# American Institute of Mathematical Sciences

November  2014, 13(6): 2445-2464. doi: 10.3934/cpaa.2014.13.2445

## Weak solutions to the equations of stationary magnetohydrodynamic flows in porous media

 1 Laboratoire de Mathématiques, CNRS UMR 6620, Université Blaise Pascal (Clermont-Ferrand 2), 63177 Aubière cedex 2 Université Blaise Pascal & CNRS UMR 6620, Laboratoire de Mathématiques, Campus des Cézeaux, B.P. 80026, F-63177 Aubière cedex 3 Laboratoire de Mathématiques, CNRS UMR 6620, Université Blaise Pascal, 63177 Aubière Cedex, France

Received  November 2013 Revised  May 2014 Published  July 2014

We study the differential system which describes the steady flow of an electrically conducting fluid in a saturated porous medium, when the fluid is subjected to the action of a magnetic field. The system consists of the stationary Brinkman-Forchheimer equations and the stationary magnetic induction equation. We prove existence of weak solutions to the system posed in a bounded domain of $\mathbb{R}^3$ and equipped with boundary conditions. We also prove uniqueness in the class of small solutions, and regularity of weak solutions. Then we establish a convergence result, as the Brinkman coefficient (viscosity) tends to 0, of the weak solutions to a solution of the system formed by the Darcy-Forchheimer equations and the magnetic induction equation.
Citation: Youcef Amirat, Laurent Chupin, Rachid Touzani. Weak solutions to the equations of stationary magnetohydrodynamic flows in porous media. Communications on Pure & Applied Analysis, 2014, 13 (6) : 2445-2464. doi: 10.3934/cpaa.2014.13.2445
##### References:
 [1] Y. Amirat, Écoulements en milieux poreux n'obéissant pas à la loi de Darcy (French) [Flows in porous media not obeying the Darcy law], ESAIM: Math. Mod. Numer. Anal. - Modél. Math. Anal. Numér., 25 (1991), 273-306.  Google Scholar [2] L. Cattabriga, Su un problema al contorno relativo al sistema di equazioni di Stokes (Italian), Rend. Sem. Mat. Univ. Padova, 31 (1961), 308-340.  Google Scholar [3] O. Celebi, V. Kalantarov and D. Ugurlu, On continuous dependence on solutions of the Brinkman-Forchheimer equations, Appl. Math. Lett., 19 (2006), 801-807. doi: 10.1016/j.aml.2005.11.002.  Google Scholar [4] P. E. Druet, Existence for the stationary MHD-equations coupled to heat transfer with nonlocal radiation effects, Czechoslovak Math. J., 59 (2009), 791-825. doi: 10.1007/s10587-009-0048-9.  Google Scholar [5] G. Duvaut and J. L. Lions, Les inéquations en mécanique et en physique, Dunod, 1972.  Google Scholar [6] G. Duvaut and J. L. Lions, Inéquations en thermoélasticité et magnétohydrodynamique, Arch. Rational Mech. Anal., 46 (1972), 241-279.  Google Scholar [7] P. Fabrie, Régularité de la solution de l'équation de Darcy-Forchheimer, Nonlinear Anal., 13 (1989), 1025-1051. doi: 10.1016/0362-546X(89)90093-X.  Google Scholar [8] M. Firdaouss, J. L. Guermond and P. Le Quéré, Nonlinear corrections to Darcy's law at low Reynolds numbers, J. Fluid Mech., 343 (1997), 331-350. doi: 10.1017/S0022112097005843.  Google Scholar [9] F. Franchi and B. Straughan, Continuous dependence and decay for the Forchheimer equations, R. Soc. Lond. Proc. Ser. A, Math. Phys. Eng. Sci., 459 (2003), 3195-3202. doi: 10.1098/rspa.2003.1169.  Google Scholar [10] G. P. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations. I. Linearized Steady Problems, Springer tracts in Natural Philosophy, 38, Springer Verlag, New-York, 1994. doi: 10.1007/978-1-4612-5364-8.  