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

  • * Corresponding author: Imam Wijaya

    * Corresponding author: Imam Wijaya 
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  • 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.

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


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  • Figure 1.  Representative elementary volume (REV)

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

    Figure 3.  The boundary conditions and the finite element mesh

    Figure 4.  Time evolution of velocity magnitude

    Figure 5.  Computation domain and porosity value distribution

    Figure 6.  Time evolution of magnitude velocity

    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
     | Show Table
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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
     | Show Table
    DownLoad: CSV
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