Fluid accumulation in thin-film fl ows driven by surface tension and gravity (I): Rigorous analysis of a drainage equation

TL;DR

This paper rigorously analyzes a drainage equation related to thin-film flows, revealing oscillatory behaviors of solutions and providing insights into accumulation regions driven by gravity and surface tension.

Contribution

It offers a rigorous mathematical analysis of the drainage equation, establishing conditions for oscillations and their specific patterns, advancing understanding of thin-film flow dynamics.

Findings

Solutions not tending to 1 are oscillatory.

Solutions oscillate in a specific, predictable manner.

The analysis informs the boundary layer problem in thin-film flows.

Abstract

We derive a boundary layer equation describing accumulation regions within a thin-film approximation framework where gravity and surface tension balance. As part of the analysis of this problem we investigate in detail and rigorously the 'drainage' equation (phi"'+1)phi^3=1. In particular, we prove that all solutions that do not tend to 1 as the independent variable goes to infinity are oscillatory, and that they oscillate in a very specific way. This result and the method of proof will be used in the analysis of solutions of the afore mentioned boundary layer problem.