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
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.

References [Enhancements On Off] (What's this?)

  • [1] Robert A. Adams, Sobolev spaces, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1975. Pure and Applied Mathematics, Vol. 65. MR 0450957
  • [2] Chérif Amrouche and Vivette Girault, Decomposition of vector spaces and application to the Stokes problem in arbitrary dimension, Czechoslovak Math. J. 44(119) (1994), no. 1, 109–140. MR 1257940
  • [3] Guy Bayada and Michèle Chambat, The transition between the Stokes equations and the Reynolds equation: a mathematical proof, Appl. Math. Optim. 14 (1986), no. 1, 73–93. MR 826853, https://doi.org/10.1007/BF01442229
  • [4] Mahdi Boukrouche and Rachid El Mir, On a non-isothermal, non-Newtonian lubrication problem with Tresca law: existence and the behavior of weak solutions, Nonlinear Anal. Real World Appl. 9 (2008), no. 2, 674–692. MR 2382968, https://doi.org/10.1016/j.nonrwa.2006.12.012
  • [5] Haïm Brezis, Analyse fonctionnelle, Collection Mathématiques Appliquées pour la Maîtrise. [Collection of Applied Mathematics for the Master’s Degree], Masson, Paris, 1983 (French). Théorie et applications. [Theory and applications]. MR 697382
  • [6] A. Cameron,
    The viscous wedge,
    Trans. ASME 248 (1958), no. 1.
  • [7] W. F. Cope,
    The hydrodynamic theory of film lubrication,
    Proc. R. Soc. London, Ser. A 197 (1949), 201-217.
  • [8] G. Duvaut and J.-L. Lions, Inequalities in mechanics and physics, Springer-Verlag, Berlin-New York, 1976. Translated from the French by C. W. John; Grundlehren der Mathematischen Wissenschaften, 219. MR 0521262
  • [9] B. O. Jacobson,
    At the boundary between lubrication and wear, First World Tribology Conference, London (1997), pp. 291-298.
  • [10] B. O. Jacobson and B. J. Hamrock,
    Non-Newtonian fluid incorporated into elastohydrodynamic lubrication of rectangular contacts, J. Tribol 106 (1984), 275-284.
  • [11] V. A. Kondrat′ev and O. A. Oleĭnik, Boundary value problems for a system in elasticity theory in unbounded domains. Korn inequalities, Uspekhi Mat. Nauk 43 (1988), no. 5(263), 55–98, 239 (Russian); English transl., Russian Math. Surveys 43 (1988), no. 5, 65–119. MR 971465, https://doi.org/10.1070/RM1988v043n05ABEH001945
  • [12] Nader Masmoudi, Some uniform elliptic estimates in a porous medium, C. R. Math. Acad. Sci. Paris 339 (2004), no. 12, 849–854 (English, with English and French summaries). MR 2111721, https://doi.org/10.1016/j.crma.2004.10.007
  • [13] SergueïA. Nazarov and Juha H. Videman, A modified nonlinear Reynolds equation for thin viscous flows in lubrication, Asymptot. Anal. 52 (2007), no. 1-2, 1–36. MR 2337025
  • [14] R. Pitt, H. Hervet and L. Léger,
    Direct experimental evidences for flow with slip at hexadecane solid interfaces,
    La revue de métallurgie-CIT/Science, February 2001.
  • [15] 0. Reynolds,
    On the theory of lubrication and its application to Mr. Beauchamp Tower's experiment, Phil. Trans. R. Soc. London 177 (1886), no. 1, 157-234.
  • [16] James Serrin, Mathematical principles of classical fluid mechanics, Handbuch der Physik (herausgegeben von S. Flügge), Bd. 8/1, Strömungsmechanik I (Mitherausgeber C. Truesdell), Springer-Verlag, Berlin-Göttingen-Heidelberg, 1959, pp. 125–263. MR 0108116
  • [17] J. Shieh and B. J. Hamrock, Film collapse in EHL and micro-EHL, J. Tribol. 113 (1991), 372-377.
  • [18] Jacques Simon, Compact sets in the space 𝐿^{𝑝}(0,𝑇;𝐵), Ann. Mat. Pura Appl. (4) 146 (1987), 65–96. MR 916688, https://doi.org/10.1007/BF01762360
  • [19] Jon Wilkening, Practical error estimates for Reynolds’ lubrication approximation and its higher order corrections, SIAM J. Math. Anal. 41 (2009), no. 2, 588–630. MR 2507463, https://doi.org/10.1137/070695447
  • [20] 0. C. Zienkiewicz, A note on theory of hydrodynamic lubrication of parallel surface thrust bearings, Proc. 9th Int. Conf. on Applied Mechanics, Brussels 4 (1957), 251-258.
  • [21] 0. C. Zienkiewicz,
    Temperature distribution within lubricating films between parallel surfaces and its effect on the pressure developed, Proc. Conf. on Lubrication and Wear, Inst. Mech. Eng., London, Pap. 81 (1957), p. 135.

Similar Articles

Retrieve articles in Quarterly of Applied Mathematics with MSC (2010): 35L05, 35L85, 65N30, 74M15

Retrieve articles in all journals with MSC (2010): 35L05, 35L85, 65N30, 74M15

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

Brown University The Quarterly of Applied Mathematics
is distributed by the American Mathematical Society
for Brown University
Online ISSN 1552-4485; Print ISSN 0033-569X
© 2017 Brown University
Comments: qam-query@ams.org
AMS Website