# American Institute of Mathematical Sciences

March  2022, 21(3): 845-865. doi: 10.3934/cpaa.2022001

## Analysis of Brinkman-Forchheimer extended Darcy's model in a fluid saturated anisotropic porous channel

 1 Department of Mathematics, National Institute of Technology Meghalaya, Shillong-793003, India 2 Department of Mathematics, Mahindra University, Hyderabad- 500043 - Telangana, INDIA 3 Department of Mathematics, Indian Institute of Technology Kharagpur, West Bengal-721302, India

* Corresponding author

Received  May 2021 Revised  November 2021 Published  March 2022 Early access  December 2021

We present asymptotic analysis of Couette flow through a channel packed with porous medium. We assume that the porous medium is anisotropic and the permeability varies along all the directions so that it appears as a positive semidefinite matrix in the momentum equation. We developed existence and uniqueness results corresponding to the anisotropic Brinkman-Forchheimer extended Darcy's equation in case of fully developed flow using the Browder-Minty theorem. Complemented with the existence and uniqueness analysis, we present an asymptotic solution by taking Darcy number as the perturbed parameter. For a high Darcy number, the corresponding problem is dealt with regular perturbation expansion. For low Darcy number, the problem of interest is a singular perturbation. We use matched asymptotic expansion to treat this case. More generally, we obtained an approximate solution for the nonlinear problem, which is uniformly valid irrespective of the porous medium parameter values. The analysis presented serves a dual purpose by providing the existence and uniqueness of the anisotropic nonlinear Brinkman-Forchheimer extended Darcy's equation and provide an approximate solution that shows good agreement with the numerical solution.

