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Steady state solutions for a lubrication multi-fluid flow

Published online by Cambridge University Press:  19 July 2011

LAURENT CHUPIN  and
BÉRÉNICE GREC
LAURENT CHUPIN
Affiliation:
Laboratoire de mathématiques, CNRS UMR 6620, Université Blaise Pascal, Clermont-Ferrand II, Campus des Cézeaux, F-63177 Aubière Cedex, France email: Laurent.Chupin@math.univ-bpclermont.fr
BÉRÉNICE GREC
Affiliation:
MAP5, CNRS UMR 8145, 45 rue des Saint Pères, F-75270 Paris Cedex 06, France email: berenice.grec@parisdescartes.fr
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Abstract

We describe possible solutions for a stationary flow of two superposed fluids between two close surfaces in relative motion. Physically, this study is within the lubrication framework, in which it is of interest to predict the relative positions of the lubricant and the air in the device. Mathematically, we observe that this problem corresponds to finding the interface between the two fluids, and we prove that this interface can be viewed as a square root of a polynomial of degree at most 6. We solve this equation using an original method. First, we check that our results are consistent with previous work. Next, we use this solution to answer some physically relevant questions related to the lubrication setting. For instance, we obtain theoretical and numerical results, which can predict the occurrence of a full film with respect to physical parameters (fluxes, shear velocity, viscosities). In particular, we present a figure giving the number of stationary solutions depending on the physical parameters. Moreover, we give some indications for a better understanding of the multi-fluid case.

Keywords

CavitationLubricationMulti-fluidSaturationSteady stateStokes equationsReynolds equation
Type
Papers
Information
European Journal of Applied Mathematics , Volume 22 , Issue 6 , December 2011 , pp. 581 - 612
Copyright
Copyright © Cambridge University Press 2011

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