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About a cavitation model including bubbles in thin film lubrication: A first mathematical analysis

Part of: Two-phase and multiphase flows Equations of mathematical physics and other areas of application Compressible fluids and gas dynamics, general Incompressible viscous fluids

Published online by Cambridge University Press:  14 October 2019

ALFREDO JARAMILLO Open the ORCID record for ALFREDO JARAMILLO [Opens in a new window] ,
GUY BAYADA ,
IONEL CIUPERCA  and
MOHAMMED JAI
ALFREDO JARAMILLO
Affiliation:
Instituto de Ciências Matemáticas e de Computação, Universidade de São Paulo, 13560-970 São Carlos, Brazil email: ajaramillopalma@gmail.com
GUY BAYADA
Affiliation:
CNRS, INSA de Lyon, Institut Camille Jordan UMR 5208, Université de Lyon, F-69621 Villeurbanne, France emails: guy.bayada@insa-lyon.fr; mohammed.jai@insa-lyon.fr
IONEL CIUPERCA
Affiliation:
CNRS, Institut Camille Jordan UMR 5208, Université de Lyon, F-69622 Villeurbanne, France email: ciuperca@math.univ-lyon1.fr
MOHAMMED JAI
Affiliation:
CNRS, INSA de Lyon, Institut Camille Jordan UMR 5208, Université de Lyon, F-69621 Villeurbanne, France emails: guy.bayada@insa-lyon.fr; mohammed.jai@insa-lyon.fr
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Abstract

In lubrication problems, which concern 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 partial differential equation (PDE) (that describes the flow) and the Rayleigh–Plesset ordinary differential equation (that describes micro-bubbles evolution). A simplified form without bubbles convection is studied here. 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.

Keywords

Cavitation modellingthin film lubricationReynolds equationRayleigh–Plesset equation

MSC classification

Primary: 76D08: Lubrication theory
Secondary: 76N10: Existence, uniqueness, and regularity theory 76T10: Liquid-gas two-phase flows, bubbly flows 35Q35: PDEs in connection with fluid mechanics
Type
Papers
Information
European Journal of Applied Mathematics , Volume 31 , Issue 5 , October 2020 , pp. 828 - 853
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Copyright
© Cambridge University Press 2019

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