Hostname: page-component-586b7cd67f-l7hp2 Total loading time: 0 Render date: 2024-11-25T19:50:47.679Z Has data issue: false hasContentIssue false
  • English
  • Français

Homogenisation of the Stokes problem with a pure non-homogeneous slip boundary condition by the periodic unfolding method

Published online by Cambridge University Press:  21 February 2011

ANCA CAPATINA  and
HORIA ENE
ANCA CAPATINA
Affiliation:
Institute of Mathematics of the Romanian Academy, P.O. Box 1-764, RO-70700 Bucharest, Romania email: [email protected]
HORIA ENE
Affiliation:
Institute of Mathematics of the Romanian Academy, P.O. Box 1-764, RO-70700 Bucharest, Romania email: [email protected] University of Piteşti, Department of Mathematics, 110040 Piteşti, România email: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We study the homogenisation of the Stokes system with a non-homogeneous Fourier boundary condition on the boundary of the holes, depending on a parameter γ. Such systems arise in the modelling of the flow of an incompressible viscous fluid through a porous medium under the influence of body forces. At the limit, by using the periodic unfolding method in perforated domains, we obtain, following the values of γ, different Darcy's laws of type Mu = −N∇p + F with suitable matrices M and N with F depending on the right-hand side in the bulk term and in the boundary condition.

Keywords

HomogenisationPeriodic unfoldingStokes systemNon-homogeneous Fourier boundary condition
Type
Papers
Information
European Journal of Applied Mathematics , Volume 22 , Issue 4 , August 2011 , pp. 333 - 345
Copyright
Copyright © Cambridge University Press 2011

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Allaire, G. (1989) Homogenization of Navier-Stokes equations with a slip boundary condition. Commun. Pure Appl. Math. XLIV, 605–642.Google Scholar
[2]Cioranescu, D. & Donato, P. (1988) Homogénéisation du problème du Neumann non homogène dans des ouverts perforés. Asymptotic Anal. 1, 115–138.Google Scholar
[3]Cioranescu, D., Donato, P. & Ene, H. (1996) Homogenization of the Stokes problem with non homogeneous slip boundary conditions. Math. Methods Appl. Sci. 19, 857–881.3.0.CO;2-D>CrossRefGoogle Scholar
[4]Cioranescu, D., Donato, P. & Griso, G. (2002) Periodic unfolding and homogenization. C. R. Math. Acad. Sci., Paris 335, 99–104.Google Scholar
[5]Cioranescu, D., Donato, P. & Zaki, R. (2006) Periodic unfolding and Robin problems in perforated domains. C. R. Math. Acad. Sci., Paris, Ser. VII 342, 469–474.CrossRefGoogle Scholar
[6]Cioranescu, D., Donato, P. & Zaki, R. (2007) Asymptotic behaviour of elliptic problems in perforated domains with nonlinear conditions. Asymptotic Anal. 53 (4), 209–235.Google Scholar
[7]Conca, C. (1985) On the application of the homogenization theory to a class of problems arising in fluid mechanics. J. Math. Pures Appl. 64, 31–75.Google Scholar
[8]Damlamian, A. (2006) An elementary introduction to periodic unfolding. Gakuto Int. Ser., Math. Sci. Appl. 24, 119–136.Google Scholar
[9]Ene, H. & Sanchez-Palencia, E. (1975) Equation et phénomenes de surface pour l'écoulement dans un modèle de milieux poreux. J. Mech. 14, 73–108.Google Scholar
[10]Melcher, J. R. (1981) Continuum Electromechanics, MIT Press, Cambridge, MA and London, England.Google Scholar
[11]Sanchez-Palencia, E. (1980) Non homogeneous media and vibration theory. Lecture Notes in Physics, Vol. 127, Springer, Berlin.Google Scholar
[12]Tartar, L. (1976) Quelques remarques sur l'homogénéisation. In: Functional Analysis and Numerical Analysis, Proc. Japan-France séminar édition, Japanese Society for the Promotion of Science, pp. 468–482.Google Scholar
[13]Tartar, L. (1980) Incompressible fluid flow in a porous medium convergence of the homogenization process. In: Appendix to Lecture Notes in Physics, Vol. 127, Springer-Velag, Berlin.Google Scholar
[14]Temam, R. (1979) Navier-Stokes Equations: Theory and Numerical Analysis, North-Holland, Amsterdam.Google Scholar
[15]Vanninathan, M. (1981) Homogenization of eigenvalues problems in perforated domains. Proc. Indian Acad. Sci. 90, 239–271.CrossRefGoogle Scholar
[16]Zaki, R. (2007) La méthode d'éclatement périodique pour l'xshomogénéisation dans les domaines perforés, Thèse, Univerité Pierre et Marie Curie, Paris.Google Scholar