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
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.
KeywordsFluid Flow Electromagnetism Dirichlet Boundary Dirichlet Boundary Condition Limit Problem
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- 1.G. Allaire, Homogénéisation des équations de Navier-Stokes, Thèse, Université Paris 6 (1989).Google Scholar
- 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
- 10.C. Conca, The Stokes sieve problem, Comm. in Appl. Num. Meth., vol. 4, pp. 113–121 (1988).Google Scholar
- 14.H. Kacimi, Thèse de troisième cycle, Université Paris 6 (1988).Google Scholar
- 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
- 20.J. L. Lions, Some Methods in the Mathematical Analysis of Systems and their Control, Beijing, Gordon and Breach, New York (1981).Google Scholar
- 25.E. Sanchez-Palencia, Non Homogeneous Media and Vibration Theory, Lecture Notes in Physics 127, Springer-Verlag (1980).Google Scholar
- 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
- 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
- 28.L. Tartar, Convergence of the homogenization process, Appendix of .Google Scholar
- 29.L. Tartar, Cours Peccot au Collège de France, Unpublished (mars 1977).Google Scholar