Archive for Rational Mechanics and Analysis

, Volume 113, Issue 3, pp 261–298 | Cite as

Homogenization of the Navier-Stokes equations in open sets perforated with tiny holes II: Non-critical sizes of the holes for a volume distribution and a surface distribution of holes

  • Grégoire Allaire


This paper is devoted to the homogenization of the Stokes or Navier-Stokes equations with a Dirichlet boundary condition in a domain containing many tiny solid obstacles, periodically distributed in each direction of the axes. For obstacles of critical size it was established in Part I that the limit problem is described by a law of Brinkman type. Here we prove that for smaller obstacles, the limit problem reduces to the Stokes or Navier-Stokes equations, and for larger obstacles, to Darcy's law. We also apply the abstract framework of Part I to the case of a domain containing tiny obstacles, periodically distributed on a surface. (For example, in three dimensions, consider obstacles of size ε2, located at the nodes of a regular plane mesh of period ε.) This provides a mathematical model for fluid flows through mixing grids, based on a special form of the Brinkman law in which the additional term is concentrated on the plane of the grid.


Fluid Flow Electromagnetism Dirichlet Boundary Dirichlet Boundary Condition Limit Problem 
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  1. 1.
    G. Allaire, Homogénéisation des équations de Navier-Stokes, Thèse, Université Paris 6 (1989).Google Scholar
  2. 9.
    D. Cioranescu & F. Murat, Un terme étrange venu d'ailleurs, Nonlinear Partial Differential Equations and their Applications, Collège de France Seminar, Vol. 2 & 3, ed. by H. Brezis & J. L. Lions, Research Notes in Mathematics 60, pp. 98–138, and 70, pp. 154–178, Pitman, London (1982).Google Scholar
  3. 10.
    C. Conca, The Stokes sieve problem, Comm. in Appl. Num. Meth., vol. 4, pp. 113–121 (1988).Google Scholar
  4. 14.
    H. Kacimi, Thèse de troisième cycle, Université Paris 6 (1988).Google Scholar
  5. 16.
    J. B. Keller, Darcy's law for flow in porous media and the two-space method, Lecture Notes in Pure and Appl. Math. 54, Dekker, New York (1980).Google Scholar
  6. 20.
    J. L. Lions, Some Methods in the Mathematical Analysis of Systems and their Control, Beijing, Gordon and Breach, New York (1981).Google Scholar
  7. 25.
    E. Sanchez-Palencia, Non Homogeneous Media and Vibration Theory, Lecture Notes in Physics 127, Springer-Verlag (1980).Google Scholar
  8. 26.
    E. Sanchez-Palencia, Problèmes mathématiques liés à l'écoulement d'un fluide visqueux à travers une grille, Ennio de Giorgi Colloquium, ed. by P. Krée, Research Notes in Mathematics 125, pp. 126–138, Pitman, London (1985).Google Scholar
  9. 27.
    E. Sanchez-Palencia, Boundary-value problems in domains containing perforated walls, Nonlinear Partial Differential Equations and their Applications, Collège de France Seminar, Vol. 3, ed. by H. Brezis & J. L. Lions, Research Notes in Mathematics 70, pp. 309–325, Pitman, London (1982).Google Scholar
  10. 28.
    L. Tartar, Convergence of the homogenization process, Appendix of [25].Google Scholar
  11. 29.
    L. Tartar, Cours Peccot au Collège de France, Unpublished (mars 1977).Google Scholar

Copyright information

© Springer-Verlag 1991

Authors and Affiliations

  • Grégoire Allaire
    • 1
  1. 1.Commissariat à l'Energie AtomiqueC.E.N. SaclayGif sur YvetteFrance

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