# American Institute of Mathematical Sciences

March  2021, 14(3): 1197-1212. doi: 10.3934/dcdss.2020234

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

 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

Received  January 2019 Revised  May 2019 Published  December 2019

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.

Citation: Imam Wijaya, Hirofumi Notsu. Stability estimates and a Lagrange-Galerkin scheme for a Navier-Stokes type model of flow in non-homogeneous porous media. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 1197-1212. doi: 10.3934/dcdss.2020234
##### References:

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##### References:
Representative elementary volume (REV)
The order of convergence for scheme (22)
The boundary conditions and the finite element mesh
Time evolution of velocity magnitude
Computation domain and porosity value distribution
Time evolution of magnitude velocity
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
 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
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
 $N$ $Er1$ $Er2$ 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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