A generalization of the Darcy-Forchheimer equation involving an implicit, pressure-dependent relation between the drag force and the velocity. (English) Zbl 1308.35198
Summary: We study mathematical properties of steady flows described by the system of equations generalizing the classical porous media models of Darcy and Forchheimer. The considered generalizations are outlined by implicit relations between the drag force and the velocity, that are in addition parametrized by the pressure. We analyze such drag force-velocity relations which are described through a maximal monotone graph varying continuously with the pressure. Large-data existence of a solution to this system is established, whereupon we show that under certain assumptions on data, the pressure satisfies a maximum or minimum principle, even if the drag coefficient depends on the pressure exponentially.
MSC:
| 35Q35 | PDEs in connection with fluid mechanics |
| 76S05 | Flows in porous media; filtration; seepage |
| 35B50 | Maximum principles in context of PDEs |
Keywords:
Darcy-Forchheimer equation; pressure dependent material coefficient; implicit constitutive theory; maximal monotone graph; existence theory; maximum/minimum principleReferences:
| [1] | Alberti, G.; Ambrosio, L., A geometrical approach to monotone functions in \(R^n\), Math. Z., 230, 2, 259-316 (1999) · Zbl 0934.49025 |
| [2] | Aulisa, E.; Bloshanskaya, L.; Hoang, L.; Ibragimov, A., Analysis of generalized Forchheimer flows of compressible fluids in porous media, J. Math. Phys., 50, 10, 103102 (2009), 44 pp · Zbl 1236.76067 |
| [3] | Barus, C., Isotherms, isopiestics and isometrics relative to viscosity, Amer. J. Sci., 45, 3, 87-96 (1893) |
| [4] | Bulíček, M.; Gwiazda, P.; Málek, J.; Rajagopal, K. R.; Świerczewska-Gwiazda, A., On flows of fluids described by an implicit constitutive equation characterized by a maximal monotone graph, (Robinson, J. C.; Rodrigo, J. L.; Sadowski, W., Mathematical Aspects of Fluid Mechanics. Mathematical Aspects of Fluid Mechanics, London Math. Soc. Lecture Note Ser., vol. 402 (2012), Cambridge University Press), 26-54 · Zbl 1296.35137 |
| [5] | Bulíček, M.; Gwiazda, P.; Málek, J.; Świerczewska-Gwiazda, A., On unsteady flows of implicitly constituted incompressible fluids, SIAM J. Math. Anal., 44, 4, 2756-2801 (2012) · Zbl 1256.35074 |
| [6] | Bulíček, M.; Majdoub, M.; Málek, J., Unsteady flows of fluids with pressure dependent viscosity in unbounded domains, Nonlinear Anal. Real World Appl., 11, 5, 3968-3983 (2010) · Zbl 1201.35156 |
| [7] | Bulíček, M.; Málek, J.; Rajagopal, K. R., Navier’s slip and evolutionary Navier-Stokes-like systems with pressure and shear-rate dependent viscosity, Indiana Univ. Math. J., 56, 1, 51-85 (2007) · Zbl 1129.35055 |
| [8] | Bulíček, M.; Málek, J.; Rajagopal, K. R., Mathematical analysis of unsteady flows of fluids with pressure, shear-rate, and temperature dependent material moduli that slip at solid boundaries, SIAM J. Math. Anal., 41, 2, 665-707 (2009) · Zbl 1195.35239 |
| [9] | Chiadò Piat, V.; Dal Maso, G.; Defranceschi, A., \(G\)-convergence of monotone operators, Ann. Inst. H. Poincaré Anal. Non Linéaire, 7, 3, 123-160 (1990) · Zbl 0731.35033 |
| [10] | Fabrie, P., Regularity of the solution of Darcy-Forchheimer’s equation, Nonlinear Anal., 13, 9, 1025-1049 (1989) · Zbl 0719.35070 |
| [11] | Francfort, G.; Murat, F.; Tartar, L., Monotone operators in divergence form with \(x\)-dependent multivalued graphs, Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat. (8), 7, 1, 23-59 (2004) · Zbl 1115.35047 |
| [12] | Franta, M.; Málek, J.; Rajagopal, K. R., On steady flows of fluids with pressure- and shear-dependent viscosities, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 461, 2055, 651-670 (2005) · Zbl 1145.76311 |
| [13] | Galdi, G. P., An Introduction to the Mathematical Theory of the Navier-Stokes Equations. Steady-State Problems, Springer Monogr. Math. (2011), Springer: Springer New York · Zbl 1245.35002 |
| [14] | Gazzola, F., A note on the evolution Navier-Stokes equations with a pressure-dependent viscosity, Z. Angew. Math. Phys., 48, 5, 760-773 (1997) · Zbl 0895.76018 |
