American Institute of Mathematical Sciences

Advanced
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

Youcef Amirat 1, , Laurent Chupin 2, and  Rachid Touzani 3,

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.
Keywords: Magnetohydrodynamic flows in porous media, Darcy-Forchheimer equations, Brinkman-Forchheimer equations, weak solutions., Lorentz force.
Mathematics Subject Classification: Primary: 76S05, 76W05; Secondary: 35Q3.
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],, \emph{ESAIM: Math. Mod. Numer. Anal. - Modél. Math. Anal. Num\'er., 25 (1991), 273.   Google Scholar

[2]

L. Cattabriga, Su un problema al contorno relativo al sistema di equazioni di Stokes (Italian),, \emph{Rend. Sem. Mat. Univ. Padova, 31 (1961), 308.   Google Scholar

[3]

O. Celebi, V. Kalantarov and D. Ugurlu, On continuous dependence on solutions of the Brinkman-Forchheimer equations,, \emph{Appl. Math. Lett., 19 (2006), 801.  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,, \emph{Czechoslovak Math. J., 59 (2009), 791.  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,, \emph{Arch. Rational Mech. Anal., 46 (1972), 241.   Google Scholar

[7]

P. Fabrie, Régularité de la solution de l'équation de Darcy-Forchheimer,, \emph{Nonlinear Anal., 13 (1989), 1025.  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,, \emph{J. Fluid Mech., 343 (1997), 331.  doi: 10.1017/S0022112097005843.  Google Scholar

[9]

F. Franchi and B. Straughan, Continuous dependence and decay for the Forchheimer equations,, \emph{R. Soc. Lond. Proc. Ser. A, 459 (2003), 3195.  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, (1994).  doi: 10.1007/978-1-4612-5364-8.  Google Scholar

[11]

C. Geindreau and J. L. Auriault, Magnetohydrodynamic flows in porous media,, \emph{J. Fluid Mech., 466 (2002), 343.  doi: 10.1017/S0022112002001404.  Google Scholar

[12]

V. Georgescu, Some boundary value problems for differential forms on compact Riemannian manifolds,, \emph{Ann. Mat. Pura Appl., 4 (1979), 159.  doi: 10.1007/BF02411693.  Google Scholar

[13]

J. F. Gerbeau and C. Le Bris, A coupled system arising in magnetohydrodynamics,, \emph{Appl. Math. Lett., 12 (1999), 53.  doi: 10.1016/S0893-9659(98)00172-4.  Google Scholar

[14]

T. Giorgi, Derivation of the Forchheimer law via matched asymptotic expansions,, \emph{Transport in Porous Media, 29 (1997), 191.   Google Scholar

[15]

C. He and Z. Xin, On the regularity of weak solutions to the magnetohydrodynamic equations,, \emph{J. Differential Equations, 213 (2005), 235.  doi: 10.1016/j.jde.2004.07.002.  Google Scholar

[16]

J. D. Jackson, Classical Electrodynamics,, Second Edition, (1975).   Google Scholar

[17]

O. A. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flow,, Gordon and Breach, (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,, \emph{Acta Materialia, 46 (1998), 4067.   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,, \emph{Nonlinear anal., 47 (2001), 3281.  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,, \emph{R. Soc. Lond. Proc. Ser. A, 455 (1999), 2173.  doi: 10.1098/rspa.1999.0398.  Google Scholar

[25]

L. E. Payne and B. Straugham, Convergence and continuous dependence for the Brinkman-Forchheimer equations,, \emph{Stud. Appl. Math., 102 (1999), 419.  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,, \emph{Chemical Engineering Journal, 173 (2011), 598.   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], (1994).   Google Scholar

[28]

M. Sermange and R. Temam, Some mathematical questions related to the MHD equations,, \emph{Comm. Pure Appl. Math., 36 (1983), 635.  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],, \emph{Ann. Fac. Sci. Toulouse Math., 3 (1981), 247.   Google Scholar

[30]

L. Tartar, Topics in Nonlinear Analysis,, Publications Mathématiques d'Orsay, (1978).   Google Scholar

[31]

R. Temam, Navier-Stokes equations,, 3rd Edition, (1984).   Google Scholar

[32]

S. Whitaker, The Forchheimer equation: a theoretical development,, \emph{Transport in Porous Media, 25 (1996), 27.   Google Scholar

[33]

K. Zaïdat, Influence d'un champ magnétique glissant sur la solidification dirigée des alliages métalliques binaires,, PhD Thesis, (2005).   Google Scholar

show all references

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],, \emph{ESAIM: Math. Mod. Numer. Anal. - Modél. Math. Anal. Num\'er., 25 (1991), 273.   Google Scholar

[2]

