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June  2019, 14(2): 289-316. doi: 10.3934/nhm.2019012

Newtonian fluid flow in a thin porous medium with non-homogeneous slip boundary conditions

María Anguiano 1, and  Francisco Javier Suárez-Grau 2,,

1. 

Departamento de Análisis Matemático, Facultad de Matemáticas, Universidad de Sevilla, P. O. Box 1160, 41080-Sevilla, Spain
2. 

Departamento de Ecuaciones Diferenciales y Análisis Numérico, Facultad de Matemáticas, Universidad de Sevilla, 41012-Sevilla, Spain

* Corresponding author: Francisco Javier Suárez-Grau

Received  April 2018 Revised  September 2018 Published  April 2019

Fund Project: María Anguiano is supported by Junta de Andalucía (Spain), Proyecto de Excelencia P12-FQM-2466. Francisco Javier Suárez-Grau is supported by Ministerio de Economía y Competitividad (Spain), Proyecto Excelencia MTM2014-53309-P.

We consider the Stokes system in a thin porous medium $ \Omega_\varepsilon $ of thickness $ \varepsilon $ which is perforated by periodically distributed solid cylinders of size $ \varepsilon $. On the boundary of the cylinders we prescribe non-homogeneous slip boundary conditions depending on a parameter $ \gamma $. The aim is to give the asymptotic behavior of the velocity and the pressure of the fluid as $ \varepsilon $ goes to zero. Using an adaptation of the unfolding method, we give, following the values of $ \gamma $, different limit systems.

Keywords: Homogenization, Stokes system, Darcy's law, thin porous medium, non-homogeneous slip boundary condition.
Mathematics Subject Classification: Primary: 76A20, 76M50, 35B27; Secondary: 76S05.
Citation: María Anguiano, Francisco Javier Suárez-Grau. Newtonian fluid flow in a thin porous medium with non-homogeneous slip boundary conditions. Networks and Heterogeneous Media, 2019, 14 (2) : 289-316. doi: 10.3934/nhm.2019012
References:
[1]

J. N. L. Albert and T. H. Epps, Self-assembly of block copolymer thin films, Materials Today, 13 (2010), 24-33.  doi: 10.1016/S1369-7021(10)70106-1.

[2]

G. Allaire, Homogenization of the Navier-Stokes equations with a slip boundary condition, Comm. Pure Appl. Math., 44 (1989), 605-642.  doi: 10.1002/cpa.3160440602.

[3]

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

[4]

M. Anguiano and F. J. Suárez-Grau, Homogenization of an incompressible non-Newtonian flow through a thin porous medium, Z. Angew. Math. Phys., 68 (2017), Art. 45, 25 pp. doi: 10.1007/s00033-017-0790-z.

[5]

T. Arbogast, J. 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.

[6]

A. Bourgeat, S. Luckhaus and A. Mikelić, Convergence of the homogenization process for a double-porosity model of immiscible two-phase flow, SIAM J. Math.Anal., 27 (1996), 1520-1543.  doi: 10.1137/S0036141094276457.

[7]

A. Capatina and H. Ene, Homogenisation of the Stokes problem with a pure non-homogeneous slip boundary condition by the periodic unfolding method, Euro. J. of Applied Mathematics, 22 (2011), 333-345.  doi: 10.1017/S0956792511000088.

[8]

D. Cioranescu and J. Saint Jean Paulin, Homogenization in open sets with holes, J. Math. Anal. Appl., 71 (1979), 590-607.  doi: 10.1016/0022-247X(79)90211-7.

[9]

D. Cioranescu and P. Donato, Homogénéisation du problème du Neumann non homogène dans des ouverts perforés, Asymptotic Analysis, 1 (1988), 115-138. 

[10]

D. Cioranescu and P. Donato, Exact internal controllability in perforated domains, J. Math. Pures Appl., 68 (1989), 185-213. 

[11]

D. Cioranescu and J. Saint Jean Paulin, Truss structures: Fourier conditions and eigenvalue problems, in Boundary Control and Boundary Variation (Ed. J.P. Zolezio), Springer-Verlag, 178 (1992), 125-141. doi: 10.1007/BFb0006691.

