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Existence results for a monophasic compressible Darcy–Brinkman’s flow in porous media

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Abstract

In this paper, we are interested in the displacement of a single compressible phase in Darcy–Brinkman’s flow in porous media. The equations are obtained by the conservation of mass and by considering the Brinkman regularization velocity of the standard Darcy infiltration velocity. This model is treated in its general form with the whole nonlinear terms. First, we prove the existence and uniqueness of a strong solution in one dimensional space for the Darcy–Brinkman system. Second, we treat this system with Bear hypothesis in multidimensional spaces.

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References

  1. Adimurthi, M.I.S.H.R.A.S., Gowda, G.V.: Optimal entropy solutions for conservation laws with discontinuous flux-functions. J. Hyperbolic Differ. Equ. 2.04, 783–837 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  2. Alinhac, S., Gérard, P.: Opérateurs Pseudo-différentiels et théorème de Nash-Moser. Inter-Editions et Editions du CNRS, Paris (1991)

    MATH  Google Scholar 

  3. Allaire, Grégoire: Homogenization of the Navier–Stokes equations in open sets perforated with tiny holes I. Abstract framework, a volume distribution of holes. Arch. Rational Mech. Anal. 113.3, 209–259 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  4. Amiri, A., Vafai, K.: Analysis of dispersion effects and non-thermal equilibrium, non-Darcian, variable porosity incompressible flow through porous media. Int. J. Heat Mass Transf. 37(6), 939–954 (1994)

    Article  Google Scholar 

  5. Andreianov, B., Karlsen, K.H., Risebro, N.H.: A theory of \(L^1\)-dissipative solvers for scalar conservation laws with discontinuous flux. Arch. Rational Mech. Anal. 201(1), 27–86 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  6. Angirasa, D.: Forced convective heat transfer in metallic fibrous materials. Trans.-Ame. Soc. Mech. Eng. J. Heat Trans. 124(4), 739–745 (2002)

    Article  Google Scholar 

  7. Auriault, J.L., Geindreau, C., Boutin, C.: Filtration law in porous media with poor separation of scales. Transp. Porous Media 60(1), 89–108 (2005)

    Article  MathSciNet  Google Scholar 

  8. Bear, J.: Dynamics of Fluids in Porous Media. Courier Corporation, Chelmsford (2013)

    MATH  Google Scholar 

  9. Aziz, K., Settari, A.: Petroleum reservoir simulation. Applied Science Publisher LTD, London (1979)

    Google Scholar 

  10. Bresch, D., Jabin, P.E.: Global existence of weak solutions for compressible Navier-Stokes equations: thermodynamically unstable pressure and anisotropic viscous stress tensor. Ann. Math. 188(2), 577–684 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  11. Bresch, D., Jabin, P.E.: Global Weak Solutions of PDEs for Compressible Media: A Compactness Criterion to Cover New Physical Situations. Shocks, Singularities and Oscillations in Nonlinear Optics and Fluid Mechanics, pp. 33–54. Springer, Cham (2017)

    Chapter  Google Scholar 

  12. Brinkman, H.C.: A calculation of the viscous force exerted by a flowing fluid on a dense swarm of particles. Appl. Sci. Res. 1(1), 27–34 (1949)

    Article  MATH  Google Scholar 

  13. Calmidi, V.V., Mahajan, R.L.: Forced convection in high porosity metal foams. Trans.-Am. Soc. Mech. Eng. J. Heat Transf. 122(3), 557–565 (2000)

    Article  Google Scholar 

  14. Chavent, G., Jaffré, G.: Mathematical Models and Finite Elements for Reservoir Simulation: Single Phase, Multiphase and Multicomponent Flows through Porous Media, vol. 17. Elsevier, Amsterdam (1986)

    MATH  Google Scholar 

  15. Coclite, G., Mishra, S., Risebro, N.: Convergence of an Engquist–Osher scheme for a multi-dimensional triangular system of conservation laws. Math. Comput. 79(269), 71–94 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  16. Coclite, G.M., Risebro, N.H.: Conservation laws with time dependent discontinuous coefficients. SIAM J. Math. Anal. 36(4), 1293–1309 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  17. Donatelli, D., Trivisa, K.: On a nonlinear model for tumor growth: global in time weak solutions. J. Math. Fluid Mech. 16(4), 787–803 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  18. Dukhan, N.: Analysis of Brinkman-extended Darcy flow in porous media and experimental verification using metal foam. J. Fluids Eng. 134(7), 071–201 (2012)

