Numerical solutions of the Orr-Sommerfeld equation for a thin liquid film on an inclined plane

Abstract

A numerical solution of the initial behavior of free surface liquid flows on inclined planes is presented. These instabilities, under some conditions, may evolve to surface-wave, that often appear on thin liquid films. Liquid films help us to remove the heat from solid surfaces, and also reduce the friction between high viscosity fluids and pipe walls, among other applications; therefore, understanding and predicting this behavior is useful in industry. The surface-waves instability phenomena are governed by the Orr-Sommerfeld equation and their boundary conditions. In this work we present a study through a numerical approach of the Orr-Sommerfeld equation. The numerical solution was based on a Galerkin method using Chebyshev polynomials for the discratization, which made it possible to express the Orr-Sommerfeld equation and their boundary conditions as a generalized eigenvalue problem. The method is compared with previous works for validation. The solution gives the critical conditions in which the liquid film turns unstable, and describes possible features that produce these instabilities. All codes, data and plots were produced in the MATLAB environment.