Full-scale waves on downflowing films

Igor L. Kliakhandler

Lawrence Berkeley National Laboratory

 

Abstract

Investigation of many nonlinear problems begins from study of its linear stability problem. It is well-known that if Reynolds number is small, and surface tension is large, the dispersion relation for liquid film flowing down an inclined plane may be approximated as Rew ~ k2 - k4 (with some rescaling). Here deviation of interface from its steady-state position h ~ eikx+wt. This dispersion relation gives rise to KS equation. For large Reynolds numbers, boundary layer theory is conventionally applied; its linear theory is more sophisticated than linear theory of KS equation.

However, full-scale numerical investigation of real dispersion relation in the wide range of parameters was never done to check how the nonlinear theories describe linear wavy dynamics on the surface of the film. We found that in a wide range of parameters real dispersion relation exhibits remarkably different behavior than that predicted by conventional theories. Reconstructing the pertinent dispersion relation, and combining this with associated nonlinear terms leads to a new type of evolution equation for description of dynamics of downflowing films. Numerical simulations of this new equation provide an alternative approach for description of scallop waves observed in many experiments.