On weak solutions of a control-volume model for liquid films flowing down a fibre

TL;DR

This paper analytically investigates weak solutions and traveling waves in a control-volume model for liquid films flowing down a fibre, combining mathematical proofs with numerical simulations.

Contribution

It establishes the existence of weak and traveling wave solutions for a nonlinear PDE system modeling liquid film flow on fibres, advancing theoretical understanding.

Findings

Existence of weak solutions proven using energy-entropy estimates

Traveling wave solutions are demonstrated analytically

Numerical simulations illustrate dynamic and wave solutions

Abstract

This paper presents an analytical investigation of the solutions to a control volume model for liquid films flowing down a vertical fibre. The evolution of the free surface is governed by a coupled system of degenerate nonlinear partial differential equations, which describe the fluid film's radius and axial velocity. We demonstrate the existence of weak solutions to this coupled system by applying a priori estimates derived from energy-entropy functionals. Additionally, we establish the existence of traveling wave solutions for the system. To illustrate our analytical findings, we present numerical studies that showcase the dynamic solutions of the partial differential equations as well as the traveling wave solutions.