# American Institute of Mathematical Sciences

## 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, doi: 10.3934/dcdss.2020234
##### References:

show all references

##### 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
 [1] Yingwen Guo, Yinnian He. Fully discrete finite element method based on second-order Crank-Nicolson/Adams-Bashforth scheme for the equations of motion of Oldroyd fluids of order one. Discrete & Continuous Dynamical Systems - B, 2015, 20 (8) : 2583-2609. doi: 10.3934/dcdsb.2015.20.2583 [2] Franck Boyer, Pierre Fabrie. Outflow boundary conditions for the incompressible non-homogeneous Navier-Stokes equations. Discrete & Continuous Dynamical Systems - B, 2007, 7 (2) : 219-250. doi: 10.3934/dcdsb.2007.7.219 [3] Wojciech M. Zajączkowski. Long time existence of regular solutions to non-homogeneous Navier-Stokes equations. Discrete & Continuous Dynamical Systems - S, 2013, 6 (5) : 1427-1455. doi: 10.3934/dcdss.2013.6.1427 [4] Yinnian He, R. M.M. Mattheij. Reformed post-processing Galerkin method for the Navier-Stokes equations. Discrete & Continuous Dynamical Systems - B, 2007, 8 (2) : 369-387. doi: 10.3934/dcdsb.2007.8.369 [5] Kaitai Li, Yanren Hou. Fourier nonlinear Galerkin method for Navier-Stokes equations. Discrete & Continuous Dynamical Systems - A, 1996, 2 (4) : 497-524. doi: 10.3934/dcds.1996.2.497 [6] Hi Jun Choe, Do Wan Kim, Yongsik Kim. Meshfree method for the non-stationary incompressible Navier-Stokes equations. Discrete & Continuous Dynamical Systems - B, 2006, 6 (1) : 17-39. doi: 10.3934/dcdsb.2006.6.17 [7] Evrad M. D. Ngom, Abdou Sène, Daniel Y. Le Roux. Boundary stabilization of the Navier-Stokes equations with feedback controller via a Galerkin method. Evolution Equations & Control Theory, 2014, 3 (1) : 147-166. doi: 10.3934/eect.2014.3.147 [8] Ana Bela Cruzeiro. Navier-Stokes and stochastic Navier-Stokes equations via Lagrange multipliers. Journal of Geometric Mechanics, 2019, 11 (4) : 553-560. doi: 10.3934/jgm.2019027 [9] Michele Coti Zelati. Remarks on the approximation of the Navier-Stokes equations via the implicit Euler scheme. Communications on Pure & Applied Analysis, 2013, 12 (6) : 2829-2838. doi: 10.3934/cpaa.2013.12.2829 [10] Hi Jun Choe, Hyea Hyun Kim, Do Wan Kim, Yongsik Kim. Meshless method for the stationary incompressible Navier-Stokes equations. Discrete & Continuous Dynamical Systems - B, 2001, 1 (4) : 495-526. doi: 10.3934/dcdsb.2001.1.495 [11] Takayuki Kubo, Ranmaru Matsui. On pressure stabilization method for nonstationary Navier-Stokes equations. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2283-2307. doi: 10.3934/cpaa.2018109 [12] Zhendong Luo. A reduced-order SMFVE extrapolation algorithm based on POD technique and CN method for the non-stationary Navier-Stokes equations. Discrete & Continuous Dynamical Systems - B, 2015, 20 (4) : 1189-1212. doi: 10.3934/dcdsb.2015.20.1189 [13] Boris Haspot, Ewelina Zatorska. From the highly compressible Navier-Stokes equations to the porous medium equation -- rate of convergence. Discrete & Continuous Dynamical Systems - A, 2016, 36 (6) : 3107-3123. doi: 10.3934/dcds.2016.36.3107 [14] Corentin Audiard. On the non-homogeneous boundary value problem for Schrödinger equations. Discrete & Continuous Dynamical Systems - A, 2013, 33 (9) : 3861-3884. doi: 10.3934/dcds.2013.33.3861 [15] María Anguiano, Francisco Javier Suárez-Grau. Newtonian fluid flow in a thin porous medium with non-homogeneous slip boundary conditions. Networks & Heterogeneous Media, 2019, 14 (2) : 289-316. doi: 10.3934/nhm.2019012 [16] Li Li, Yanyan Li, Xukai Yan. Homogeneous solutions of stationary Navier-Stokes equations with isolated singularities on the unit sphere. Ⅲ. Two singularities. Discrete & Continuous Dynamical Systems - A, 2019, 39 (12) : 7163-7211. doi: 10.3934/dcds.2019300 [17] Pavel I. Plotnikov, Jan Sokolowski. Compressible Navier-Stokes equations. Conference Publications, 2009, 2009 (Special) : 602-611. doi: 10.3934/proc.2009.2009.602 [18] Jan W. Cholewa, Tomasz Dlotko. Fractional Navier-Stokes equations. Discrete & Continuous Dynamical Systems - B, 2018, 23 (8) : 2967-2988. doi: 10.3934/dcdsb.2017149 [19] Yinnian He, Kaitai Li. Nonlinear Galerkin approximation of the two dimensional exterior Navier-Stokes problem. Discrete & Continuous Dynamical Systems - A, 1996, 2 (4) : 467-482. doi: 10.3934/dcds.1996.2.467 [20] Jian Su, Yinnian He. The almost unconditional convergence of the Euler implicit/explicit scheme for the three dimensional nonstationary Navier-Stokes equations. Discrete & Continuous Dynamical Systems - B, 2017, 22 (9) : 3421-3438. doi: 10.3934/dcdsb.2017173

2019 Impact Factor: 1.233