Quarterly of Applied Mathematics

			Journals Home Search Author Info Subscribe Tech Support My Subscriptions Help
		Online ISSN 1552-4485; Print ISSN 0033-569X

Asymptotic behaviour of solutions of lubrication problem in a thin domain with a rough boundary and Tresca fluid-solid interface law

Author(s): Mahdi Boukrouche; Ionel Ciuperca
Journal: Quart. Appl. Math. 64 (2006), 561-591.
MSC (2000): Primary 35R35, 35J85, 78M35, 78M40, 74K35
Posted: July 18, 2006
MathSciNet review: 2259055
Retrieve article in: PDF

Abstract | References | Similar articles | Additional information

Abstract: We study the asymptotic behavior of the solution of a Stokes flow in a thin domain, with a thickness of order $\varepsilon$ , and a rough surface. The roughness is defined by a quasi-periodic function with period $\varepsilon$ . We suppose that the flow is subject to a Tresca fluid-solid interface condition. We prove a new result on the lower-semicontinuity for the two-scale convergence, which allows us to obtain rigorously the limit problem and to establish the uniqueness of its solution.

References:

1.: R. Pit Mesure locale de la vitesse à l'interface solide-liquide simple: Glissement et rôle des interactions. Physics thesis, Univ. Paris XI, (1999).
2.: R. Pit, H. Hervet, L. Léger Direct experimental evidences for flow with slip at hexadecane solid interfaces. La revue de Métallurgie-CIT/Science, February (2001).
3.: G. Bayada, M. Boukrouche On a free boundary problem for the Reynolds equation derived from the Stokes system with Tresca boundary conditions. J. Math. Anal. Appl. (2003); 282, 212-231. MR 2000339 (2004g:35237)
4.: M. Boukrouche, G. $\L$ ukaszewicz Asymptotic analysis of solutions of a thin film lubrication problem with Coulomb fluid-solid interface law. Internat. J. Engrg. Sci. (2003); 41, 521-537. MR 1954918 (2003j:76032)
5.: M. Boukrouche and G. $\L$ ukaszewicz On a lubrication problem with Fourier and Tresca boundary conditions. Math. Models Methods Appl. Sci. (2004); 14, (6), 913-941. MR 2069499 (2005d:35277)
6.: M. Boukrouche, R. El Mir Asymptotic analysis of a non-Newtonian fluid in a thin domain with Tresca law. Nonlinear Anal. (2004), 59, no.1-2, 85-105. MR 2092079 (2005f:35331)
7.: M. Boukrouche, F. Saidi Non-isothermal lubrication problem with Tresca fluid-solid interface law. Part I. To appear in Nonlinear Anal. Real World Applications.
8.: G. Duvaut, J.L. Lions Les inéquations en mécanique et en physique. Dunod, Paris, (1972). MR 0464857 (57:4778)
9.: N. Benhaboucha, M. Chambat, I. Ciuperca Asymptotic behaviour of pressure and stresses in a thin film flow with a rough boundary. Quart. Appl. Math. (2005), 63, 369-400. MR 2150781 (2006h:35207)
10.: G. Bayada, I. Ciuperca, M. Jai Homogenized elliptic equations and variational inequalities with oscillating parameters. Application to the study of thin flow behavior with rough surfaces. To appear in Nonlinear Anal.: Real World Applications.
11.: G. Buscaglia, I. Ciuperca, M. Jai Existence and uniqueness for several non-linear elliptic problems arising in lubrication theory. J. Differential Equations, 218, Issue 1, 1 November 2005, 187-215. MR 2174972
12.: G. Buscaglia, I. Ciuperca and M. Jai On nano-scale hydrodynamic lubrication models. CRAS Mécanique, Vol. 333, Issue 6, June 2005, Pages 453-458.
13.: V. Girault, P.A. Raviart Finite Element Approximation of the Navier-Stokes Equations. Springer-Verlag, (1979). MR 0548867 (83b:65122)
14.: R. Temam Navier-Stokes Equations. Theory and Numerical Analysis. North-Holland, Amsterdam, (1979). MR 0603444 (82b:35133)
15.: G. Allaire Homogenization and two-scale convergence. SIAM J. Math. Anal. (1992), 23, no.6, 1482-1518. MR 1185639 (93k:35022)
16.: D. Lukkassen, G. Nguetseng, P. Wall Two-scale convergence. Int. J. Pure Appl. Math. (2002), 2, no. 1, 35-86. MR 1912819 (2003f:35019)
17.: I. Ekeland, R. Temam Analyse convexe et problèmes variationnels. Dunod 1974. MR 0463993 (57:3931a)
18.: U. Hornung, W. Jäger, A. Mikelic Reactive transport through an array of cells with semi-permeable membranes. RAIRO Modl. Math. Anal. Numér. (1994), 28, no.1, 59-94. MR 1259268 (94m:76118)
19.: A. Bourgeat, A. Mikelic A Note on the Homogenization of Bingham Flow through a Porous Medium. J. Math. Pures et Appl., 72 (1993), 405-414. MR 1228999 (94d:76085)
20.: A. Bourgeat, A. Mikelic, S. Wright On the Stochastic Two-Scale Convergence in the Mean and Applications. J. Reine Angew. Math. 456 (1994), 19-51. MR 1301450 (99b:35008)
21.: V. Barbu, T. Precupanu Convexity and Optimization in Banach Spaces. Mathematics and its applications. Ets. Acdemiei Bucuresti, Roumania (1986).
22.: H. Brézis Opérateurs Maximaux Monotones et semi-groupes de contractions dans les espaces de Hilbert. North-Holand, Mathematics Studies (1973). MR 0348562 (50:1060)

	Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
\|