Numerical solution for Kapitza waves on a thin liquid film

TL;DR

This paper introduces a spectral Galerkin method using Chebyshev polynomials to accurately and efficiently solve the Orr-Sommerfeld equation for Kapitza waves on thin liquid films, with validation against analytical and experimental data.

Contribution

The paper presents a stable, fast spectral method that effectively incorporates boundary conditions for solving free surface flow instabilities, improving upon existing numerical approaches.

Findings

Method shows excellent agreement with analytical solutions in long-wave regime.

Results align well with experimental data, confirming accuracy.

Spectral approach offers stability and efficiency for initial instability analysis.

Abstract

The flow of a liquid film over an inclined plane is frequently found in nature and industry, and, under some conditions, instabilities in the free surface may appear. These instabilities are initially two-dimensional surface waves, known as Kapitza waves. Surface waves are important to many industrial applications. For example, liquid films with surface waves are employed to remove heat from solid surfaces. The initial phase of the instability is governed by the Orr-Sommerfeld equation and the appropriate boundary conditions; therefore, the fast and accurate solution of this equation is useful for industry. This paper presents a spectral method to solve the Orr-Sommerfeld equation with free surface boundary conditions. Our numerical approach is based on a Galerkin method with Chebyshev polynomials of the first kind, making it possible to express the Orr-Sommerfeld equation and their boundary conditions as a generalized eigenvalue problem. The main advantages of the present spectral method when compared to others such as, for instance, spectral collocation, are its stability and its readiness in including the boundary conditions in the discretized equations. We compare our numerical results with analytical solutions based on Perturbation Methods, which are valid only for long wave instabilities, and show that the results agree in the region of validity of the long-wave hypothesis. Far from this region, our results are still valid. In addition, we compare our results with published experimental results and the agreement is very good. The method is stable, fast, and capable to solve initial instabilities in free surface flows.