Asymptotic behaviour of solutions of lubrication problem in a thin domain with a rough boundary and Tresca fluid-solid interface law
Authors:
Mahdi Boukrouche and Ionel Ciuperca
Journal:
Quart. Appl. Math. 64 (2006), 561-591
MSC (2000):
Primary 35R35, 35J85, 78M35, 78M40, 74K35
DOI:
https://doi.org/10.1090/S0033-569X-06-01030-3
Published electronically:
July 18, 2006
MathSciNet review:
2259055
Full-text PDF Free Access
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.
Pit99 R. Pit Mesure locale de la vitesse à l’interface solide-liquide simple: Glissement et rôle des interactions. Physics thesis, Univ. Paris XI, (1999).
Pit2001 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).
- Guy Bayada and Mahdi Boukrouche, On a free boundary problem for the Reynolds equation derived from the Stokes systems with Tresca boundary conditions, J. Math. Anal. Appl. 282 (2003), no. 1, 212–231. MR 2000339, DOI https://doi.org/10.1016/S0022-247X%2803%2900140-9
- Mahdi Boukrouche and Grzegorz Łukaszewicz, Asymptotic analysis of solutions of a thin film lubrication problem with Coulomb fluid-solid interface law, Internat. J. Engrg. Sci. 41 (2003), no. 6, 521–537. MR 1954918, DOI https://doi.org/10.1016/S0020-7225%2802%2900282-3
- Mahdi Boukrouche and Grzegorz Łukaszewicz, On a lubrication problem with Fourier and Tresca boundary conditions, Math. Models Methods Appl. Sci. 14 (2004), no. 6, 913–941. MR 2069499, DOI https://doi.org/10.1142/S0218202504003490
- Mahdi Boukrouche and Rachid El Mir, Asymptotic analysis of a non-Newtonian fluid in a thin domain with Tresca law, Nonlinear Anal. 59 (2004), no. 1-2, 85–105. MR 2092079, DOI https://doi.org/10.1016/j.na.2004.07.003
bs1 M. Boukrouche, F. Saidi Non-isothermal lubrication problem with Tresca fluid-solid interface law. Part I. To appear in Nonlinear Anal. Real World Applications.
- G. Duvaut and J.-L. Lions, Les inéquations en mécanique et en physique, Dunod, Paris, 1972 (French). Travaux et Recherches Mathématiques, No. 21. MR 0464857
- Nadia Benhaboucha, Michèle Chambat, and Ionel Ciuperca, Asymptotic behaviour of pressure and stresses in a thin film flow with a rough boundary, Quart. Appl. Math. 63 (2005), no. 2, 369–400. MR 2150781, DOI https://doi.org/10.1090/S0033-569X-05-00963-3
Ciuperca2 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.
- G. Buscaglia, I. Ciuperca, and M. Jai, Existence and uniqueness for several non-linear elliptic problems arising in lubrication theory, J. Differential Equations 218 (2005), no. 1, 187–215. MR 2174972, DOI https://doi.org/10.1016/j.jde.2005.06.018
Ciuperca4 G. Buscaglia, I. Ciuperca and M. Jai On nano-scale hydrodynamic lubrication models. CRAS Mécanique, Vol. 333, Issue 6, June 2005, Pages 453-458.
- 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
- 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
- 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
- Dag Lukkassen, Gabriel Nguetseng, and Peter Wall, Two-scale convergence, Int. J. Pure Appl. Math. 2 (2002), no. 1, 35–86. MR 1912819
- Ivar Ekeland and Roger Temam, Analyse convexe et problèmes variationnels, Dunod; Gauthier-Villars, Paris-Brussels-Montreal, Que., 1974 (French). Collection Études Mathématiques. MR 0463993
- U. Hornung, W. Jäger, and A. Mikelić, Reactive transport through an array of cells with semi-permeable membranes, RAIRO Modél. Math. Anal. Numér. 28 (1994), no. 1, 59–94 (English, with English and French summaries). MR 1259268, DOI https://doi.org/10.1051/m2an/1994280100591
- A. Bourgeat and A. Mikelić, A note on homogenization of Bingham flow through a porous medium, J. Math. Pures Appl. (9) 72 (1993), no. 4, 405–414 (English, with English and French summaries). MR 1228999
- Alain Bourgeat, Andro Mikelić, and Steve Wright, Stochastic two-scale convergence in the mean and applications, J. Reine Angew. Math. 456 (1994), 19–51. MR 1301450
Barbu V. Barbu, T. Precupanu Convexity and Optimization in Banach Spaces. Mathematics and its applications. Ets. Acdemiei Bucuresti, Roumania (1986).
