Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

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



Asymptotic behaviour of pressure and stresses in a thin film flow with a rough boundary

Authors: Nadia Benhaboucha, Michèle Chambat and Ionel Ciuperca
Journal: Quart. Appl. Math. 63 (2005), 369-400
MSC (2000): Primary 76D07, 76D08, 78M35, 78M40
DOI: https://doi.org/10.1090/S0033-569X-05-00963-3
Published electronically: May 4, 2005
MathSciNet review: 2150781
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Abstract: We study the asymptotic behaviour of an incompressible viscous flow in a narrow gap with a thickness of order $\eta$ and a rough surface. The roughness is defined by a quasi-periodical function with period $\epsilon$. In both cases, when $\eta$ is smaller or at the same order as $\epsilon$, we obtain different Reynolds equations for the pressure. We have also studied the convergence of the stresses on the rough boundary and we discuss the different cases.

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  • 1. Yves Achdou, O. Pironneau, and F. Valentin, Effective boundary conditions for laminar flows over periodic rough boundaries, J. Comput. Phys. 147 (1998), no. 1, 187–218. MR 1657773, https://doi.org/10.1006/jcph.1998.6088
  • 2. A. Ait Moussa, C. Licht, P. Suquet, Modélisation et singularités de contraintes d'un joint collé mince, Actes $9^{e}$Congrès français de Mécanique (Metz) (1989), 258-259.
  • 3. Grégoire Allaire, Homogenization and two-scale convergence, SIAM J. Math. Anal. 23 (1992), no. 6, 1482–1518. MR 1185639, https://doi.org/10.1137/0523084
  • 4. Youcef Amirat, Didier Bresch, Jérôme Lemoine, and Jacques Simon, Effect of rugosity on a flow governed by stationary Navier-Stokes equations, Quart. Appl. Math. 59 (2001), no. 4, 769–785. MR 1866556, https://doi.org/10.1090/qam/1866556
  • 5. Guy Bayada and Michèle Chambat, Homogenization of the Stokes system in a thin film flow with rapidly varying thickness, RAIRO Modél. Math. Anal. Numér. 23 (1989), no. 2, 205–234 (English, with French summary). MR 1001328, https://doi.org/10.1051/m2an/1989230202051
  • 6. G. Bayada, M. Chambat, New models in the theory of the hydrodynamic lubrication of rough surfaces, Trans. of the AMS., J. of Trib. 110 (1988), 402-407.
  • 7. G. Bayada, M. Chambat and J. B. Faure, Some effects of the boundary roughness in a thin film flow, Preprint no. 58 1986, Lyon1. MR 0942450 (89i:76044)
  • 8. G. Bayada, M. Chambat, and K. Lhalouani, Asymptotic analysis of a thin-layer device with Tresca’s contact law in elasticity, Math. Methods Appl. Sci. 22 (1999), no. 10, 811–836. MR 1700180, https://doi.org/10.1002/(SICI)1099-1476(19990710)22:10<811::AID-MMA63>3.0.CO;2-K
  • 9. H. Brézis, Analyse fonctionnelle, Théorie et applications, Dunod, Paris, 1999. MR 0697382 (85a:46001)
  • 10. I. Charpentier and J. Saint Jean Paulin, Limit behaviour of a three-dimensional thin tall structure depending on three small parameters, Ricerche Mat. 44 (1995), no. 2, 459–488 (1996). MR 1469715
  • 11. H. Christensen and K. Tonder, The hydrodynamic lubrication of rough bearing surfaces of finite width, J. of Lub. Tech., Trans. ASME, F 93 (1971), no. 3, 324-330.
  • 12. M. Dauge, Stationary Stokes and Navier-Stokes systems on two- or three-dimensional domains with corners, Part 1 - linearized equations, SIAM J. Math. Anal. 20 (1989) 74-97. MR 0977489 (90b:35191)
  • 13. A. Dyson, Hydrodynamic lubrication of rough surfaces a review work, Proceedings of the 4th Leeds-Lyon Symposium on surfaces roughness on lubrication, 1977, IME, 61-69.
  • 14. H. G. Elrod, A review of theories for the fluid dynamic effects of roughness on laminar lubricating films, Proceedings of the 4th Leeds-Lyon Symposium on surfaces roughness on lubrication, 1977, IME, 11-26.
  • 15. J. B. Faure, Application des techniques d'homogénéisation à la prise en compte des phénomènes de rugosité en lubrification hydrodynamique, thèse, Lyon 1, 1986.
  • 16. V. Girault and P.-A. Raviart, Finite Element Approximation of the Navier-Stokes Equation, Springer-Verlag, 1979. MR 0548867 (83b:65122)
  • 17. Willi Jäger and Andro Mikelić, Couette flows over a rough boundary and drag reduction, Comm. Math. Phys. 232 (2003), no. 3, 429–455. MR 1952473, https://doi.org/10.1007/s00220-002-0738-8
  • 18. Christian Licht, Comportement asymptotique d’une bande dissipative mince de faible rigidité, C. R. Acad. Sci. Paris Sér. I Math. 317 (1993), no. 4, 429–433 (French, with English and French summaries). MR 1235462
  • 19. G. Nguetseng, A general Convergence Result for a Functional related to the Theory of Homogenization, SIAM J. Math. Anal. 20 (1989), no. 3, 608-623. MR 0990867 (90j:35030)
  • 20. N. Phan Thien, Hydrodynamic lubrication of rough surfaces, Proc. R. Soc. London, A383 (1982) 439-446.
  • 21. E. Sanchez-Palencia, Non homogeneous media and vibration theory, Lecture Notes in Physics 127, Springer Verlag, Berlin, 1978.
  • 22. R. Temam, Navier-Stokes Equations, North Holland, Amsterdam, 1979. MR 0603444 (82b:35133)
  • 23. R. Temam, Autour de la Mécanique des fluides, Cours de Dea, ENS de Lyon, Mars 2001.

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Additional Information

Nadia Benhaboucha
Affiliation: CNRS UMR 5585, Maply, Bat J. Braconnier, Univ. Lyon1, 69622 Villeurbanne Cedex France
Email: Nadia.Benhaboucha@maply.univ-lyon1.fr

Michèle Chambat
Affiliation: CNRS UMR 5585, Maply, Bat J. Braconnier, Univ. Lyon1, 69622 Villeurbanne Cedex France
Email: chambat@maply.univ-lyon1.fr

Ionel Ciuperca
Affiliation: CNRS UMR 5585, Maply, Bat J. Braconnier, Univ. Lyon1, 69622 Villeurbanne Cedex France
Email: ciuperca@maply.univ-lyon1.fr

DOI: https://doi.org/10.1090/S0033-569X-05-00963-3
Keywords: Homogenization, rough boundary, Reynolds equation, asymptotic stresses
Received by editor(s): January 3, 2005
Published electronically: May 4, 2005
Article copyright: © Copyright 2005 Brown University

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