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February  2021, 20(2): 533-545. doi: 10.3934/cpaa.2020279

Homogenization and singular perturbation in porous media

Eduard Marušić-Paloka , and  Igor Pažanin

Departmant of mathematics, University of Zagreb, Bijenička 30, 10000, Zagreb, Croatia

* Corresponding author

Received  April 2020 Revised  September 2020 Published  February 2021 Early access  December 2020

Fund Project: The authors of this work have been supported by the Croatian Science Foundation (grant: 2735 Asymptotic analysis of the boundary value problems in continuum mechanics - AsAn)

We study a Dirichlet problem in periodic porous medium depending on two small parameters, the hydraulic permeability of the porous inclusions $ \delta $ and the period $ \varepsilon $. We study the situation as $ \delta\to 0 \;, \; \varepsilon\to 0 $ and $ \varepsilon \to 0 \;, \; \delta\to 0 $ and prove that the two limits do not commute.

Keywords: Homogenization, porous medium, Laplace equation, singular perturbation.
Mathematics Subject Classification: Primary: 35B27, 76M50; Secondary: 35Q30.
Citation: Eduard Marušić-Paloka, Igor Pažanin. Homogenization and singular perturbation in porous media. Communications on Pure & Applied Analysis, 2021, 20 (2) : 533-545. doi: 10.3934/cpaa.2020279
References:
[1]

G. Allaire, Homogenization and two-scale convergence, SIAM J. Math. Anal., 23 (1992), 1482-1518.  doi: 10.1137/0523084.  Google Scholar

[2]

G. Allaire, Homogenization of the Navier-Stokes equations in open sets perforated with tiny holes. Ⅰ. Abstract framework, a volume distribution of holes, Arch. Ration. Mech. Anal., 113 (1990), 209-259.  doi: 10.1007/BF00375065.  Google Scholar

[3]

T. Arbogast, J. Jr. Douglas and U. Hornung, Derivation of the double porosity model of single phase flow via homogenization theory, SIAM J. Math. Anal., 21 (1990), 823-836.  doi: 10.1137/0521046.  Google Scholar

[4]

J. M. Arrieta and M. Pereira, Homogenization in a thin domain with an oscillatory boundary, J. Math. Pures Appl., 96 (2011), 29-57.  doi: 10.1016/j.matpur.2011.02.003.  Google Scholar

[5]

N. S. Bakhvalov and G. Panasenko, Homogenisation: Averaging Processes in Porous Media Periodic, Springer, 1989. doi: 10.1007/978-94-009-2247-1.  Google Scholar

[6]

F. A. Bornemann, Homogenization in Time of Singularly Perturbed Conservative Mechanical Systems, Springer, 1998. doi: 10.1007/BFb0092091.  Google Scholar

[7]

A. Bourgeat, E. Marušić-Paloka and A. Mikelić, Effective Fluid Flow in a Porous Medium Containing a Thin Fissure, Asymptotic Analysis, 11 (1995), 241-262.   Google Scholar

[8]

A. Bourgeat, S. Luckhaus and A. Mikelić, A rigorous result for a double porosity model of immiscible two-phase flows, SIAM J. Math. Anal., 27 (1996), 1520-1543.  doi: 10.1137/S0036141094276457.  Google Scholar

[9]

D. Caillerie, Homogénéisation des équations de la diffusion stationnaire dans les domaines cylindriques aplatis, RAIRO Analyse Numeérique, 15 (1981), 295-319.   Google Scholar

[10]

D. Caillerie, Thin elastic and perodic plates, Math. Methods Appl. Sci., 6 (1984), 159-191. doi: 10.1002/mma.1670060112.  Google Scholar

[11]

G. S. Chechkin, Averaging of boundary value problems with a singular perturbation of the boundary conditions, Mat. Sb., 186 (1993), 191-222.  doi: 10.1070/SM1994v079n01ABEH003608.  Google Scholar

[12]

D. Cioranescu and J. Saint Jean Paulin, Homogenization of Reticulated Structures, Springer, 1999. doi: 10.1007/978-1-4612-2158-6.  Google Scholar

[13]

