A Mathematical Justification of a Thin Film Approximation for the Flow down an Inclined Plane

TL;DR

This paper rigorously justifies the use of thin film approximations for flow down inclined planes by providing error estimates between Navier-Stokes solutions and simplified models.

Contribution

It offers a mathematical proof validating the accuracy of thin film approximations derived from the Navier-Stokes equations.

Findings

Error estimates between Navier-Stokes and approximate models

Validation of thin film approximation accuracy

Mathematical framework for stability analysis

Abstract

We consider a two-dimensional motion of a thin film flowing down an inclined plane under the influence of the gravity and the surface tension. In order to investigate the stability of such flow, we often use a thin film approximation, which is an approximation obtained by the perturbation expansion with respect to the aspect ratio of the film. The famous example of the approximate equations are the Burgers equation, Kuramoto--Sivashinsky equation, KdV--Burgers equation, KdV--Kuramoto--Sivashinsky equation, and so on. In this paper, we give a mathematically rigorous justification of a thin film approximation by establishing an error estimate between the solution of the Navier--Stokes equations and those of approximate equations.