    Article  Google Scholar 

  19. Gimse, T., Risebro, N.H.: Solution of the Cauchy problem for a conservation law with discontinuous flux function. SIAM J. Math. Anal. 23(3), 635–648 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  20. Gagneux, G., Madaune-Tort, M.: Analyse Mathématique de Modèles Nonlinéaires de l’ingénierie Pétrolière, vol. 22. Springe, Berlin (1995)

    MATH  Google Scholar 

  21. Holden, H., Risebro, N.H.: Front Tracking for Hyperbolic Conservation Laws. Springer, Berlin (2011)

    Book  MATH  Google Scholar 

  22. Israwi, S.: Large time existence for 1D Green-Naghdi equations. Nonlinear Anal. Theory Methods Appl. 74(1), 81–93 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  23. Kato, T., Ponce, G.: Commutator estimates and the euler and Navier–Stokes equations. Commun. Pure Appl. Math. 41(7), 891–907 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  24. Karlsen, K.H., Risebro, N.H., Towers, J.D.: \(L^1\) Stability for entropy solutions of degenerate parabolic convection-diffusion equations with discontinuous coefficients. Skr. K. Nor. Vidensk. Selsk. 3, 1–49 (2003)

    MATH  Google Scholar 

  25. Ladyzhenskaia, O.A., Solonnikov, V.A., Uraltseva, N.: Linear and quasi-linear equations of parabolic type, vol. 23. American Mathematical Society, Providence (1988)

    Google Scholar 

  26. Lannes, D.: Sharp estimates for pseudo-differential operators with symbols of limited smoothness and commutators. J. Funct. Anal. 232(2), 495–539 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  27. Lions, J.L.: Quelques méthodes de résolution des problèmes aux limites non-linéaires. Dunod-Gauthier-Villars, Paris (1969)

    MATH  Google Scholar 

  28. Lions, P.-L.: Mathematical topics in fluid mechanics, vol. I: incompressible models. Oxford Lect. Ser. Math. Appl. 3 (1996)

  29. Lions, P.-L.: Mathematical topics in fluid mechanics, Vol. II: compressible models. Oxford Lect. Ser. Math. Appl. (1998)

  30. Marusic-Paloka, E., Pazanin, I., Marusic, S.: Comparison between Darcy and Brinkman laws in a fracture. Appl. Math. Comput. 218(14), 7538–7545 (2012)

    MathSciNet  MATH  Google Scholar 

  31. Valdes-Parada, F.J., Ochoa-Tapia, J.A., Alvarez-Ramirez, J.: On the effective viscosity for the Darcy–Brinkman equation. Phys. A Stat. Mech. Appl. 385(1), 69–79 (2007)

    Article  MathSciNet  Google Scholar 

  32. Perthame, B., Vauchelet, N.: Incompressible limit of a mechanical model of tumour growth with viscosity. Phil. Trans. R. Soc. A 373(2050), 20140283 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  33. Renken, K.J., Poulikakos, D.: Forced convection in channel filled with porous medium, including the effects of flow inertia, variable porosity, and Brinkman friction. ASME J. Heat Transf. 109, 880–888 (1987)

    Article  Google Scholar 

  34. Simon, J.: Compact sets in the space \(L^p (0, T; B)\). Annal. Mat. Appl. 146(1), 65–96 (1986)

    Article  MATH  Google Scholar 

  35. Taylor, M.: Partial Differential Equations II: Qualitative Studies of Linear Equations, vol. 116. Springer, Berlin (2013)

    Google Scholar 

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Authors and Affiliations

  1. Laboratoire de Mathématiques Jean Leray, École Centrale de Nantes, UMR 6629 CNRS, 44321, Nantes, France

    Houssein Nasser El Dine & Mazen Saad

  2. Laboratory of Mathematics, Faculty of Sciences and Doctoral School of Sciences and Technology (EDST), Lebanese University, Hadath, Beirut, Lebanon

    Houssein Nasser El Dine & Raafat Talhouk

Authors
  1. Houssein Nasser El Dine
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  2. Mazen Saad
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  3. Raafat Talhouk
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Correspondence to Mazen Saad.

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Nasser El Dine, H., Saad, M. & Talhouk, R. Existence results for a monophasic compressible Darcy–Brinkman’s flow in porous media. J Elliptic Parabol Equ 5, 125–147 (2019). https://doi.org/10.1007/s41808-019-00035-y

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  • DOI: https://doi.org/10.1007/s41808-019-00035-y

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