Google Scholar [11] C. Geindreau and J. L. Auriault, Magnetohydrodynamic flows in porous media, J. Fluid Mech., 466 (2002), 343-63. doi: 10.1017/S0022112002001404.  Google Scholar [12] V. Georgescu, Some boundary value problems for differential forms on compact Riemannian manifolds, Ann. Mat. Pura Appl., 4 (1979), 159-198. doi: 10.1007/BF02411693.  Google Scholar [13] J. F. Gerbeau and C. Le Bris, A coupled system arising in magnetohydrodynamics, Appl. Math. Lett., 12 (1999), 53-57. doi: 10.1016/S0893-9659(98)00172-4.  Google Scholar [14] T. Giorgi, Derivation of the Forchheimer law via matched asymptotic expansions, Transport in Porous Media, 29 (1997), 191-206. Google Scholar [15] C. He and Z. Xin, On the regularity of weak solutions to the magnetohydrodynamic equations, J. Differential Equations, 213 (2005), 235-254. doi: 10.1016/j.jde.2004.07.002.  Google Scholar [16] J. D. Jackson, Classical Electrodynamics, Second Edition, John Wiley & Son, New York, 1975, (Third Edition, 1999).  Google Scholar [17] O. A. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flow, Gordon and Breach, Rev. second edition, 1969.  Google Scholar [18] P. Lehmann, R. Moreau, D. Camel and R. Bolcato, Modification of interdendritic convection in directional solidifcation by a uniform magnetic field, Acta Materialia, 46 (1998), 4067-4079. Google Scholar [19] P. Lehmann, R. Moreau, D. Camel and R. Bolcato, A simple analysis of the effect of convection on the structure of the mushy zone in the case of horizontal Bridgman solidification. Comparison with experimental results,, \emph{J. Cryst. Growth, 183 (): 690.   Google Scholar [20] J. L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires, Dunod-Gauthier-Villars, 1969.  Google Scholar [21] P. L. Lions, Mathematical topics in fluid mechanics. Volume 1. Incompressible models, Oxford Science Publications, 1996.  Google Scholar [22] A. J. Meir and P. G. Schmidt, On electromagnetically and thermally driven liquid-metal flows, Nonlinear anal., 47 (2001), 3281-3294. doi: 10.1016/S0362-546X(01)00445-X.  Google Scholar [23] R. J. Moreau, Magnetohydrodynamics, Kluwer Academic Publishers, 1990. doi: 10.1007/978-94-015-7883-7.  Google Scholar [24] L. E. Payne, J. C. Song and B. Straugham, Continuous dependence and convergence results for Brinkman and Forchheimer models with variable viscosity, R. Soc. Lond. Proc. Ser. A, Math. Phys. Eng. Sci., 455 (1999), 2173-2190. doi: 10.1098/rspa.1999.0398.  Google Scholar [25] L. E. Payne and B. Straugham, Convergence and continuous dependence for the Brinkman-Forchheimer equations, Stud. Appl. Math., 102 (1999), 419-439. doi: 10.1111/1467-9590.00116.  Google Scholar [26] V. R. Prasad, A. Beg and B. Vasu, Thermo-diffusion and diffusion-thermo effects on MHD free convection flow past a vertical porous plate in a non-Darcy porous medium, Chemical Engineering Journal, 173 (2011) 598-606. Google Scholar [27] B. Saramito, Stabilité d'un Plasma: Modélisation mathématique et simulation numérique (French) [Stability of a plasma: mathematical modelling and numerical simulation], Recherches en Mathématiques Appliquées [Research in Applied Mathematics], Masson, 1994.  Google Scholar [28] M. Sermange and R. Temam, Some mathematical questions related to the MHD equations, Comm. Pure Appl. Math., 36 (1983), 635-664. doi: 10.1002/cpa.3160360506.  Google Scholar [29] J. Simon, Régularité de la solution d'un problème aux limites non linéaires (French) [Regularity of the solution of a nonlinear boundary problem], Ann. Fac. Sci. Toulouse Math., 3 (1981), 247-274.  