L. Cattabriga, Su un problema al contorno relativo al sistema di equazioni di Stokes (Italian),, \emph{Rend. Sem. Mat. Univ. Padova, 31 (1961), 308.   Google Scholar

[3]

O. Celebi, V. Kalantarov and D. Ugurlu, On continuous dependence on solutions of the Brinkman-Forchheimer equations,, \emph{Appl. Math. Lett., 19 (2006), 801.  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,, \emph{Czechoslovak Math. J., 59 (2009), 791.  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,, \emph{Arch. Rational Mech. Anal., 46 (1972), 241.   Google Scholar

[7]

P. Fabrie, Régularité de la solution de l'équation de Darcy-Forchheimer,, \emph{Nonlinear Anal., 13 (1989), 1025.  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,, \emph{J. Fluid Mech., 343 (1997), 331.  doi: 10.1017/S0022112097005843.  Google Scholar

[9]

F. Franchi and B. Straughan, Continuous dependence and decay for the Forchheimer equations,, \emph{R. Soc. Lond. Proc. Ser. A, 459 (2003), 3195.  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, (1994).  doi: 10.1007/978-1-4612-5364-8.  Google Scholar

[11]

C. Geindreau and J. L. Auriault, Magnetohydrodynamic flows in porous media,, \emph{J. Fluid Mech., 466 (2002), 343.  doi: 10.1017/S0022112002001404.  Google Scholar

[12]

V. Georgescu, Some boundary value problems for differential forms on compact Riemannian manifolds,, \emph{Ann. Mat. Pura Appl., 4 (1979), 159.  doi: 10.1007/BF02411693.  Google Scholar

[13]

J. F. Gerbeau and C. Le Bris, A coupled system arising in magnetohydrodynamics,, \emph{Appl. Math. Lett., 12 (1999), 53.  doi: 10.1016/S0893-9659(98)00172-4.  Google Scholar

[14]

T. Giorgi, Derivation of the Forchheimer law via matched asymptotic expansions,, \emph{Transport in Porous Media, 29 (1997), 191.   Google Scholar

[15]

C. He and Z. Xin, On the regularity of weak solutions to the magnetohydrodynamic equations,, \emph{J. Differential Equations, 213 (2005), 235.  doi: 10.1016/j.jde.2004.07.002.  Google Scholar

[16]

J. D. Jackson, Classical Electrodynamics,, Second Edition, (1975).   Google Scholar

[17]

O. A. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flow,, Gordon and Breach, (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,, \emph{Acta Materialia, 46 (1998), 4067.   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,, \emph{Nonlinear anal., 47 (2001), 3281.  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,, \emph{R. Soc. Lond. Proc. Ser. A, 455 (1999), 2173.  doi: 10.1098/rspa.1999.0398.  Google Scholar

[25]

L. E. Payne and B. Straugham, Convergence and continuous dependence for the Brinkman-Forchheimer equations,, \emph{Stud. Appl. Math., 102 (1999), 419.  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,, \emph{Chemical Engineering Journal, 173 (2011), 598.   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], (1994).   Google Scholar

[28]

M. Sermange and R. Temam, Some mathematical questions related to the MHD equations,, \emph{Comm. Pure Appl. Math., 36 (1983), 635.  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],, \emph{Ann. Fac. Sci. Toulouse Math., 3 (1981), 247.   Google Scholar

[30]

L. Tartar, Topics in Nonlinear Analysis,, Publications Mathématiques d'Orsay, (1978).   Google Scholar

[31]

R. Temam, Navier-Stokes equations,, 3rd Edition, (1984).   Google Scholar

[32]

S. Whitaker, The Forchheimer equation: a theoretical development,, \emph{Transport in Porous Media, 25 (1996), 27.   Google Scholar

[33]

K. Zaïdat, Influence d'un champ magnétique glissant sur la solidification dirigée des alliages métalliques binaires,, PhD Thesis, (2005).   Google Scholar

[1]

Varga K. Kalantarov, Sergey Zelik. Smooth attractors for the Brinkman-Forchheimer equations with fast growing nonlinearities. Communications on Pure & Applied Analysis, 2012, 11 (5) : 2037-2054. doi: 10.3934/cpaa.2012.11.2037
[2]

Yuncheng You, Caidi Zhao, Shengfan Zhou. The existence of uniform attractors for 3D Brinkman-Forchheimer equations. Discrete & Continuous Dynamical Systems - A, 2012, 32 (10) : 3787-3800. doi: 10.3934/dcds.2012.32.3787
[3]

Elder Jesús Villamizar-Roa, Henry Lamos-Díaz, Gilberto Arenas-Díaz. Very weak solutions for the magnetohydrodynamic type equations. Discrete & Continuous Dynamical Systems - B, 2008, 10 (4) : 957-972. doi: 10.3934/dcdsb.2008.10.957
[4]