- H. Brézis, Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert, North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., Inc., New York, 1973 (French). North-Holland Mathematics Studies, No. 5. Notas de Matemática (50). MR 0348562
Pit99 R. Pit Mesure locale de la vitesse à l’interface solide-liquide simple: Glissement et rôle des interactions. Physics thesis, Univ. Paris XI, (1999).
Pit2001 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).
BB03 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.
Boukrouche03 M. Boukrouche, G. Łukaszewicz Asymptotic analysis of solutions of a thin film lubrication problem with Coulomb fluid-solid interface law. Internat. J. Engrg. Sci. (2003); 41, 521-537.
Boukrouche04 M. Boukrouche and G. Łukaszewicz On a lubrication problem with Fourier and Tresca boundary conditions. Math. Models Methods Appl. Sci. (2004); 14, (6), 913-941.
Boukrouche004 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.
bs1 M. Boukrouche, F. Saidi Non-isothermal lubrication problem with Tresca fluid-solid interface law. Part I. To appear in Nonlinear Anal. Real World Applications.
Duvaut72 G. Duvaut, J.L. Lions Les inéquations en mécanique et en physique. Dunod, Paris, (1972).
bcc1 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.
Ciuperca2 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.
Ciuperca3 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.
Ciuperca4 G. Buscaglia, I. Ciuperca and M. Jai On nano-scale hydrodynamic lubrication models. CRAS Mécanique, Vol. 333, Issue 6, June 2005, Pages 453-458.
Girault79 V. Girault, P.A. Raviart Finite Element Approximation of the Navier-Stokes Equations. Springer-Verlag, (1979).
Temam79 R. Temam Navier-Stokes Equations. Theory and Numerical Analysis. North-Holland, Amsterdam, (1979).
Allaire92 G. Allaire Homogenization and two-scale convergence. SIAM J. Math. Anal. (1992), 23, no.6, 1482-1518.
Negu D. Lukkassen, G. Nguetseng, P. Wall Two-scale convergence. Int. J. Pure Appl. Math. (2002), 2, no. 1, 35–86.
Ekeland I. Ekeland, R. Temam Analyse convexe et problèmes variationnels. Dunod 1974.
Jager U. Hornung, W. Jäger, A. Mikelić Reactive transport through an array of cells with semi-permeable membranes. RAIRO Modl. Math. Anal. Numér. (1994), 28, no.1, 59-94.
Mikelic93 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.
Bourgeat94 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.
Barbu V. Barbu, T. Precupanu Convexity and Optimization in Banach Spaces. Mathematics and its applications. Ets. Acdemiei Bucuresti, Roumania (1986).
Brezis H. Brézis Opérateurs Maximaux Monotones et semi-groupes de contractions dans les espaces de Hilbert. North-Holand, Mathematics Studies (1973).
Similar Articles
Retrieve articles in Quarterly of Applied Mathematics
with MSC (2000):
35R35,
35J85,
78M35,
78M40,
74K35
Retrieve articles in all journals
with MSC (2000):
35R35,
35J85,
78M35,
78M40,
74K35
Additional Information
Mahdi Boukrouche
Affiliation:
Laboratory of Mathematics, University of Saint-Etienne, EA 3989, 23 rue Dr. Paul Michelon, 42023 Saint-Etienne, France
MR Author ID:
335804
Email:
Mahdi.Boukrouche@univ-st-etienne.fr
Ionel Ciuperca
Affiliation:
Institut Camille Jordan, Université Lyon 1, UMR 5208, France
Email:
ciuperca@math.univ-lyon1.fr
Keywords:
Free boundary problem,
Lubrication,
Rough boundary,
Tresca fluid-solid conditions,
Homogenization,
Lower-semicontinuity for the two-scale convergence,
Reynolds equation.
Received by editor(s):
January 27, 2006
Published electronically:
July 18, 2006
Article copyright:
© Copyright 2006
Brown University
The copyright for this article reverts to public domain 28 years after publication.