Google Scholar [30] L. Tartar, Topics in Nonlinear Analysis, Publications Mathématiques d'Orsay, 78.13, 1978.  Google Scholar [31] R. Temam, Navier-Stokes equations, 3rd Edition, North-Holland, Amsterdam, 1984. Reedited in the AMS-Chelsea Series, Amer. Math. Soc., Providence, RI, 2001.  Google Scholar [32] S. Whitaker, The Forchheimer equation: a theoretical development, Transport in Porous Media, 25 (1996), 27-62. Google Scholar [33] K. Zaïdat, Influence d'un champ magnétique glissant sur la solidification dirigée des alliages métalliques binaires, PhD Thesis, Institut National Polytechnique de Grenoble, 2005. Available from: http://tel.archives-ouvertes.fr/tel-00011040 Google Scholar

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##### References:
 [1] Y. Amirat, Écoulements en milieux poreux n'obéissant pas à la loi de Darcy (French) [Flows in porous media not obeying the Darcy law], ESAIM: Math. Mod. Numer. Anal. - Modél. Math. Anal. Numér., 25 (1991), 273-306.  Google Scholar [2] L. Cattabriga, Su un problema al contorno relativo al sistema di equazioni di Stokes (Italian), Rend. Sem. Mat. Univ. Padova, 31 (1961), 308-340.  Google Scholar [3] O. Celebi, V. Kalantarov and D. Ugurlu, On continuous dependence on solutions of the Brinkman-Forchheimer equations, Appl. Math. Lett., 19 (2006), 801-807. doi: 10.1016/j.aml.2005.11.002.  Google Scholar [4] P. E. Druet, Existence for the stationary MHD-equations coupled to heat transfer with nonlocal radiation effects, Czechoslovak Math. J., 59 (2009), 791-825. doi: 10.1007/s10587-009-0048-9.  Google Scholar [5] G. Duvaut and J. L. Lions, Les inéquations en mécanique et en physique, Dunod, 1972.  Google Scholar [6] G. Duvaut and J. L. Lions, Inéquations en thermoélasticité et magnétohydrodynamique, Arch. Rational Mech. Anal., 46 (1972), 241-279.  Google Scholar [7] P. Fabrie, Régularité de la solution de l'équation de Darcy-Forchheimer, Nonlinear Anal., 13 (1989), 1025-1051. doi: 10.1016/0362-546X(89)90093-X.  Google Scholar [8] M. Firdaouss, J. L. Guermond and P. Le Quéré, Nonlinear corrections to Darcy's law at low Reynolds numbers, J. Fluid Mech., 343 (1997), 331-350. doi: 10.1017/S0022112097005843.  Google Scholar [9] F. Franchi and B. Straughan, Continuous dependence and decay for the Forchheimer equations, R. Soc. Lond. Proc. Ser. A, Math. Phys. Eng. Sci., 459 (2003), 3195-3202. doi: 10.1098/rspa.2003.1169.  Google Scholar [10] G. P. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations. I. Linearized Steady Problems, Springer tracts in Natural Philosophy, 38, Springer Verlag, New-York, 1994. doi: 10.1007/978-1-4612-5364-8.  Google Scholar [11] C. Geindreau and J. L. Auriault, Magnetohydrodynamic flows in porous media, J. Fluid Mech., 466 (2002), 343-63. doi: 10.1017/S0022112002001404.  Google Scholar [12] V. Georgescu, Some boundary value problems for differential forms on compact Riemannian manifolds, Ann. Mat. Pura Appl., 4 (1979), 159-198. doi: 10.1007/BF02411693.  Google Scholar [13] J. F. Gerbeau and C. Le Bris, A coupled system arising in magnetohydrodynamics, Appl. Math. Lett., 12 (1999), 53-57. doi: 10.1016/S0893-9659(98)00172-4.  Google Scholar [14] T. Giorgi, Derivation of the Forchheimer law via matched asymptotic expansions, Transport in Porous Media, 29 (1997), 191-206. Google Scholar [15] C. He and Z. Xin, On the regularity of weak solutions to the magnetohydrodynamic equations, J. Differential Equations, 213 (2005), 235-254. doi: 10.1016/j.jde.2004.07.002.  