On a diffusion equation with rupture

TL;DR

This paper introduces a mathematical model for bubble cluster evolution with rupture, analyzing conditions for periodic solutions and suggesting rupture dynamics depend on bubble size and liquid layer thickness.

Contribution

It develops a novel diffusion-based model for bubble rupture, providing analytical and numerical insights into periodic behavior and rupture conditions.

Findings

Existence of periodic solutions when rupture occurs only in the largest bubble

Numerical evidence suggests no periodic solutions if rupture occurs in smaller bubbles

Model links rupture dynamics to liquid layer thickness and bubble size

Abstract

We propose a model to describe an evolution of a bubble cluster with rupture. In a special case, the equation is reduced to a single parabolic equation with evaporation for the thickness of a liquid layer covering bubbles. We postulate that a bubble collapses if this liquid layer becomes thin. We call this collapse a rupture. We prove for our model that there is a periodic-in-time solution if the place of rupture occurs only in the largest bubble. Numerical tests indicate that there may not exist a periodic solution if such an assumption is violated.