Citation: Timir Karmakar, Meraj Alam, G. P. Raja Sekhar. Analysis of Brinkman-Forchheimer extended Darcy's model in a fluid saturated anisotropic porous channel. Communications on Pure and Applied Analysis, 2022, 21 (3) : 845-865. doi: 10.3934/cpaa.2022001
##### References:
 [1] A. A. Avramenko, Igor V. Shevchuk, M. M. Kovetskaya and Y. Y. Kovetska, Darcy–Brinkman–Forchheimer model for film boiling in porous media, Transp. Porous Med., 134 (2020), 503-536.  doi: 10.1007/s11242-020-01452-7. [2] S. Berg, A. W. Cense, J. P. Hofman and R. M. M. Smits, Flow in Porous Media with Slip Boundary Condition, Society of Core Analysts, Calgary, Canada. SCA-2007-13. [3] A. Bush, Perturbation Methods for Engineers and Scientists, Routledge, 2018. [4] S. Caucao, G. N. Gatica, R. Oyarzúa and N. Sánchez, A fully-mixed formulation for the steady double-diffusive convection system based upon Brinkman–Forchheimer equations, J. Sci. Comput., 85 (2020), 1-37.  doi: 10.1007/s10915-020-01305-x. [5] S. Chellam and M. R. Wiesner, Slip flow through porous media with permeable boundaries: implications for the dimensional scaling of packed beds, Water Environ. Research, 65 (1993), 744-749. [6] M. Delavar and M. Azimi, I using porous material for heat transfer enhancement in heat exchangers, J. Eng. Sci. Tech. Rev., 6 (2013), 14-16. [7] P. Forchheimer, Water movement through the ground, Z. Ver. German, Ing., 45 (1901), 1782-1788. [8] T. Ghosh and G. P. Raja Sekhar, A note on Mellin-Fourier integral transform technique to solve Stokes' problem analogue to flow through a composite layer of free flow and porous medium, J. Math. Anal. Appl., 483 (2020), 123578.  doi: 10.1016/j.jmaa.2019.123578. [9] R. Givler and S. Altobelli, A determination of the effective viscosity for the Brinkman–Forchheimer flow model, J. Fluid Mech., 258 (1994), 355-370. [10] A. A. Hill and B. Straughan, Poiseuille flow in a fluid overlying a porous medium, J. Fluid Mech., 603 (2008), 137-149.  doi: 10.1017/S0022112008000852. [11] K. Hooman, A perturbation solution for forced convection in a porous-saturated duct, J. Comput. Appl. Math., 211 (2008), 57-66.  doi: 10.1016/j.cam.2006.11.005. [12] R. A. Horn and C. R. Johnson, Matrix Analysis, Cambridge University Press, econd edition, 2013. [13] B. K. Jha and M. L. Kaurangini, Approximate analytical solutions for the nonlinear Brinkman-Forchheimer-extended Darcy flow model, Appl. Math., 2 (2011), 1432-1436. [14] V. Kalantarov and S. Zelik, Smooth attractors for the Brinkman-Forchheimer equations with fast growing nonlinearities, Commun. Pure Appl. Anal., 11 (2012), 2037-2054.  doi: 10.3934/cpaa.2012.11.2037. [15] P. Kaloni and J. Guo, Steady nonlinear double-diffusive convection in a porous medium based upon the Brinkman–Forchheimer model, J. Math. Anal. Appl., 204 (1996), 138-155.  doi: 10.1006/jmaa.1996.0428. [16] T. Karmakar, Physics of unsteady Couette flow in an anisotropic porous medium, J. Eng. Math., 130 (2021), 26 pp. doi: 10.1007/s10665-021-10165-9. [17] T. Karmakar, M. Reza and G. P. Raja Sekhar, Forced convection in a fluid saturated anisotropic porous channel with isoflux boundaries, Phys. Fluid., 31 (2019), 117109. [18] N. Kladias and V. Prasad, Experimental verification of Darcy-Brinkman-Forchheimer flow model for natural convection in porous media, J. Thermophysics heat transfer, 5 (1991), 560-576. [19] P. Kumar and G. P. Raja Sekhar, Elastohydrodynamics of a deformable porous packing in a channel competing under shear and pressure gradient, Phys. Fluid., 32 (2020), 061901, 22 pp. [20] B. Lai, J. L. Miskimins and Y. S. Wu, Non-Darcy porous-media flow according to the Barree and Conway model: laboratory and numerical-modeling studies, SPE j., 17 (2012), 70-79. [21] D. A. Nield and A. Bejan, Convection in Porous Media, Vol. 3, Springer, 2006. [22] D. A. Nield, S. Junqueira and J. L. Lage, Forced convection in a fluid-saturated porous-medium channel with isothermal or isoflux boundaries, J. Fluid Mech., 322 (1996), 201-214. [23] S. Salsa, Partial Differential Equations in Action: From Modelling to Theory, Springer, 2016. doi: 10.1007/978-3-319-31238-5. [24] P. Skrzypacz and D. Wei, Solvability of the Brinkman-Forchheimer-Darcy equation, J. Appl. Math., 2017 (2017), 10 pp. doi: 10.1155/2017/7305230. [25] K. Vafai and S. J. Kim, Forced convection in a channel filled with a porous medium: an exact solution, ASME J. Heat Transfer 111 (1989), 1103–1106. [26] K. Vafai and C. L. Tien, Boundary and inertia effects on flow and heat transfer in porous media, Int. J. Heat Mass Trans., 24 (1981), 195-203. [27] E. Zeidler, Nonlinear Functional Analysis and Its Applications: Ⅱ/B: Nonlinear Monotone Operators, Springer Science & Business Media, 2013. doi: 10.1007/978-1-4612-0985-0.

