numerically continued into the more important high-Re and strongly nonlinear regions. It is shown that the solitary wave speed approaches 1.67 times the Nusselt velocity, and the thickness of the substrate film approaches 0.47 times the Nusselt film thickness at large Re. These results are favourably compared to experimental data of Chu & Dukler (1974, 1975). We also confirm the period-doubling scenario of Brauner & Maron by showing that short finite-amplitude monochromatic waves are unstable to subharmonic instability.