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

Benhaboucha, Nadia; Chambat, Michèle; Ciuperca, Ionel

doi:10.1090/S0033-569X-05-00963-3

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.

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, DOI https://doi.org/10.1006/jcph.1998.6088

Ait 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.

Grégoire Allaire, Homogenization and two-scale convergence, SIAM J. Math. Anal. 23 (1992), no. 6, 1482–1518. MR 1185639, DOI https://doi.org/10.1137/0523084
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, DOI https://doi.org/10.1090/qam/1866556
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, DOI https://doi.org/10.1051/m2an/1989230202051

B-Chchap2 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.

M. Chambat, G. Bayada, and J. B. Faure, Some effects of the boundary roughness in a thin film, Boundary control and boundary variations (Nice, 1986) Lecture Notes in Comput. Sci., vol. 100, Springer, Berlin, 1988, pp. 96–115. MR 942450, DOI https://doi.org/10.1007/BFb0041913
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, DOI https://doi.org/10.1002/%28SICI%291099-1476%2819990710%2922%3A10%3C811%3A%3AAID-MMA63%3E3.0.CO%3B2-K
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
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

christensenchap2 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.

Monique Dauge, Stationary Stokes and Navier-Stokes systems on two- or three-dimensional domains with corners. I. Linearized equations, SIAM J. Math. Anal. 20 (1989), no. 1, 74–97. MR 977489, DOI https://doi.org/10.1137/0520006

Dyson 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.

Elrod 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.

TheseFaurechap2 J. B. Faure, Application des techniques d’homogénéisation à la prise en compte des phénomènes de rugosité en lubrification hydrodynamique, thèse, Lyon 1, 1986.

V. Girault and P.-A. Raviart, Finite element approximation of the Navier-Stokes equations, Lecture Notes in Mathematics, vol. 749, Springer-Verlag, Berlin-New York, 1979. MR 548867
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, DOI https://doi.org/10.1007/s00220-002-0738-8
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
Gabriel 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 990867, DOI https://doi.org/10.1137/0520043

phanchap2 N. Phan Thien, Hydrodynamic lubrication of rough surfaces, Proc. R. Soc. London, A383 (1982) 439–446.

Sanchezchap2 E. Sanchez-Palencia, Non homogeneous media and vibration theory, Lecture Notes in Physics 127, Springer Verlag, Berlin, 1978.

Roger Temam, Navier-Stokes equations, Revised edition, Studies in Mathematics and its Applications, vol. 2, North-Holland Publishing Co., Amsterdam-New York, 1979. Theory and numerical analysis; With an appendix by F. Thomasset. MR 603444

courschap2 R. Temam, Autour de la Mécanique des fluides, Cours de Dea, ENS de Lyon, Mars 2001.