# American Institute of Mathematical Sciences

October  2014, 7(5): 1065-1077. doi: 10.3934/dcdss.2014.7.1065

## A nonlinear effective slip interface law for transport phenomena between a fracture flow and a porous medium

 1 Institute of Applied Mathematics, Interdisciplinary Center of Scienti c Computing and BIOQUANT, University of Heidelberg, Im Neuenheimer Feld 294, 69120 Heidelberg, Germany 2 Université de Lyon, CNRS UMR 5208, Université Lyon 1, Institut Camille Jordan, 43, blvd. du 11 novembre 1918, 69622 Villeurbanne Cedex, France

Received  March 2013 Published  May 2014

We present modeling of an incompressible viscous flow through a fracture adjacent to a porous medium. A fast stationary flow, predominantly tangential to the porous medium is considered. Slow flow in such setting can be described by the Beavers-Joseph-Saffman slip. For fast flows, a nonlinear filtration law in the porous medium and a non- linear interface law are expected. In this paper we rigorously derive a quadratic effective slip interface law which holds for a range of Reynolds numbers and fracture widths. The porous medium flow is described by the Darcy law. The result shows that the interface slip law can be nonlinear, independently of the regime for the bulk flow. Since most of the interface and boundary slip laws are obtained via upscaling of complex systems, the result indicates that studying the inviscid limits for the Navier-Stokes equations with linear slip law at the boundary should be rethought.
Citation: Anna Marciniak-Czochra, Andro Mikelić. A nonlinear effective slip interface law for transport phenomena between a fracture flow and a porous medium. Discrete & Continuous Dynamical Systems - S, 2014, 7 (5) : 1065-1077. doi: 10.3934/dcdss.2014.7.1065
##### References:
 [1] G. Allaire, One-phase Newtonian flow,, in Homogenization and Porous Media (ed. U. Hornung), (1997), 45. doi: 10.1007/978-1-4612-1920-0_3. Google Scholar [2] G. S. Beavers and D. D. Joseph, Boundary conditions at a naturally permeable wall,, J. Fluid Mech., 30 (1967), 197. doi: 10.1017/S0022112067001375. Google Scholar [3] A. Bourgeat, E. Marušić-Paloka and A. Mikelić, Effective behavior of porous medium containing a thin fissure,, in Calculus of Variations, (1994), 69. Google Scholar [4] A. Bourgeat, E. Marušić-Paloka and A. Mikelić, Effective behavior for a fluid flow in porous medium containing a thin fissure,, Asymptotic Anal., 11 (1995), 241. Google Scholar [5] A. Bourgeat, E. Marušić-Paloka and A. Mikelić, Weak non-linear corrections for Darcy's law,, $M^3$ AS : Math. Models Methods Appl. Sci., 6 (1996), 1143. doi: 10.1142/S021820259600047X. Google Scholar [6] T. Carraro, C. Goll, A. Marciniak-Czochra and A. Mikelić, Pressure jump interface law for the Stokes-Darcy coupling: Confirmation by direct numerical simulations,, Journal of Fluid Mechanics, 732 (2013), 510. doi: 10.1017/jfm.2013.416. Google Scholar [7] M. Discacciati and A. Quarteroni, Navier-Stokes/Darcy coupling: Modeling, analysis, and numerical approximation,, Rev. Mat. Complut., 22 (2009), 315. Google Scholar [8] H. I. Ene and E. Sanchez-Palencia, Equations et phénomènes de surface pour l'écoulement dans un modèle de milieu poreux,, J. Mécan., 14 (1975), 73. Google Scholar [9] O. Iliev and V. Laptev, On numerical simulation of flow through oil filters,, Computing and Visualization in Science, 6 (2004), 139. doi: 10.1007/s00791-003-0118-8. Google Scholar [10] W. Jäger and A. Mikelić, On the boundary conditions at the contact interface between a porous medium and a free fluid,, Ann. Sc. Norm. Super. Pisa, 23 (1996), 403. Google Scholar [11] W. Jäger and A. Mikelić, On the interface boundary conditions by Beavers, Joseph and Saffman,, SIAM J. Appl. Math., 60 (2000), 1111. doi: 10.1137/S003613999833678X. Google Scholar [12] W. Jäger, A. Mikelić and N. Neuß, Asymptotic analysis of the laminar viscous flow over a porous bed,, SIAM J. on Scientific and Statistical Computing, 22 (2001), 2006. Google Scholar [13] W. Jäger and A. Mikelić, Modeling effective interface laws for transport phenomena between an unconfined fluid and a porous medium using homogenization,, Transport in Porous Media, 78 (2009), 489. doi: 10.1007/s11242-009-9354-9. Google Scholar [14] M. Kaviany, Principles of Heat Transfer in Porous Media,, 2nd Revised edition, (1995). Google Scholar [15] Q. Liu and A. Prosperetti, Pressure-driven flow in a channel with porous walls,, Journal of Fluid Mechanics, 679 (2011), 77. doi: 10.1017/jfm.2011.124. Google Scholar [16] A. Marciniak-Czochra and A. Mikelić, Effective pressure interface law for transport phenomena between an unconfined fluid and a porous medium using homogenization,, SIAM: Multiscale Modeling and Simulation, 10 (2012), 285. doi: 10.1137/110838248. Google Scholar [17] A. Mikelić, Homogenization theory and applications to filtration through porous media,, in Filtration in Porous Media and Industrial Applications, (1734), 127. doi: 10.1007/BFb0103977. Google Scholar [18] P. G. Saffman, On the boundary condition at the interface of a porous medium,, Studies in Applied Mathematics, 1 (1971), 93. Google Scholar [19] M. Sahraoui and M. Kaviany, Slip and no-slip velocity boundary conditions at interface of porous, plain media,, Int. J. Heat Mass Transfer, 35 (1992), 927. doi: 10.1016/0017-9310(92)90258-T. Google Scholar [20] E. Sanchez-Palencia, Non-Homogeneous Media and Vibration Theory,, Springer Verlag, (1980). Google Scholar [21] L. Tartar, Convergence of the homogenization process,, Appendix of [20]., (). Google Scholar [22] R. Temam, Navier-Stokes Equations,, 3rd revised edition, (1984). Google Scholar