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##### References:
 [1] A. A. Avramenko, Igor V. Shevchuk, M. M. Kovetskaya and Y. Y. Kovetska, Darcy–Brinkman–Forchheimer model for film boiling in porous media, Transp. Porous Med., 134 (2020), 503-536.  doi: 10.1007/s11242-020-01452-7. [2] S. Berg, A. W. Cense, J. P. Hofman and R. M. M. Smits, Flow in Porous Media with Slip Boundary Condition, Society of Core Analysts, Calgary, Canada. SCA-2007-13. [3] A. Bush, Perturbation Methods for Engineers and Scientists, Routledge, 2018. [4] S. Caucao, G. N. Gatica, R. Oyarzúa and N. Sánchez, A fully-mixed formulation for the steady double-diffusive convection system based upon Brinkman–Forchheimer equations, J. Sci. Comput., 85 (2020), 1-37.  doi: 10.1007/s10915-020-01305-x. [5] S. Chellam and M. R. Wiesner, Slip flow through porous media with permeable boundaries: implications for the dimensional scaling of packed beds, Water Environ. Research, 65 (1993), 744-749. [6] M. Delavar and M. Azimi, I using porous material for heat transfer enhancement in heat exchangers, J. Eng. Sci. Tech. Rev., 6 (2013), 14-16. [7] P. Forchheimer, Water movement through the ground, Z. Ver. German, Ing., 45 (1901), 1782-1788. [8] T. Ghosh and G. P. Raja Sekhar, A note on Mellin-Fourier integral transform technique to solve Stokes' problem analogue to flow through a composite layer of free flow and porous medium, J. Math. Anal. Appl., 483 (2020), 123578.  doi: 10.1016/j.jmaa.2019.123578. [9] R. Givler and S. Altobelli, A determination of the effective viscosity for the Brinkman–Forchheimer flow model, J. Fluid Mech., 258 (1994), 355-370. [10] A. A. Hill and B. Straughan, Poiseuille flow in a fluid overlying a porous medium, J. Fluid Mech., 603 (2008), 137-149.  doi: 10.1017/S0022112008000852. [11] K. Hooman, A perturbation solution for forced convection in a porous-saturated duct, J. Comput. Appl. Math., 211 (2008), 57-66.  doi: 10.1016/j.cam.2006.11.005. [12] R. A. Horn and C. R. Johnson, Matrix Analysis, Cambridge University Press, econd edition, 2013. [13] B. K. Jha and M. L. Kaurangini, Approximate analytical solutions for the nonlinear Brinkman-Forchheimer-extended Darcy flow model, Appl. Math., 2 (2011), 1432-1436. [14] V. Kalantarov and S. Zelik, Smooth attractors for the Brinkman-Forchheimer equations with fast growing nonlinearities, Commun. Pure Appl. Anal., 11 (2012), 2037-2054.  doi: 10.3934/cpaa.2012.11.2037. [15] P. Kaloni and J. Guo, Steady nonlinear double-diffusive convection in a porous medium based upon the Brinkman–Forchheimer model, J. Math. Anal. Appl., 204 (1996), 138-155.  doi: 10.1006/jmaa.1996.0428. [16] T. Karmakar, Physics of unsteady Couette flow in an anisotropic porous medium, J. Eng. Math., 130 (2021), 26 pp. doi: 10.1007/s10665-021-10165-9. [17] T. Karmakar, M. Reza and G. P. Raja Sekhar, Forced convection in a fluid saturated anisotropic porous channel with isoflux boundaries, Phys. Fluid., 31 (2019), 117109. [18] N. Kladias and V. Prasad, Experimental verification of Darcy-Brinkman-Forchheimer flow model for natural convection in porous media, J. Thermophysics heat transfer, 5 (1991), 560-576. [19] P. Kumar and G. P. Raja Sekhar, Elastohydrodynamics of a deformable porous packing in a channel competing under shear and pressure gradient, Phys. Fluid., 32 (2020), 061901, 22 