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


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