D. Cioranescu, P. Donato and R. Zaki, The periodic unfolding method in perforated domains, Portugaliae Mathematica, 63 (2016), 467-496.   Google Scholar

[14]

A. Damlamian and M. Vogelius, Homogenization limits of diffusion equations in thin domains, Mod. Math. An. Num. M2AN, 22 (1988), 53-74.  doi: 10.1051/m2an/1988220100531.  Google Scholar

[15]

A. Damlamian and M. Vogelius, Homogenization limits of the eequations of elasticity in thin domains, SIAM J. Math. Anal., 18 (1987), 435-451.  doi: 10.1137/0518034.  Google Scholar

[16]

T. Fratrović and E. Marušić-Paloka, Nonlinear Brinkman-type law as a critical case in the polymer fluid filtration, Appl. Anal., 95 (2016), 562-583.  doi: 10.1080/00036811.2015.1022155.  Google Scholar

[17]

M. Jurak and Z. Tutek, A one-dimensional model of homogenized rod, Glas. Mat., 24 (1989), 271-290.   Google Scholar

[18]

R. Kohn and M. Vogelius, A new model for thin plates with rapidly varying thickness, Ⅱ a convergence proof, Quart. Appl. Math., 43 (1985), 1-22.  doi: 10.1090/qam/782253.  Google Scholar

[19]

S. Marušić, A note on permeability for a network of thin channels, Glas. Mat., 39 (2004), 339-346.  doi: 10.3336/gm.39.2.16.  Google Scholar

[20]

E. Marušić-Paloka and S. Marušić, Computation of the permeability tensor for the fluid flow through a periodic net of thin channels, Appl. Anal., 64 (1997), 27-37.  doi: 10.1080/00036819708840521.  Google Scholar

[21]

E. Marušić-Paloka, I. Pažanin and S. Marušić, Comparison between Darcy and Brinkman laws in a fracture, Appl. Math. Comput., 218 (2012), 7538-7545.  doi: 10.1016/j.amc.2012.01.021.  Google Scholar

[22]

G. A. Nguetseng, A General convergence result for a functional related to the theory of homogenization, SIAM J. Math. Anal., 20 (1989), 608-623.  doi: 10.1137/0520043.  Google Scholar

[23]

E. Sanchez-Palencia, Non-homogeneous Media and Vibration Theory, Springer, 1980.  Google Scholar

[24]

H. Kim and H. Shahgholian, Homogenization of a singular perturbation problem, J. Math. Sci., 243 (2019), 163-176.  doi: 10.1007/s10958-019-04472-x.  Google Scholar

[25]

P. Shi, A. Spagnuolo and S. Wright, Reiterated Homogenization and the Double-Porosity Model, Transport Porous Med., 59 (2005), 73-95.  doi: 10.1007/s11242-004-1121-3.  Google Scholar

show all references

References:
[1]

G. Allaire, Homogenization and two-scale convergence, SIAM J. Math. Anal., 23 (1992), 1482-1518.  doi: 10.1137/0523084.  Google Scholar

[2]

G. Allaire, Homogenization of the Navier-Stokes equations in open sets perforated with tiny holes. Ⅰ. Abstract framework, a volume distribution of holes, Arch. Ration. Mech. Anal., 113 (1990), 209-259.  doi: 10.1007/BF00375065.  Google Scholar

[3]

T. Arbogast, J. Jr. Douglas and U. Hornung, Derivation of the double porosity model of single phase flow via homogenization theory, SIAM J. Math. Anal., 21 (1990), 823-836.  doi: 10.1137/0521046.  Google Scholar

[4]

J. M. Arrieta and M. Pereira, Homogenization in a thin domain with an oscillatory boundary, J. Math. Pures Appl., 96 (2011), 29-57.  doi: 10.1016/j.matpur.2011.02.003.  Google Scholar

[5]

N. S. Bakhvalov and G. Panasenko, Homogenisation: Averaging Processes in Porous Media Periodic, Springer, 1989. doi: 10.1007/978-94-009-2247-1.  Google Scholar

[6]

F. A. Bornemann, Homogenization in Time of Singularly Perturbed Conservative Mechanical Systems, Springer, 1998. doi: 10.1007/BFb0092091.  Google Scholar

[7]