[12]

D. Cioranescu, P. Donato and H. Ene, Homogenization of the Stokes problem with non homogeneous slip boundary conditions, Math. Meth. Appl. Sci., 19 (1996), 857-881.  doi: 10.1002/(SICI)1099-1476(19960725)19:11<857::AID-MMA798>3.0.CO;2-D.

[13]

D. Cioranescu, A. Damlamian and G. Griso, Periodic unfolding and homogenization, C.R. Acad. Sci. Paris Ser. I, 335 (2002), 99-104.  doi: 10.1016/S1631-073X(02)02429-9.

[14]

D. Cioranescu, P. Donato and R. Zaki, Periodic unfolding and Robin problems in perforated domains, C. R. Math., 342 (2006), 469-474.  doi: 10.1016/j.crma.2006.01.028.

[15]

D. Cioranescu, P. Donato and R. Zaki, The periodic unfolding method in perforated domains, Portugaliae Mathematica, 63 (2006), 467-496. 

[16]

D. Cioranescu, A. Damlamian, P. Donato, G. Griso and R. Zaki, The periodic unfolding method in domains with holes, SIAM J. of Math. Anal., 44 (2012), 718-760.  doi: 10.1137/100817942.

[17]

C. Conca, On the application of the homogenization theory to a class of problems arising in fluid mechanics, J. Math. Pures Appl., 64 (1985), 31-75. 

[18]

P. Donato and Z. Yang, The period unfolding method for the wave equations in domains with holes, Advances in Mathematical Sciences and Applications, 22 (2012), 521-551. 

[19]

P. Donato and Z. Yang, The periodic unfolding method for the heat equation in perforated domains, Science China Mathematics, 59 (2016), 891-906.  doi: 10.1007/s11425-015-5103-4.

[20]

H. Ene and E. Sanchez-Palencia, Equation et phénomenes de surface pour l'écoulement dans un modèle de milieux poreux, J. Mech., 14 (1975), 73-108. 

[21]

R. A. Farrell, T. G. Fitzgerald, D. Borah, J. D. Holmes and M. A. Morris, Chemical Interactions and Their Role in the Microphase Separation of Block Copolymer Thin Films, Int. J. of Molecular Sci., 10 (2009), 3671-3712.  doi: 10.3390/ijms10093671.

[22]

V. Frishfelds, T. S. Lundström and A. Jakovics, Lattice gas analysis of liquid front in non-crimp fabrics, Transp. Porous Med., 84 (2011), 75-93.  doi: 10.1007/s11242-009-9485-z.

[23]

W. Jeon and C. B. Shin, Design and simulation of passive mixing in microfluidic systems with geometric variations, Chem. Eng. J., 152 (2009), 575-582.  doi: 10.1016/j.cej.2009.05.035.

[24]

J.-L. Lions and E. Magenes, Problèmes aux Limites non Homogènes et Applications, Dunod, Paris, 1968.

[25]

T. S. Lundström, S. Toll and J. M. Håkanson, Measurements of the permeability tensor of compressed fibre beds, Transp. Porous Med., 47 (2002), 363-380. 

[26]

S. Marušić and E. Marušić-Paloka, Two-scale convergence for thin domain and its applications to some lower-dimensional model in fluid mechanics, Asymptot. Anal., 23 (2000), 23-57. 

[27]

J. Nečas, Les méthodes Directes en Théorie des Équations Elliptiques, Masson, Paris, 1967.

[28]

G. 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.

[29]

M. Nordlund and T. S. Lundström, Effect of multi-scale porosity in local permeability modelling of non-crimp fabrics, Transp. Porous Med., 73 (2008), 109-124.  doi: 10.1007/s11242-007-9161-0.

[30]

C. Park, J. Yoon and E. L. Thomas, Enabling nanotechnology with self assembled block copolymer patterns, Polymer, 44 (2003), 6725-6760.  doi: 10.1016/j.polymer.2003.08.011.

[31]

F. F. Reuss, Notice sur un Nouvel Effet de L'electricité Galvanique, Mémoire Soc. Sup. Imp. de Moscou, 1809.

[32]

E. Sanchez-Palencia, Non-Homogeneous Media and Vibration Theory, Lecture Notes in Physics, 127. Springer-Verlag, Berlin-New York, 1980.

[33]

F. Singh, B. Stoeber and S.I. Green, Micro-PIV measurement of flow upstream of papermaking forming fabrics, Transp. Porous Med., 107 (2015), 435-448.  doi: 10.1007/s11242-014-0447-8.

