Comptes Rendus
A transformation approach for the derivation of boundary conditions between a curved porous medium and a free fluid
[Sur une approche de transformation pour le calcul des conditions à l'interface entre un milieu poreux courbé et un fluide libre]
Comptes Rendus. Mécanique, Volume 338 (2010) no. 2, pp. 71-77.

On considère le comportement d'un fluid libre au-dessus d'un milieu poreux avec une interface courbée. Utilisant une transformation des coordonnées, on obtient la description de l'écoulement dans un domaine avec une frontière plane. En limite à deux échelles, le comportement du fluid en milieu poreux est donné par une loi de Darcy avec une matrice de perméabilité non-constante. Ensuite, on obtient le comportement du fluid à l'interface : La vitesse est continue à travers l'interface dans le sens normal, mais une discontinuité apparaît en sens tangentiel. Par conséquent, les résultats indiquent une loi généralisée de Beavers et Joseph.

The behaviour of a free fluid flow above a porous medium, both separated by a curved interface, is investigated. By carrying out a coordinate transformation, we obtain the description of the flow in a domain with a straight interface. Using periodic homogenisation, the effective behaviour of the transformed partial differential equations in the porous part is given by a Darcy law with non-constant permeability matrix. Then the fluid behaviour at the porous-liquid interface is obtained with the help of generalised boundary-layer functions: Whereas the velocity in normal direction is continuous across the interface, a jump appears in tangential direction. Its magnitude seems to be related to the slope of the interface. Therefore the results indicate a generalised law of Beavers and Joseph.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crme.2010.02.001
Keywords: Fluid mechanics, Homogenisation, Interfacial exchange, Porous media
Mot clés : Mécanique des fluides, Homogénéisation, Transport de masse à travers une surface de séparation, Milieux poreux
Sören Dobberschütz 1 ; Michael Böhm 1

1 Centre for Industrial Mathematics, FB 3, University of Bremen, Postfach 330 440, 28334 Bremen, Germany
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Sören Dobberschütz; Michael Böhm. A transformation approach for the derivation of boundary conditions between a curved porous medium and a free fluid. Comptes Rendus. Mécanique, Volume 338 (2010) no. 2, pp. 71-77. doi : 10.1016/j.crme.2010.02.001. https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.1016/j.crme.2010.02.001/

[1] G.S. Beavers; D. Joseph Boundary conditions at a naturally permeable wall, J. Fluid Mech., Volume 30 (1967), pp. 197-207

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[4] W. Jäger; A. Mikelić On the interface boundary condition of Beavers, Joseph, and Saffman, SIAM J. Math. Anal., Volume 60 (2000) no. 4, pp. 1111-1127

[5] M. Neuss-Radu A result on the decay of the boundary layers in the homogenization theory, Asympt. Anal., Volume 23 (2000), pp. 313-328

[6] M. Neuss-Radu The boundary behaviour of a composite material, Math. Model. Num. Anal., Volume 35 (2001) no. 3, pp. 407-435

[7] S. Dobberschütz, Derivation of boundary conditions at a curved contact interface between a free fluid and a porous medium via homogenisation theory, Diploma thesis, University of Bremen, 2009

[8] T. Levi; E. Sanchez-Palencia On boundary conditions for fluid flow in porous media, Int. J. Engin. Sci., Volume 13 (1975), pp. 923-940

[9] H.I. Ene; E. Sanchez-Palencia Équations et phénomènes de surface pour l'écoulement dans un modèle de milieux poreux, J. Mécanique, Volume 14 (1975) no. 1, pp. 73-108

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