Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X



Transition to thermohydrodynamic lubrication problem

Authors: I. S. Ciuperca, E. Feireisl, M. Jai and A. Petrov
Journal: Quart. Appl. Math. 75 (2017), 391-414
MSC (2010): Primary 35L05, 35L85, 65N30, 74M15
DOI: https://doi.org/10.1090/qam/1468
Published electronically: March 17, 2017
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Abstract: We consider a non-isothermal Stokes equation used to calculate the pressure distribution in a thin layer of lubricant film between two surfaces. The problem is described in 2D and 3D settings by the Stokes and heat transfer equations. Under appropriate regularity assumptions on the data, existence results for the non-isothermal Stokes is recalled. Using a formal asymptotic expansion, we obtain a generalized Reynolds equation coupled with a limit energy equation, the so-called non-isothermal Reynolds system. Then existence and uniqueness are proved for this system by using a fixed-point argument. Finally, a rigorous justification of the convergence is established.

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Additional Information

I. S. Ciuperca
Affiliation: Université de Lyon, CNRS, Institut Camille Jordan UMR 5208, 43 boulevard du 11 novembre 1918, F–69622 Villeurbanne Cedex, France
Email: ciuperca@math.univ-lyon.fr

E. Feireisl
Affiliation: Institute of Mathematics of the Academy of Sciences of the Czech Republic, Žitná 25, CZ–115 67 Praha 1, Czech Republic
Email: feireisl@math.cas.cz

M. Jai
Affiliation: Université de Lyon, CNRS, INSA de Lyon Institut Camille Jordan UMR 5208, 20 Avenue A. Einstein, F–69621 Villeurbanne, France
Email: mohammed.jai@insa-lyon

A. Petrov
Affiliation: Université de Lyon, CNRS, INSA de Lyon Institut Camille Jordan UMR 5208, 20 Avenue A. Einstein, F–69621 Villeurbanne, France
Email: apetrov@math.univ-lyon1.fr

DOI: https://doi.org/10.1090/qam/1468
Keywords: Free boundary problems, lubrication, asymptotic approach, Stokes equation, Reynolds equation.
Received by editor(s): August 21, 2016
Published electronically: March 17, 2017
Additional Notes: The research of the second author leading to these results has received funding from the European Research Council under the European Union’s Seventh Framework Program (FP7/2007–2013)/ ERC Grant Agreement 320078. The Institute of Mathematics of the Academy of Sciences of the Czech Republic is supported by RVO: 67985840.
Article copyright: © Copyright 2017 Brown University

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