Hong Cai, Zhong Tan. Time periodic solutions to the three--dimensional equations of compressible magnetohydrodynamic flows. Discrete & Continuous Dynamical Systems - A, 2016, 36 (4) : 1847-1868. doi: 10.3934/dcds.2016.36.1847
[5]

Verena Bögelein, Frank Duzaar, Ugo Gianazza. Very weak solutions of singular porous medium equations with measure data. Communications on Pure & Applied Analysis, 2015, 14 (1) : 23-49. doi: 10.3934/cpaa.2015.14.23
[6]

Ioana Ciotir. Stochastic porous media equations with divergence Itô noise. Evolution Equations & Control Theory, 2019, 0 (0) : 0-0. doi: 10.3934/eect.2020010
[7]

Gabriele Grillo, Matteo Muratori, Maria Michaela Porzio. Porous media equations with two weights: Smoothing and decay properties of energy solutions via Poincaré inequalities. Discrete & Continuous Dynamical Systems - A, 2013, 33 (8) : 3599-3640. doi: 10.3934/dcds.2013.33.3599
[8]

Xiaoli Li, Dehua Wang. Global solutions to the incompressible magnetohydrodynamic equations. Communications on Pure & Applied Analysis, 2012, 11 (2) : 763-783. doi: 10.3934/cpaa.2012.11.763
[9]

Mattia Turra. Existence and extinction in finite time for Stratonovich gradient noise porous media equations. Evolution Equations & Control Theory, 2019, 8 (4) : 867-882. doi: 10.3934/eect.2019042
[10]

Leda Bucciantini, Angiolo Farina, Antonio Fasano. Flows in porous media with erosion of the solid matrix. Networks & Heterogeneous Media, 2010, 5 (1) : 63-95. doi: 10.3934/nhm.2010.5.63
[11]

Tong Tang, Hongjun Gao. Local strong solutions to the compressible viscous magnetohydrodynamic equations. Discrete & Continuous Dynamical Systems - B, 2016, 21 (5) : 1617-1633. doi: 10.3934/dcdsb.2016014
[12]

Zhong Tan, Leilei Tong. Asymptotic stability of stationary solutions for magnetohydrodynamic equations. Discrete & Continuous Dynamical Systems - A, 2017, 37 (6) : 3435-3465. doi: 10.3934/dcds.2017146
[13]

Weiping Yan. Existence of weak solutions to the three-dimensional density-dependent generalized incompressible magnetohydrodynamic flows. Discrete & Continuous Dynamical Systems - A, 2015, 35 (3) : 1359-1385. doi: 10.3934/dcds.2015.35.1359
[14]

Brahim Amaziane, Leonid Pankratov, Andrey Piatnitski. The existence of weak solutions to immiscible compressible two-phase flow in porous media: The case of fields with different rock-types. Discrete & Continuous Dynamical Systems - B, 2013, 18 (5) : 1217-1251. doi: 10.3934/dcdsb.2013.18.1217
[15]

Cédric Galusinski, Mazen Saad. A nonlinear degenerate system modelling water-gas flows in porous media. Discrete & Continuous Dynamical Systems - B, 2008, 9 (2) : 281-308. doi: 10.3934/dcdsb.2008.9.281
[16]

Olivier Pironneau, Alexei Lozinski, Alain Perronnet, Frédéric Hecht. Numerical zoom for multiscale problems with an application to flows through porous media. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 265-280. doi: 10.3934/dcds.2009.23.265
[17]

Yiqian Wang. Boundedness of solutions in a class of Duffing equations with a bounded restore force. Discrete & Continuous Dynamical Systems - A, 2006, 14 (4) : 783-800. doi: 10.3934/dcds.2006.14.783
[18]

Dongfen Bian, Boling Guo. Global existence and large time behavior of solutions to the electric-magnetohydrodynamic equations. Kinetic & Related Models, 2013, 6 (3) : 481-503. doi: 10.3934/krm.2013.6.481
[19]

Lucas C. F. Ferreira, Jhean E. Pérez-López, Élder J. Villamizar-Roa. On the product in Besov-Lorentz-Morrey spaces and existence of solutions for the stationary Boussinesq equations. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2423-2439. doi: 10.3934/cpaa.2018115
[20]

Alessio Pomponio. Oscillating solutions for prescribed mean curvature equations: euclidean and lorentz-minkowski cases. Discrete & Continuous Dynamical Systems - A, 2018, 38 (8) : 3899-3911. doi: 10.3934/dcds.2018169

2018 Impact Factor: 0.925

Tools

Metrics

  • PDF downloads (8)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences