Abstract
This paper treats 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 example, in the three-dimensional case, the obstacles have a size of ε3 and are located at the nodes of a regular mesh of size ε.) A suitable extension of the pressure is used to prove the convergence of the homogenization process to a Brinkman-type law (in which a linear zero-order term for the velocity is added to a Stokes or Navier-Stokes equation).
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
G. Allaire, Homogénéisation des équations de Stokes et de Navier-Stokes, Thèse, Université Paris 6 (1989).
G. Allaire, Homogenization of the Stokes flow in a connected porous medium, Asymptotic Analysis, 2, pp. 203–222 (1989).
G. Allaire, Homogénéisation des équations de Stokes dans un domaine perforé de petits trous répartis périodiquement, Comptes Rendus Acad. Sci. Paris, Série I, 11, pp. 741–746 (1989).
G. Allaire, Homogenization of the Navier-Stokes equations with a slip boundary condition, Comm. Pure Appl. Math, (to appear).
H. Attouch & C. Picard, Variational inequalities with varying obstacles: the general form of the limit problem, J. Funct. Analysis 50, pp. 329–386 (1983).
A. Bensoussan, J. L. Lions & G. Papanicolaou, Asymptotic Analysis for Periodic Structures, North-Holland (1978).
A. Brillard, Asymptotic analysis of incompressible and viscous fluid flow through porous media. Brinkman's law via epi-convergence methods, Ann. Fac. Sci. Toulouse 8, 2 pp. 225–252 (1986).
H. C. Brinkman, A calculation of the viscous force exerted by a flowing fluid on a dense swarm of particles, Appl. Sci. Res. A1, pp. 27–34 (1947).
D. Cioranescu & F. Murat, Un terme étrange venu d'ailleurs, Nonlinear Partial Differential Equations and their Applications, Collège de France Seminar, Vols. 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).
C. Conca, The Stokes sieve problem, Comm. in Appl. Num. Meth. 4, pp. 113–121 (1988).
E. De Giorgi, Sulla convergenza di alcune successioni di integrali del tipo dell' area, Rendiconti di Mat. 8, pp. 277–294 (1975).
E. De Giorgi, G-operators and Г-convergence, Proceedings of the International Congress of Mathematicians (Warsaw, August 1983), PWN Polish Scientific Publishers and North Holland, pp. 1175–1191 (1984).
F. Finn, Mathematical questions relating to viscous fluid flow in an exterior domain, Rocky Mountain J. Math. 3, pp. 107–140 (1973).
H. Kacimi, Thèse de troisième cycle, Université Paris 6 (1988).
H. Kacimi & F. Murat, Estimation de l'erreur dans des problèmes de Dirichlet ou apparait un terme étrange, Partial Differential Equations and the Calculus of Variations: Essays in Honor of Ennio De Giorgi, ed by F. Colombini, A. Marino, L. Modica & S. Spagnolo, Birkhäuser, Boston, pp. 661–696 (1989).
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).
R. V. Kohn & M. Vogelius, A new model for thin plates with rapidly varying thickness: II a convergence proof, Quart. Appl. Math. 43, pp. 1–22 (1985).
O. A. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flow, Gordon and Breach (1969).
T. Levy, Fluid flow through an array of fixed particles, Int. J. Engin. Sci. 21, pp. 11–23 (1983).
J. L. Lions, Some Methods in the Mathematical Analysis of Systems and Their Control, Beijing, Gordon and Breach, New York (1981).
R. Lipton & M. Avellaneda, A Darcy law for slow viscous flow past a stationary array of bubbles, Proc. Roy. Soc. Edinburgh 114A, pp. 71–79 (1990).
V. A. Marčenko & E. Ja. Hrouslov, Boundary Problems in Domains with Finely Granulated Boundaries (in Russian), Naukova Dumka, Kiev (1974).
J. Rubinstein, On the macroscopic description of slow viscous flow past a random array of spheres, J. Stat. Phys. 44, pp. 849–863 (1986).
E. Sanchez-Palencia, On the asymptotics of the fluid flow past an array of fixed obstacles, Int. J. Engin. Sci. 20, pp. 1291–1301 (1982).
E. Sanchez-Palencia, Non homogeneous media and vibration theory, Lecture Notes in Physics 127, Springer-Verlag (1980).
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).
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).
L. Tartar, Convergence of the homogenization process, Appendix of [25].
L. Tartar, Cours Peccot au Collège de France, Unpublished (mars 1977).
L. Tartar, Topics in Nonlinear Analysis, Publications mathématiques d'Orsay 78.13, Université de Paris-Sud (1978).
Author information
Authors and Affiliations
Additional information
Communicated by J. Ball
Rights and permissions
About this article
Cite this article
Allaire, G. Homogenization of the Navier-Stokes equations in open sets perforated with tiny holes I. Abstract framework, a volume distribution of holes. Arch. Rational Mech. Anal. 113, 209–259 (1991). https://doi.org/10.1007/BF00375065
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF00375065