The derivation of a nonlinear filtration law including the inertia effects via homogenization

R esum e. On consid ere l' ecoulement stationnaire d'un uide newtonien visqueux incompressible dans un milieu poreux rigide. Pour une structure g eom etrique p eriodique du milieu poreux form ee de cellules carr ees avec des c^ ot es de longueur ", la m ethode d'homog en eisation donne dii erents r esultats selon la relation en-tre ", le nombre de Reynolds et le nombre de Froude. Si le nombre de Reynolds et l'inverse du nombre de Froude sont d'ordre 1=", l' etude asymptotique formelle conduit a un syst eme homog en eis e du type Navier-Stokes a double pression, qui contient une loi nonlocale r egissant la ltration nonlin eaire. En supposant que les donn ees ne sont pas trop grandes, on d emontre que le probl eme homog en eis e poss ede une unique solution r eguli ere. De plus, on montre la convergence du pro-cessus d'homog en eisation et on etablit une estimation d'erreur. Abstract. We consider the stationary viscous incompressible uid ow through a rigid porous medium. For a periodic porous medium, with the period ", the ho-mogenization method gives diierent results depending on the relationship between the Reynolds number, the Froude's number and the period. If both, the Reynolds number and the inverse of the Froude's number are of order 1=", then the formal asymptotic expansion gives a homogenized system containing the fast and slow variables named Navier-Stokes system with two pressures. More precisely the l-tration law is nonlocal and nonlinear. Supposing that the data are not too large we prove the existence of a unique smooth solution for the homogenized problem. Furthermore, the convergence of the homogenization process is proved and the 1 error estimate is established.

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