pp. [20] B. Lai, J. L. Miskimins and Y. S. Wu, Non-Darcy porous-media flow according to the Barree and Conway model: laboratory and numerical-modeling studies, SPE j., 17 (2012), 70-79. [21] D. A. Nield and A. Bejan, Convection in Porous Media, Vol. 3, Springer, 2006. [22] D. A. Nield, S. Junqueira and J. L. Lage, Forced convection in a fluid-saturated porous-medium channel with isothermal or isoflux boundaries, J. Fluid Mech., 322 (1996), 201-214. [23] S. Salsa, Partial Differential Equations in Action: From Modelling to Theory, Springer, 2016. doi: 10.1007/978-3-319-31238-5. [24] P. Skrzypacz and D. Wei, Solvability of the Brinkman-Forchheimer-Darcy equation, J. Appl. Math., 2017 (2017), 10 pp. doi: 10.1155/2017/7305230. [25] K. Vafai and S. J. Kim, Forced convection in a channel filled with a porous medium: an exact solution, ASME J. Heat Transfer 111 (1989), 1103–1106. [26] K. Vafai and C. L. Tien, Boundary and inertia effects on flow and heat transfer in porous media, Int. J. Heat Mass Trans., 24 (1981), 195-203. [27] E. Zeidler, Nonlinear Functional Analysis and Its Applications: Ⅱ/B: Nonlinear Monotone Operators, Springer Science & Business Media, 2013. doi: 10.1007/978-1-4612-0985-0.
Schematic of Couette flow through a porous channel
$G'(u)$ behavior inside the channel for $F = 10$, $M = 1$, $K = 0.5$, $\phi = 0$, $U = 1$, $l_{s} = 0.1$ when (a) $Da = 0.01$ (b) $Da = 10$
Dimensionless velocity profile (a) for $Da = 0.01$, $M = 10$, $F = 10$, $l_{s} = 0.1$, $K = 1$, $U = 1$ (b) for $Da = 1$, $M = 1$, $F = 1$, $l_{s} = 0.1$, $K = 1$, $U = 1$
Dimensionless velocity profile (a) for different anisotropic ratio when $M = 1$, $Da = 0.01$, $U = 1$, $l_{s} = 0.1$ (b) for different anisotropic angle when $M = 1$, $K = 0.5$, $Da = 0.01$, $U = 1$, $l_{s} = 0.1$
Dimensionless velocity profile (a) for different Darcy number when $K = 1$, $M = 1$, $F = 1$, $l_{s} = 0.03$, $U = 1$ (b) for different Forchheimer number when $M = 1$, $K = 1$, $Da = 0.1$, $l_{s} = 0.1$, $U = 1$
Status of the pressure gradient in $y$-direction for $A_{r} = 1$, $M = 1$, $l_{s} = 0.03$, $K = 0.5$, $\phi = \pi/4$, $F = 1$
Wall shear stress as a function of Darcy number when $K = 1$, $M = 1$, $F = 1$, $l_{s} = 0.03$, $U = 1$. (b) Wall shear stress as a function of Darcy number for different $F$ when $K = 1$, $M = 1$, $l_{s} = 0.03$, $U = 1$
Wall shear stress as a function of Darcy number (a) for different anisotropic ratio when $M = 1$, $F = 1$, $l_{s} = 0.03$, $U = 1$ (b) for different anisotropic angle when $K = 0.5$, $M = 1$, $F = 1$, $U = 1$, $l_{s} = 0.03$
List of parameters range used for numerical simulation
 Parameters Description Value References Da Non-dimensional permeability $10^{-4}\leq Da\leq 10$ [22,17] F Non-dimensional inertial coefficient $0\leq F\leq 100$ [11,17] M Viscosity ratio $0.5\leq M\leq 10$ [22,9] K Anisotropic permeability ratio $0.5\leq K\leq4$ [17] $\phi$ Orientation angle $0\leq \phi\leq\frac{\pi}{2}$ [17]
 Parameters Description Value References Da Non-dimensional permeability $10^{-4}\leq Da\leq 10$ [22,17] F Non-dimensional inertial coefficient $0\leq F\leq 100$ [11,17] M Viscosity ratio $0.5\leq M\leq 10$ [22,9] K Anisotropic permeability ratio $0.5\leq K\leq4$ [17] $\phi$ Orientation angle $0\leq \phi\leq\frac{\pi}{2}$ [17]
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