Mathematical Problems in Mechanics
Modelling of a fluid: homogenization and mixed scales
[Modélisation d'un fluide par homogénéisation et échelles multiples]
Comptes Rendus. Mathématique, Tome 337 (2003) no. 12, pp. 809-813.

On étudie le comportement limite d'un mélange composé de gouttes d'un fluide immergées dans un deuxième fluide lorsque la répartition – théorique – des gouttes permet de se placer dans les conditions d'application de l'homogénéisation périodique. On suppose que la taille relative de ces gouttes par rapport à la petite période du problème a la valeur critique qui permet de faire apparaı̂tre un terme étrange caractérisant la loi de Brinkmann des fluides.

The limit behavior of a mixture made of drops of some fluid immersed in another fluid is analysed in the framework of periodic homogenization when the size of the drops is critical compared with the period of the underlying network. More precisely, a strange term is obtained in the limit thus leading to a Brinkmann law.

Reçu le :
Accepté le :
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DOI : 10.1016/j.crma.2003.10.016
Gruais, Isabelle 1

1 Université de Rennes 1, I.R.M.A.R., campus de Beaulieu, 35042 Rennes cedex, France
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Gruais, Isabelle. Modelling of a fluid: homogenization and mixed scales. Comptes Rendus. Mathématique, Tome 337 (2003) no. 12, pp. 809-813. doi : 10.1016/j.crma.2003.10.016. http://www.numdam.org/articles/10.1016/j.crma.2003.10.016/

[1] Allaire, G. Continuity of the darcy's law in the low volume fraction limit, Ann. Scuola Norm. Sup. Pisa, Volume 4 (1991) no. 18, pp. 475-499

[2] 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., Volume 113 (1991) no. 3, pp. 209-260

[3] Allaire, G. Homogenization of the Navier–Stokes equations in open sets perforated with tiny holes. ii: Non-critical size of the holes for a volume distribution and a surface distribution of holes, Arch. Rational Mech. Anal., Volume 113 (1991) no. 3, pp. 261-298

[4] Allaire, G. Homogenization of the Navier–Stokes equations with a slip boundary condition, Comm. Pure Appl. Math., Volume 44 (1991), pp. 605-641

[5] Allaire, G. Homogenization of the unsteady stokes equations in porous media (Brandle, C. et al., eds.), Progress in Partial Differential Equations: Calculus of Variations, Applications, Pitman Res. Notes in Math., 267, Longman Higher Education, New York, 1992

[6] G. Allaire, Sur quelques méthodes en homogénéisation et leurs applications, Habilitation à diriger des recherches, Université Pierre et Marie Curie, Paris, 1993

[7] Arbogast, T.; Falk, J.; Hornung, U. Derivation of the double porosity model of single phase flow via homogenization theory, SIAM J. Math. Anal., Volume 21 (1990) no. 4, pp. 823-836

[8] F. Bentalha, Communication personnelle, Thèse de doctorat, Université de Metz, 2002

[9] Casado-Diaz, J. Two-scale convergence for nonlinear Dirichlet problems in perforated domains, Proc. Roy. Soc. Edinburgh Sect. A, Volume 130 (2000), pp. 249-276

[10] Cioranescu, D.; Murat, F. Un terme étrange venu d'ailleurs, Nonlinear Partial Differential Equations and their Applications, College de France Seminar, Vol. II (Paris, 1979/1980), Pitman Res. Notes in Math., 60, Pitman, Boston, 1982, pp. 98-138

[11] Cioranescu, D.; Murat, F. Un terme étrange venu d'ailleurs ii, Nonlinear Partial Differential Equations and their Applications, College de France Seminar, Vol. III (Paris, 1980/1981), Pitman Res. Notes in Math., 70, Pitman, Boston, 1982, pp. 154-178

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