Google Scholar [16] J. D. Jackson, Classical Electrodynamics, Second Edition, John Wiley & Son, New York, 1975, (Third Edition, 1999).  Google Scholar [17] O. A. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flow, Gordon and Breach, Rev. second edition, 1969.  Google Scholar [18] P. Lehmann, R. Moreau, D. Camel and R. Bolcato, Modification of interdendritic convection in directional solidifcation by a uniform magnetic field, Acta Materialia, 46 (1998), 4067-4079. Google Scholar [19] P. Lehmann, R. Moreau, D. Camel and R. Bolcato, A simple analysis of the effect of convection on the structure of the mushy zone in the case of horizontal Bridgman solidification. Comparison with experimental results,, \emph{J. Cryst. Growth, 183 (): 690.   Google Scholar [20] J. L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires, Dunod-Gauthier-Villars, 1969.  Google Scholar [21] P. L. Lions, Mathematical topics in fluid mechanics. Volume 1. Incompressible models, Oxford Science Publications, 1996.  Google Scholar [22] A. J. Meir and P. G. Schmidt, On electromagnetically and thermally driven liquid-metal flows, Nonlinear anal., 47 (2001), 3281-3294. doi: 10.1016/S0362-546X(01)00445-X.  Google Scholar [23] R. J. Moreau, Magnetohydrodynamics, Kluwer Academic Publishers, 1990. doi: 10.1007/978-94-015-7883-7.  Google Scholar [24] L. E. Payne, J. C. Song and B. Straugham, Continuous dependence and convergence results for Brinkman and Forchheimer models with variable viscosity, R. Soc. Lond. Proc. Ser. A, Math. Phys. Eng. Sci., 455 (1999), 2173-2190. doi: 10.1098/rspa.1999.0398.  Google Scholar [25] L. E. Payne and B. Straugham, Convergence and continuous dependence for the Brinkman-Forchheimer equations, Stud. Appl. Math., 102 (1999), 419-439. doi: 10.1111/1467-9590.00116.  Google Scholar [26] V. R. Prasad, A. Beg and B. Vasu, Thermo-diffusion and diffusion-thermo effects on MHD free convection flow past a vertical porous plate in a non-Darcy porous medium, Chemical Engineering Journal, 173 (2011) 598-606. Google Scholar [27] B. Saramito, Stabilité d'un Plasma: Modélisation mathématique et simulation numérique (French) [Stability of a plasma: mathematical modelling and numerical simulation], Recherches en Mathématiques Appliquées [Research in Applied Mathematics], Masson, 1994.  Google Scholar [28] M. Sermange and R. Temam, Some mathematical questions related to the MHD equations, Comm. Pure Appl. Math., 36 (1983), 635-664. doi: 10.1002/cpa.3160360506.  Google Scholar [29] J. Simon, Régularité de la solution d'un problème aux limites non linéaires (French) [Regularity of the solution of a nonlinear boundary problem], Ann. Fac. Sci. Toulouse Math., 3 (1981), 247-274.  Google Scholar [30] L. Tartar, Topics in Nonlinear Analysis, Publications Mathématiques d'Orsay, 78.13, 1978.  Google Scholar [31] R. Temam, Navier-Stokes equations, 3rd Edition, North-Holland, Amsterdam, 1984. Reedited in the AMS-Chelsea Series, Amer. Math. Soc., Providence, RI, 2001.  Google Scholar [32] S. Whitaker, The Forchheimer equation: a theoretical development, Transport in Porous Media, 25 (1996), 27-62. Google Scholar [33] K. Zaïdat, Influence d'un champ magnétique glissant sur la solidification dirigée des alliages métalliques binaires, PhD Thesis, Institut National Polytechnique de Grenoble, 2005. Available from: http://tel.archives-ouvertes.fr/tel-00011040 Google Scholar
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