[34]

H. Tan and K. M. Pillai, Multiscale modeling of unsaturated flow in dual-scale fiber preforms of liquid composite molding I: Isothermal flows, Compos. Part A Appl. Sci. Manuf., 43 (2012), 1-13. 

[35]

L. Tartar, Incompressible fluid flow in a porous medium convergence of the homogenization process., in Appendix to Lecture Notes in Physics, 127 (1980).

[36]

M. Vanninathan, Homogenization of eigenvalues problems in perforated domains, Proc. Indian Acad. of Science, 90 (1981), 239-271.  doi: 10.1007/BF02838079.

[37]

R. Zaki, Homogenization of a Stokes problem in a porous medium by the periodic unfolding method, Asymptotic Analysis, 79 (2012), 229-250. 

show all references

References:
[1]

J. N. L. Albert and T. H. Epps, Self-assembly of block copolymer thin films, Materials Today, 13 (2010), 24-33.  doi: 10.1016/S1369-7021(10)70106-1.

[2]

G. Allaire, Homogenization of the Navier-Stokes equations with a slip boundary condition, Comm. Pure Appl. Math., 44 (1989), 605-642.  doi: 10.1002/cpa.3160440602.

[3]

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

[4]

M. Anguiano and F. J. Suárez-Grau, Homogenization of an incompressible non-Newtonian flow through a thin porous medium, Z. Angew. Math. Phys., 68 (2017), Art. 45, 25 pp. doi: 10.1007/s00033-017-0790-z.

[5]

T. Arbogast, J. 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.

[6]

A. Bourgeat, S. Luckhaus and A. Mikelić, Convergence of the homogenization process for a double-porosity model of immiscible two-phase flow, SIAM J. Math.Anal., 27 (1996), 1520-1543.  doi: 10.1137/S0036141094276457.

[7]

A. Capatina and H. Ene, Homogenisation of the Stokes problem with a pure non-homogeneous slip boundary condition by the periodic unfolding method, Euro. J. of Applied Mathematics, 22 (2011), 333-345.  doi: 10.1017/S0956792511000088.

[8]

D. Cioranescu and J. Saint Jean Paulin, Homogenization in open sets with holes, J. Math. Anal. Appl., 71 (1979), 590-607.  doi: 10.1016/0022-247X(79)90211-7.

[9]

D. Cioranescu and P. Donato, Homogénéisation du problème du Neumann non homogène dans des ouverts perforés, Asymptotic Analysis, 1 (1988), 115-138. 

[10]

D. Cioranescu and P. Donato, Exact internal controllability in perforated domains, J. Math. Pures Appl., 68 (1989), 185-213. 

[11]

D. Cioranescu and J. Saint Jean Paulin, Truss structures: Fourier conditions and eigenvalue problems, in Boundary Control and Boundary Variation (Ed. J.P. Zolezio), Springer-Verlag, 178 (1992), 125-141. doi: 10.1007/BFb0006691.

[12]

D. Cioranescu, P. Donato and H. Ene, Homogenization of the Stokes problem with non homogeneous slip boundary conditions, Math. Meth. Appl. Sci., 19 (1996), 857-881.  doi: 10.1002/(SICI)1099-1476(19960725)19:11<857::AID-MMA798>3.0.CO;2-D.

[13]

D. Cioranescu, A. Damlamian and G. Griso, Periodic unfolding and homogenization, C.R. Acad. Sci. Paris Ser. I, 335 (2002), 99-104.  doi: 10.1016/S1631-073X(02)02429-9.

[14]

D. Cioranescu, P. Donato and R. Zaki, Periodic unfolding and Robin problems in perforated domains, C. R. Math., 342 (2006), 469-474.  doi: 10.1016/j.crma.2006.01.028.

[15]

D. Cioranescu, P. Donato and R. Zaki, The periodic unfolding method in perforated domains, Portugaliae Mathematica, 63 (2006), 467-496. 

[16]

D. Cioranescu, A. Damlamian, P. Donato, G. Griso and R. Zaki, The periodic unfolding method in domains with holes, SIAM J. of Math. Anal., 44 (2012), 718-760.  doi: 10.1137/100817942.

