On travelling wave solutions of a model of a liquid film flowing down a fibre

TL;DR

This paper proves the existence and long-term behavior of travelling wave solutions in a liquid film flowing down a fibre, combining analytical proofs with numerical simulations.

Contribution

It establishes the existence of weak solutions for a full curvature thin-film model and analyzes their convergence to travelling waves under certain conditions.

Findings

Existence of non-negative weak solutions is proven.

Long-time convergence to travelling wave solutions is demonstrated.

Numerical simulations support analytical results.

Abstract

Existence of non-negative weak solutions is shown for a full curvature thin-film model of a liquid thin film flowing down a vertical fibre. The proof is based on the application of a priori estimates derived for energy-entropy functionals. Long-time behaviour of these weak solutions is analysed and, under some additional constraints for the model parameters and initial values, convergence towards a travelling wave solution is obtained. Numerical studies of energy minimizers and travelling waves are presented to illustrate analytical results.