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

Timir Karmakar; Meraj Alam; G. P. Raja Sekhar

doi:10.3934/cpaa.2022001

Article Contents

2022, Volume 21, Issue 3: 845-865. Doi: 10.3934/cpaa.2022001

This issue Previous Article Radial symmetry of nonnegative solutions for nonlinear integral systems Next Article Improved Hardy-Rellich inequalities

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
^* Corresponding author

Received: May 2021

Revised: November 2021

Early access: December 2021

Published: March 2022

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Abstract

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.

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Mathematics Subject Classification: Primary: 76S05, 34E10, 35J66, 35D30.

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Figure 1. Schematic of Couette flow through a porous channel

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Figure 2. $ 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 $

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Figure 3. 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 $

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Figure 4. 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 $

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Figure 5. 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 $

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Figure 6. 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 $

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Figure 7. 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 $

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Figure 8. 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 $

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Table 1. 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]

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