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Homogenization of the inviscid incompressible fluid flow through a 2D porous medium

Authors: Andro Mikelic and Laetitia Paoli
Journal: Proc. Amer. Math. Soc. 127 (1999), 2019-2028
MSC (1991): Primary 35B27, 76C05, 76S05
Published electronically: February 26, 1999
MathSciNet review: 1626446
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Abstract: We consider the non-stationary incompressible Euler equations in a 2D porous medium. We suppose a periodic porous medium, with the period proportional to the characteristic pore size $\varepsilon$ and with connected fluid part. The flow is subject to an external force, corresponding to an inflow. We start from an initial irrotational velocity and prove that the effective filtration velocity satisfies a transient filtration law. It has similarities with Darcy's law, but it now connects the time derivative of the filtration velocity with the pressure gradient. The viscosity does not appear in the filtration law any more and the permeability tensor is determined through auxiliary problems of decomposition type. Using the limit problem, we construct the correction for the fluid velocity and prove that $C^1 ( [0,T]; L^2(\Omega)^2 )$-norm of the error is of order $\varepsilon$. Similarly, we estimate the difference between the fluid pressure and its correction in $C ( [0,T]; L^1(\Omega) )$ as $C \varepsilon$.

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  • 1. Allaire G., One-phase newtonian flow, p. 45-68, in Homogenization and porous media, edited by U.Hornung, Springer, New York, 1997. CMP 97:08
  • 2. Alain Bourgeat, Eduard Marušić-Paloka, and Andro Mikelić, Weak nonlinear corrections for Darcy’s law, Math. Models Methods Appl. Sci. 6 (1996), no. 8, 1143–1155. MR 1428149,
  • 3. Jäger W., Mikeli\'{c} A., On the boundary conditions at the contact interface between a porous medium and a free fluid, Ann. Scuola Norm. Sup. Pisa, Cl. Fisiche e Matematiche (4), 23(1996), 403-465. CMP 97:10
  • 4. V. I. Judovič, A two-dimensional non-stationary problem on the flow of an ideal incompressible fluid through a given region, Mat. Sb. (N.S.) 64 (106) (1964), 562–588 (Russian). MR 0177577
  • 5. Pierre-Louis Lions, Mathematical topics in fluid mechanics. Vol. 1, Oxford Lecture Series in Mathematics and its Applications, vol. 3, The Clarendon Press, Oxford University Press, New York, 1996. Incompressible models; Oxford Science Publications. MR 1422251
  • 6. Carlo Marchioro and Mario Pulvirenti, Mathematical theory of incompressible nonviscous fluids, Applied Mathematical Sciences, vol. 96, Springer-Verlag, New York, 1994. MR 1245492
  • 7. Maru\v{s}i\'{c}-Paloka E., Mikeli\'{c} A., The derivation of a non-linear filtration law including the inertia effects via homogenization, Nonlinear Analysis, Theory, Methods and Applications, 1998, accepted for publication.
  • 8. Andro Mikelić, Effets inertiels pour un écoulement stationnaire visqueux incompressible dans un milieu poreux, C. R. Acad. Sci. Paris Sér. I Math. 320 (1995), no. 10, 1289–1294 (French, with English and French summaries). MR 1336272
  • 9. Tartar L., Convergence of the homogenization process, appendix to Lecture Notes in Physics 127, Springer, Berlin, 1980.
  • 10. Zhikov V.V., Kozlov S.M., Oleinik O.A., Homogenization of differential operators and integral functionals, Springer, New York, 1994. MR 96h:350036

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Additional Information

Andro Mikelic
Affiliation: Equipe d’Analyse Numérique Lyon-Saint Etienne, UMR 5585 CNRS, Université Lyon 1, 43 boulevard du 11 novembre 1918, 69622 Villeurbanne Cedex, France

Laetitia Paoli
Affiliation: Equipe d’Analyse Numérique Lyon-Saint Etienne, UMR 5585 CNRS, Université de Saint Etienne, 23 rue du Docteur P.Michelon, 42023 Saint Etienne Cedex, France

Received by editor(s): September 15, 1997
Published electronically: February 26, 1999
Communicated by: James Glimm
Article copyright: © Copyright 1999 American Mathematical Society