| [15] | Gazzola, F.; Secchi, P., Some results about stationary Navier-Stokes equations with a pressure-dependent viscosity, (Navier-Stokes Equations: Theory and Numerical Methods. Navier-Stokes Equations: Theory and Numerical Methods, Varenna, 1997. Navier-Stokes Equations: Theory and Numerical Methods. Navier-Stokes Equations: Theory and Numerical Methods, Varenna, 1997, Pitman Res. Notes Math. Ser., vol. 388 (1998), Longman: Longman Harlow), 31-37 · Zbl 0940.35156 |
| [16] | Hron, J.; Málek, J.; Nečas, J.; Rajagopal, K. R., Numerical simulations and global existence of solutions of two-dimensional flows of fluids with pressure- and shear-dependent viscosities, MODELLING 2001. MODELLING 2001, Pilsen. MODELLING 2001. MODELLING 2001, Pilsen, Math. Comput. Simulation, 61, 3-6, 297-315 (2003) · Zbl 1205.76159 |
| [17] | Hron, J.; Málek, J.; Průša, V.; Rajagopal, K. R., Further remarks on simple flows of fluids with pressure-dependent viscosities, Nonlinear Anal. Real World Appl., 12, 1, 394-402 (2011) · Zbl 1206.35207 |
| [18] | Kannan, K.; Rajagopal, K. R., Flow through porous media due to high pressure gradients, Appl. Math. Comput., 199, 2, 748-759 (2008) · Zbl 1228.76161 |
| [19] | Kato, T., Demicontinuity, hemicontinuity and monotonicity, Bull. Amer. Math. Soc., 70, 548-550 (1964) · Zbl 0123.10701 |
| [20] | Lions, J.-L., Quelques méthodes de résolution des problèmes aux limites non linéaires (1969), Dunod · Zbl 0189.40603 |
| [21] | Málek, J.; Nečas, J.; Rajagopal, K. R., Global analysis of the flows of fluids with pressure-dependent viscosities, Arch. Ration. Mech. Anal., 165, 3, 243-269 (2002) · Zbl 1022.76011 |
| [22] | Málek, J.; Rajagopal, K. R., Mathematical issues concerning the Navier-Stokes equations and some of its generalizations, (Evolutionary Equations, vol. II. Evolutionary Equations, vol. II, Handb. Differ. Equ. (2005), Elsevier/North-Holland: Elsevier/North-Holland Amsterdam), 371-459 · Zbl 1095.35027 |
| [23] | Minty, G. J., Monotone (nonlinear) operators in Hilbert space, Duke Math. J., 29, 341-346 (1962) · Zbl 0111.31202 |
| [24] | Nakshatrala, K. B.; Rajagopal, K. R., A numerical study of fluids with pressure-dependent viscosity flowing through a rigid porous medium, Internat. J. Numer. Methods Fluids, 67, 3, 342-368 (2011) · Zbl 1308.76273 |
| [25] | Nield, D. A.; Bejan, A., Convection in Porous Media (1999), Springer-Verlag: Springer-Verlag New York · Zbl 0924.76001 |
| [26] | Průša, V., Revisiting Stokes first and second problems for fluids with pressure-dependent viscosities, Internat. J. Engrg. Sci., 48, 12, 2054-2065 (2010) |
| [27] | Rajagopal, K. R., On implicit constitutive theories for fluids, J. Fluid Mech., 550, 243-249 (2006) · Zbl 1097.76009 |
| [28] | Rajagopal, K. R., On a hierarchy of approximate models for flows of incompressible fluids through porous solids, Math. Models Methods Appl. Sci., 17, 2, 215-252 (2007) · Zbl 1123.76066 |
| [29] | Rajagopal, K. R.; Saccomandi, G.; Vergori, L., Unsteady flows of fluids with pressure dependent viscosity, J. Math. Anal. Appl., 404, 2, 362-372 (2013) · Zbl 1304.76018 |
| [30] | Renardy, M., Some remarks on the Navier-Stokes equations with a pressure-dependent viscosity, Comm. Partial Differential Equations, 11, 7, 779-793 (1986) · Zbl 0597.35097 |
| [31] | Srinivasan, S.; Rajagopal, K. R., A thermodynamic basis for the derivation of the Darcy, Forchheimer and Brinkman models for flows through porous media and their generalizations, Internat. J. Non-Linear Mech., 58, 162-166 (2014) |
| [32] | Subramanian, S. C.; Rajagopal, K. R., A note on the flow through porous solids at high pressures, Comput. Math. Appl., 53, 2, 260-275 (2007) · Zbl 1129.76054 |
| [33] | Suslov, S. A.; Tran, T. D., Revisiting plane Couette-Poiseuille flows of a piezo-viscous fluid, J. Non-Newton. Fluid Mech., 154, 170-178 (2008) |
| [34] | Suslov, S. A.; Tran, T. D., Stability of plane Poiseuille-Couette flows of a piezo-viscous fluid, J. Non-Newton. Fluid Mech., 156, 139-149 (2009) · Zbl 1274.76214 |
| [35] | Uğurlu, D., On the existence of a global attractor for the Brinkman-Forchheimer equations, Nonlinear Anal., 68, 7, 1986-1992 (2008) · Zbl 1137.35316 |
| [36] | Vasudevaiah, M.; Rajagopal, K. R., On fully developed flows of fluids with a pressure dependent viscosity in a pipe, Appl. Math., 50, 4, 341-353 (2005) · Zbl 1099.76019 |
| [37] | Zhao, C.; You, Y., Approximation of the incompressible convective Brinkman-Forchheimer equations, J. Evol. Equ., 12, 4, 767-788 (2012) · Zbl 1259.35170 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