[17]

C. Conca, On the application of the homogenization theory to a class of problems arising in fluid mechanics, J. Math. Pures Appl., 64 (1985), 31-75. 

[18]

P. Donato and Z. Yang, The period unfolding method for the wave equations in domains with holes, Advances in Mathematical Sciences and Applications, 22 (2012), 521-551. 

[19]

P. Donato and Z. Yang, The periodic unfolding method for the heat equation in perforated domains, Science China Mathematics, 59 (2016), 891-906.  doi: 10.1007/s11425-015-5103-4.

[20]

H. Ene and E. Sanchez-Palencia, Equation et phénomenes de surface pour l'écoulement dans un modèle de milieux poreux, J. Mech., 14 (1975), 73-108. 

[21]

R. A. Farrell, T. G. Fitzgerald, D. Borah, J. D. Holmes and M. A. Morris, Chemical Interactions and Their Role in the Microphase Separation of Block Copolymer Thin Films, Int. J. of Molecular Sci., 10 (2009), 3671-3712.  doi: 10.3390/ijms10093671.

[22]

V. Frishfelds, T. S. Lundström and A. Jakovics, Lattice gas analysis of liquid front in non-crimp fabrics, Transp. Porous Med., 84 (2011), 75-93.  doi: 10.1007/s11242-009-9485-z.

[23]

W. Jeon and C. B. Shin, Design and simulation of passive mixing in microfluidic systems with geometric variations, Chem. Eng. J., 152 (2009), 575-582.  doi: 10.1016/j.cej.2009.05.035.

[24]

J.-L. Lions and E. Magenes, Problèmes aux Limites non Homogènes et Applications, Dunod, Paris, 1968.

[25]

T. S. Lundström, S. Toll and J. M. Håkanson, Measurements of the permeability tensor of compressed fibre beds, Transp. Porous Med., 47 (2002), 363-380. 

[26]

S. Marušić and E. Marušić-Paloka, Two-scale convergence for thin domain and its applications to some lower-dimensional model in fluid mechanics, Asymptot. Anal., 23 (2000), 23-57. 

[27]

J. Nečas, Les méthodes Directes en Théorie des Équations Elliptiques, Masson, Paris, 1967.

[28]

G. 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.

[29]

M. Nordlund and T. S. Lundström, Effect of multi-scale porosity in local permeability modelling of non-crimp fabrics, Transp. Porous Med., 73 (2008), 109-124.  doi: 10.1007/s11242-007-9161-0.

[30]

C. Park, J. Yoon and E. L. Thomas, Enabling nanotechnology with self assembled block copolymer patterns, Polymer, 44 (2003), 6725-6760.  doi: 10.1016/j.polymer.2003.08.011.

[31]

F. F. Reuss, Notice sur un Nouvel Effet de L'electricité Galvanique, Mémoire Soc. Sup. Imp. de Moscou, 1809.

[32]

E. Sanchez-Palencia, Non-Homogeneous Media and Vibration Theory, Lecture Notes in Physics, 127. Springer-Verlag, Berlin-New York, 1980.

[33]

F. Singh, B. Stoeber and S.I. Green, Micro-PIV measurement of flow upstream of papermaking forming fabrics, Transp. Porous Med., 107 (2015), 435-448.  doi: 10.1007/s11242-014-0447-8.

[34]

H. Tan and K. M. Pillai, Multiscale modeling of unsaturated flow in dual-scale fiber preforms of liquid composite molding I: Isothermal flows, Compos. Part A Appl. Sci. Manuf., 43 (2012), 1-13. 

[35]

L. Tartar, Incompressible fluid flow in a porous medium convergence of the homogenization process., in Appendix to Lecture Notes in Physics, 127 (1980).

[36]

M. Vanninathan, Homogenization of eigenvalues problems in perforated domains, Proc. Indian Acad. of Science, 90 (1981), 239-271.  doi: 10.1007/BF02838079.

[37]

R. Zaki, Homogenization of a Stokes problem in a porous medium by the periodic unfolding method, Asymptotic Analysis, 79 (2012), 229-250. 

Figure 1.  Views of a periodic cell in 2D (left) and 3D (right)
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Figure 2.  View of $ \omega_\varepsilon $
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Figure 3.  Views of the domain $ \Omega_\varepsilon $ (left) and $ \Lambda_\varepsilon $ (right)
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