Analysis of a cavitation model including bubbles in thin film lubrication

TL;DR

This paper investigates a novel coupled cavitation model combining flow dynamics and micro-bubbles in thin film lubrication, proving local existence, stability, and providing numerical validation of the mathematical properties.

Contribution

It introduces the first mathematical analysis of a coupled PDE-ODE cavitation model in thin film lubrication, establishing existence and stability results.

Findings

Proved local existence of solutions for the coupled model.

Established stability theorems based on spectral continuity.

Numerical results support theoretical findings.

Abstract

In the lubrication area, which is concerned with thin film flow, cavitation has been considered as a fundamental element to correctly describe the characteristics of lubricated mechanisms. Here, the well-posedness of a cavitation model that can explain the interaction between viscous effects and micro-bubbles of gas is studied. This cavitation model consists of a coupled problem between the compressible Reynolds PDE (that describes the flow) and the Rayleigh-Plesset ODE (that describes micro-bubbles evolution). This coupled model seems never to be studied before from its mathematical aspects. Local times existence results are proved and stability theorems are obtained based on the continuity of the spectrum for bounded linear operators. Numerical results are presented to illustrate these theoretical results.