A. Bourgeat, E. Marušić-Paloka and A. Mikelić, Effective Fluid Flow in a Porous Medium Containing a Thin Fissure, Asymptotic Analysis, 11 (1995), 241-262.   Google Scholar

[8]

A. Bourgeat, S. Luckhaus and A. Mikelić, A rigorous result for a double porosity model of immiscible two-phase flows, SIAM J. Math. Anal., 27 (1996), 1520-1543.  doi: 10.1137/S0036141094276457.  Google Scholar

[9]

D. Caillerie, Homogénéisation des équations de la diffusion stationnaire dans les domaines cylindriques aplatis, RAIRO Analyse Numeérique, 15 (1981), 295-319.   Google Scholar

[10]

D. Caillerie, Thin elastic and perodic plates, Math. Methods Appl. Sci., 6 (1984), 159-191. doi: 10.1002/mma.1670060112.  Google Scholar

[11]

G. S. Chechkin, Averaging of boundary value problems with a singular perturbation of the boundary conditions, Mat. Sb., 186 (1993), 191-222.  doi: 10.1070/SM1994v079n01ABEH003608.  Google Scholar

[12]

D. Cioranescu and J. Saint Jean Paulin, Homogenization of Reticulated Structures, Springer, 1999. doi: 10.1007/978-1-4612-2158-6.  Google Scholar

[13]

D. Cioranescu, P. Donato and R. Zaki, The periodic unfolding method in perforated domains, Portugaliae Mathematica, 63 (2016), 467-496.   Google Scholar

[14]

A. Damlamian and M. Vogelius, Homogenization limits of diffusion equations in thin domains, Mod. Math. An. Num. M2AN, 22 (1988), 53-74.  doi: 10.1051/m2an/1988220100531.  Google Scholar

[15]

A. Damlamian and M. Vogelius, Homogenization limits of the eequations of elasticity in thin domains, SIAM J. Math. Anal., 18 (1987), 435-451.  doi: 10.1137/0518034.  Google Scholar

[16]

T. Fratrović and E. Marušić-Paloka, Nonlinear Brinkman-type law as a critical case in the polymer fluid filtration, Appl. Anal., 95 (2016), 562-583.  doi: 10.1080/00036811.2015.1022155.  Google Scholar

[17]

M. Jurak and Z. Tutek, A one-dimensional model of homogenized rod, Glas. Mat., 24 (1989), 271-290.   Google Scholar

[18]

R. Kohn and M. Vogelius, A new model for thin plates with rapidly varying thickness, Ⅱ a convergence proof, Quart. Appl. Math., 43 (1985), 1-22.  doi: 10.1090/qam/782253.  Google Scholar

[19]

S. Marušić, A note on permeability for a network of thin channels, Glas. Mat., 39 (2004), 339-346.  doi: 10.3336/gm.39.2.16.  Google Scholar

[20]

E. Marušić-Paloka and S. Marušić, Computation of the permeability tensor for the fluid flow through a periodic net of thin channels, Appl. Anal., 64 (1997), 27-37.  doi: 10.1080/00036819708840521.  Google Scholar

[21]

E. Marušić-Paloka, I. Pažanin and S. Marušić, Comparison between Darcy and Brinkman laws in a fracture, Appl. Math. Comput., 218 (2012), 7538-7545.  doi: 10.1016/j.amc.2012.01.021.  Google Scholar

[22]

G. A. Nguetseng, A General convergence result for a functional related to the theory of homogenization, SIAM J. Math. Anal., 20 (1989), 608-623.  doi: 10.1137/0520043.  Google Scholar

[23]

E. Sanchez-Palencia, Non-homogeneous Media and Vibration Theory, Springer, 1980.  Google Scholar

[24]

H. Kim and H. Shahgholian, Homogenization of a singular perturbation problem, J. Math. Sci., 243 (2019), 163-176.  doi: 10.1007/s10958-019-04472-x.  Google Scholar

[25]

P. Shi, A. Spagnuolo and S. Wright, Reiterated Homogenization and the Double-Porosity Model, Transport Porous Med., 59 (2005), 73-95.  doi: 10.1007/s11242-004-1121-3.  Google Scholar

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