## Homogenization of the Navier-Stokes equations in open sets perforated with tiny holes. I: Abstract framework, a volume distribution of holes.(English)Zbl 0724.76020

The homogenization of the Stokes (or Navier-Stokes) equations with a no- slip (Dirichlet) boundary condition in open sets perforated with tiny holes (obstacles) is considered for flows in porous media as well as in mixing grids. For flows in a porous medium, modeled as a periodic repetition of an elementary cell of size $$\epsilon$$, the cirtical size of a solid obstacle (hole) is defined as $$\exp (-1/\epsilon^ 2)$$ in the two-dimensional case and $$\epsilon^{N/(N-2)}$$ in the N-dimensional case (N$$\geq 3)$$. The homogenization of the Stokes equations converges to the Brinkman law for the critical hole size, to the Darcy law if the hole size is asymptotically larger than the critical size, and to the Stokes law if it is asymptotically smaller than the critical size. For flows in mixing grids, the homogenization of the Stokes equations converges to the Darcy law if the holes have the same size as the period, and to a particular form of the Brinkman law in a mixing grid represented by its vanes of size $$\epsilon^ 2$$ in the three-dimensional case periodically distributed at the nodes of a regular mesh of size $$\epsilon$$. This part of the article deals with the abstract formulation for these two kinds of flows. The Brinkman law is derived in this part, and it is proved that in the two-dimensional case the limiting Brinkman-type law is independent of the shape of the holes. Effective equations associated with holes are established on a hypersurface rather than throughout the medium. Optimal $$L^ 2$$-estimates of the pressure, leading to a proof of the convergence of the homogenization process and some new results, including correctors and error estimates, are derived.

### MSC:

 76D05 Navier-Stokes equations for incompressible viscous fluids 35Q30 Navier-Stokes equations

Zbl 0724.76021
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### References:

 [1] G. Allaire, Homogénéisation des équations de Stokes et de Navier-Stokes, Thèse, Université Paris 6 (1989). [2] G. Allaire, Homogenization of the Stokes flow in a connected porous medium, Asymptotic Analysis, 2, pp. 203-222 (1989). · Zbl 0682.76077 [3] 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). · Zbl 0696.73010 [4] G. Allaire, Homogenization of the Navier-Stokes equations with a slip boundary condition, Comm. Pure Appl. Math, (to appear). · Zbl 0738.35059 [5] H. Attouch & C. Picard, Variational inequalities with varying obstacles: the general form of the limit problem, J. Funct. Analysis 50, pp. 329-386 (1983). [6] A. Bensoussan, J. L. Lions & G. Papanicolaou, Asymptotic Analysis for Periodic Structures, North-Holland (1978). · Zbl 0404.35001 [7] 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). · Zbl 0628.76093 [8] 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). · Zbl 0041.54204 [9] 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). [10] C. Conca, The Stokes sieve problem, Comm. in Appl. Num. Meth. 4, pp. 113-121 (1988). · Zbl 0646.76041 [11] E. De Giorgi, Sulla convergenza di alcune successioni di integrali del tipo dell’ area, Rendiconti di Mat. 8, pp. 277-294 (1975). · Zbl 0316.35036 [12] E. De Giorgi, G-operators and ?-convergence, Proceedings of the International Congress of Mathematicians (Warsaw, August 1983), PWN Polish Scientific Publishers and North Holland, pp. 1175-1191 (1984). [13] F. Finn, Mathematical questions relating to viscous fluid flow in an exterior domain, Rocky Mountain J. Math. 3, pp. 107-140 (1973). · Zbl 0261.35068 [14] H. Kacimi, Thèse de troisième cycle, Université Paris 6 (1988). [15] 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). [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). · Zbl 0439.76017 [17] 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). · Zbl 0565.73046 [18] O. A. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flow, Gordon and Breach (1969). · Zbl 0184.52603 [19] T. Levy, Fluid flow through an array of fixed particles, Int. J. Engin. Sci. 21, pp. 11-23 (1983). · Zbl 0539.76092 [20] J. L. Lions, Some Methods in the Mathematical Analysis of Systems and Their Control, Beijing, Gordon and Breach, New York (1981). · Zbl 0542.93034 [21] 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). · Zbl 0850.76778 [22] V. A. Mar?enko & E. Ja. Hrouslov, Boundary Problems in Domains with Finely Granulated Boundaries (in Russian), Naukova Dumka, Kiev (1974). [23] J. Rubinstein, On the macroscopic description of slow viscous flow past a random array of spheres, J. Stat. Phys. 44, pp. 849-863 (1986). · Zbl 0629.76104 [24] 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). · Zbl 0501.76086 [25] E. Sanchez-Palencia, Non homogeneous media and vibration theory, Lecture Notes in Physics 127, Springer-Verlag (1980). · Zbl 0432.70002 [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). [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). [28] L. Tartar, Convergence of the homogenization process, Appendix of [25]. · Zbl 1188.35004 [29] L. Tartar, Cours Peccot au Collège de France, Unpublished (mars 1977). [30] L. Tartar, Topics in Nonlinear Analysis, Publications mathématiques d’Orsay 78.13, Université de Paris-Sud (1978).
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