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##### References:
 [1] G. Allaire, One-phase Newtonian flow,, in Homogenization and Porous Media (ed. U. Hornung), (1997), 45. doi: 10.1007/978-1-4612-1920-0_3. Google Scholar [2] G. S. Beavers and D. D. Joseph, Boundary conditions at a naturally permeable wall,, J. Fluid Mech., 30 (1967), 197. doi: 10.1017/S0022112067001375. Google Scholar [3] A. Bourgeat, E. Marušić-Paloka and A. Mikelić, Effective behavior of porous medium containing a thin fissure,, in Calculus of Variations, (1994), 69. Google Scholar [4] A. Bourgeat, E. Marušić-Paloka and A. Mikelić, Effective behavior for a fluid flow in porous medium containing a thin fissure,, Asymptotic Anal., 11 (1995), 241. Google Scholar [5] A. Bourgeat, E. Marušić-Paloka and A. Mikelić, Weak non-linear corrections for Darcy's law,, $M^3$ AS : Math. Models Methods Appl. Sci., 6 (1996), 1143. doi: 10.1142/S021820259600047X. Google Scholar [6] T. Carraro, C. Goll, A. Marciniak-Czochra and A. Mikelić, Pressure jump interface law for the Stokes-Darcy coupling: Confirmation by direct numerical simulations,, Journal of Fluid Mechanics, 732 (2013), 510. doi: 10.1017/jfm.2013.416. Google Scholar [7] M. Discacciati and A. Quarteroni, Navier-Stokes/Darcy coupling: Modeling, analysis, and numerical approximation,, Rev. Mat. Complut., 22 (2009), 315. Google Scholar [8] H. I. Ene and E. Sanchez-Palencia, Equations et phénomènes de surface pour l'écoulement dans un modèle de milieu poreux,, J. Mécan., 14 (1975), 73. Google Scholar [9] O. Iliev and V. Laptev, On numerical simulation of flow through oil filters,, Computing and Visualization in Science, 6 (2004), 139. doi: 10.1007/s00791-003-0118-8. Google Scholar [10] W. Jäger and A. Mikelić, On the boundary conditions at the contact interface between a porous medium and a free fluid,, Ann. Sc. Norm. Super. Pisa, 23 (1996), 403. Google Scholar [11] W. Jäger and A. Mikelić, On the interface boundary conditions by Beavers, Joseph and Saffman,, SIAM J. Appl. Math., 60 (2000), 1111. doi: 10.1137/S003613999833678X. Google Scholar [12] W. Jäger, A. Mikelić and N. Neuß, Asymptotic analysis of the laminar viscous flow over a porous bed,, SIAM J. on Scientific and Statistical Computing, 22 (2001), 2006. Google Scholar [13] W. Jäger and A. Mikelić, Modeling effective interface laws for transport phenomena between an unconfined fluid and a porous medium using homogenization,, Transport in Porous Media, 78 (2009), 489. doi: 10.1007/s11242-009-9354-9. Google Scholar [14] M. Kaviany, Principles of Heat Transfer in Porous Media,, 2nd Revised edition, (1995). Google Scholar [15] Q. Liu and A. Prosperetti, Pressure-driven flow in a channel with porous walls,, Journal of Fluid Mechanics, 679 (2011), 77. doi: 10.1017/jfm.2011.124. Google Scholar [16] A. Marciniak-Czochra and A. Mikelić, Effective pressure interface law for transport phenomena between an unconfined fluid and a porous medium using homogenization,, SIAM: Multiscale Modeling and Simulation, 10 (2012), 285. doi: 10.1137/110838248. Google Scholar [17] A. Mikelić, Homogenization theory and applications to filtration through porous media,, in Filtration in Porous Media and Industrial Applications, (1734), 127. doi: 10.1007/BFb0103977. Google Scholar [18] P. G. Saffman, On the boundary condition at the interface of a porous medium,, Studies in Applied Mathematics, 1 (1971), 93. Google Scholar [19] M. Sahraoui and M. Kaviany, Slip and no-slip velocity boundary conditions at interface of porous, plain media,, Int. J. Heat Mass Transfer, 35 (1992), 927. doi: 10.1016/0017-9310(92)90258-T. Google Scholar [20] E. Sanchez-Palencia, Non-Homogeneous Media and Vibration Theory,, Springer Verlag, (1980). Google Scholar [21] L. Tartar, Convergence of the homogenization process,, Appendix of [20]., (). Google Scholar [22] R. Temam, Navier-Stokes Equations,, 3rd revised edition, (1984). Google Scholar
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