## 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.(English)Zbl 0724.76021

This article is a continuation of Part I on the homogenization of the Stokes (Navier-Stokes) equations [see the foregoing entry (Zbl 0724.76020)]. The convergence of the problem to the Stokes and the Darcy laws, depending on the hole size being smaller and larger, respectively, than the critical size as defined in the first part, is established. Moreover, a different geometric distribution of holes in considered on a hyperplane H which intersects the fluid medium, with the typical hole size $$\epsilon^ 2$$ for $$N=2$$ and $$e^{-1/\epsilon}$$ for $$N=3$$. The method developed in first part has been used to derive a weaker estimate satisfied by the pressure in this case. The results also provide an effective method for computing viscous fluid flows through porous walls, or mixing grids.

### MSC:

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

### Keywords:

Stokes law; Darcy law; convergence; porous walls; mixing grids

Zbl 0724.76020
Full Text:

### References:

 [1] G. Allaire, Homogénéisation des équations de Navier-Stokes, Thèse, Université Paris 6 (1989). [2] 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). [3] C. Conca, The Stokes sieve problem, Comm. in Appl. Num. Meth., vol. 4, pp. 113-121 (1988). · Zbl 0646.76041 [4] H. Kacimi, Thèse de troisième cycle, Université Paris 6 (1988). [5] 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 [6] J. L. Lions, Some Methods in the Mathematical Analysis of Systems and their Control, Beijing, Gordon and Breach, New York (1981). · Zbl 0542.93034 [7] E. Sanchez-Palencia, Non Homogeneous Media and Vibration Theory, Lecture Notes in Physics 127, Springer-Verlag (1980). · Zbl 0432.70002 [8] 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). [9] 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). [10] L. Tartar, Convergence of the homogenization process, Appendix of [25]. · Zbl 1188.35004 [11] L. Tartar, Cours Peccot au Collège de France, Unpublished (mars 1977).
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.