Stability estimates and a Lagrange-Galerkin scheme for a Navier-Stokes type model of flow in non-homogeneous porous media

Imam Wijaya; Hirofumi Notsu

doi:10.3934/dcdss.2020234

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2021, Volume 14, Issue 3: 1197-1212. Doi: 10.3934/dcdss.2020234

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Stability estimates and a Lagrange-Galerkin scheme for a Navier-Stokes type model of flow in non-homogeneous porous media

Imam Wijaya^1, , and
Hirofumi Notsu^2,3,

1.
Division of Mathematical and Physics Science, Kanazawa University, Kakuma, Kanazawa 920-1192, Japan
2.
Faculty of Mathematical and Physics, Kanazawa University, Kakuma, Kanazawa 920-1192, Japan
3.
Japan Science and Technology Agency, PRESTO, Kawaguchi 332-0012, Japan

^* Corresponding author: Imam Wijaya
^* Corresponding author: Imam Wijaya

Received: December 31, 2018

Revised: April 30, 2019

Early access: December 2019

Published: March 2021

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Abstract

The purposes of this work are to study the $ L^{2} $-stability of a Navier-Stokes type model for non-stationary flow in porous media proposed by Hsu and Cheng in 1989 and to develop a Lagrange-Galerkin scheme with the Adams-Bashforth method to solve that model numerically. The stability estimate is obtained thanks to the presence of a nonlinear drag force term in the model which corresponds to the Forchheimer term. We derive the Lagrange-Galerkin scheme by extending the idea of the method of characteristics to overcome the difficulty which comes from the non-homogeneous porosity. Numerical experiments are conducted to investigate the experimental order of convergence of the scheme. For both simple and complex designs of porosities, our numerical simulations exhibit natural flow profiles which well describe the flow in non-homogeneous porous media.

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Mathematics Subject Classification: Primary: 76S05, 35Q30; Secondary: 65M25, 74S05.

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Figure 1. Representative elementary volume (REV)

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Figure 2. The order of convergence for scheme (22)

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Figure 3. The boundary conditions and the finite element mesh

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Figure 4. Time evolution of velocity magnitude

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Figure 5. Computation domain and porosity value distribution

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Figure 6. Time evolution of magnitude velocity

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Table 1. The unit of important symbols

No	Symbol	Unit	Name of the symbol
[0.5ex] 1	$ u $	$ {\rm m} \cdot {\rm s}^{-1} $	Darcy velocity
2	$ p $	$ {\rm kg} \cdot {\rm m}^{-1}\cdot {\rm s}^{-2} $	Pressure
3	$ \phi $	-	porosity
4	$ k_{D} $	$ {\rm kg}^{-1} \cdot {\rm m}^3 \cdot {\rm s} $	Hydraulic conductivity
5	$ K $	$ {\rm m}^2 $	Permeability
6	$ \mu $	$ {\rm kg} \cdot {\rm m}^{-1}\cdot {\rm s}^{-1} $	Dynamic viscosity
7	$ \rho $	$ {\rm kg} \cdot {\rm m}^{-3} $	Density
8	$ d_{p} $	$ {\rm m} $	Particle diameter
9	$ F $	-	Forchheimer constant
10	$ B $	$ {\rm kg} \cdot {\rm m}^{-2} \cdot {\rm s}^{-2} $	Drag force per unit volume

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Table 2. Values of $ Er1 $ and $ Er2 $, their slopes, and CPU times for the problem in Subsection 6.1 by scheme (22)

$ N $	$ Er1 $	$ Er2 $	Slope of $ Er1 $	Slope of $ Er2 $	CPU time [s]
4	$ 3.4\times10^{-1} $	$ 1.6\times10^{-1} $	$ - $	$ - $	1.9
8	$ 7.1\times10^{-2} $	$ 5.8\times10^{-3} $	2.26	4.76	16.4
16	$ 1.4\times10^{-2} $	$ 1.2\times10^{-3} $	2.34	2.30	174.8
32	$ 3.5\times10^{-3} $	$ 2.9\times10^{-4} $	2.00	2.05	577.4
64	$ 1.0\times10^{-3} $	$ 6.3\times10^{-5} $	1.81	2.20	5,953.9
128	$ 2.8\times10^{-4} $	$ 1.5\times10^{-5} $	1.84	2.07	58,150.9

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