Construction of stationary waves on a falling film

Abstract

Waves on a film falling down a vertical wall exhibit many distinct features. They tend to be locally stationary over several wavelengths, viz. they travel with constant speeds and shapes over a long distance. In the limit of very long (solitary) waves, these stationary waves also exhibit two length scales with small and short capillary waves running ahead of a large tear-drop shaped hump. We present a spectral-element method for this difficult multi-scale free surface problem. A boundary layer approximation of the equation of motion allows a Fourier expansion in the streamwise direction in conjunction with a domain decomposition in the direction normal to the wall that eliminates numerical instability. This mixed method hence enjoys both the exponential convergence rate of a spectral technique and the numerical advantage provided by a compactly supported basis which yields sparse projected differential operators. All stationary wave families, parameterized by the wavelength, are then constructed using a Newton continuation scheme. The constructed waves are favorably compared to experimentally measured wave shapes.