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Homogenization of a non-stationary non-Newtonian flow in a porous medium containing a thin fissure

Published online by Cambridge University Press:  05 February 2018

MARÍA ANGUIANO
MARÍA ANGUIANO*
Affiliation:
Departamento de Análisis Matemático, Facultad de Matemáticas, Universidad de Sevilla, P. O. Box 1160, 41080-Sevilla, Spain email: anguiano@us.es
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Abstract

We consider a non-stationary incompressible non-Newtonian Stokes system in a porous medium with characteristic size of the pores ϵ and containing a thin fissure of width ηϵ. The viscosity is supposed to obey the power law with flow index $\frac{5}{3}\leq q\leq 2$. The limit when size of the pores tends to zero gives the homogenized behaviour of the flow. We obtain three different models depending on the magnitude ηϵ with respect to ϵ: if ηϵ ≪ $\varepsilon^{q\over 2q-1}$ the homogenized fluid flow is governed by a time-dependent non-linear Darcy law, while if ηϵ ≫ $\varepsilon^{q\over 2q-1}$ is governed by a time-dependent non-linear Reynolds problem. In the critical case, ηϵ ≈ $\varepsilon^{q\over 2q-1}$, the flow is described by a time-dependent non-linear Darcy law coupled with a time-dependent non-linear Reynolds problem.

Keywords

Non-Newtonian flowNon-stationary Stokes equationDarcy–Reynolds equationporous mediumfissure
Type
Papers
Information
European Journal of Applied Mathematics , Volume 30 , Issue 2 , April 2019 , pp. 248 - 277
Copyright
Copyright © Cambridge University Press 2018 

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References

[1] Allaire, G. (1989) Homogenization of the Stokes flow in a connected porous medium. Asymptot. Anal. 2, 203–222.Google Scholar
[2] Allaire, G. (1992) Homogenization and two-scale convergence. SIAM J. Math. Anal. 23, 1482–1518.Google Scholar
[3] Bourgeat, A., ElAmri, H. & Tapiero, R. (1991) Existence d'une taille critique pour une fissure dans un milieu poreux. Second Colloque Franco Chilien de Mathematiques Appliquées, Cepadués Edts, Tolouse, 67–80.Google Scholar
[4] Bourgeat, A. & Tapiero, R. (1993) Homogenization in a perforated domain including a thin full interlayer. Int. Ser. Num. Math. 114, 25–36.Google Scholar
[5] Bourgeat, A., Marušic-Paloka, E. & Mikelić, A. (1995) Effective fluid flow in a porous medium containing a thin fissure. Asymptot. Anal. 11, 241–262.Google Scholar
[6] Bourgeat, A. & Mikelić, A. (1996) Homogenization of a polymer flow through a porous medium. Nonlinear Anal. 26, 1221–1253.Google Scholar
[7] Bourgeat, A., Gipouloux, O. & Marušić-Paloka, E. (2003) Filtration law for polymer flow through porous media. Multiscale Model. Simul. 1, 432–457.Google Scholar
[8] Clopeau, T. & Mikelić, A. (1998) On the non-stationary quasi-Newtonian flow through a thin slab. In: Navier–Stokes Equations: Theory and Numerical Methods, Pitman Res. Notes Math. Ser., 288, Longman, Harlow.Google Scholar
[9] Cloud, J. E. & Clark, P. E. (1985) Alternatives to the power-law fluid model for crosslinked fluids. Soc. Pet. Eng. J. 935-942.Google Scholar
[10] Galdi, G. P. 1994 An introduction to the mathematical theory of the Navier–Stokes equations, Springer-Verlag.Google Scholar
[11] Lions, J. L. 1969 Quelques méthodes de résolution des problèmes aux limites non linéaires, Dunod, Gauthier-Villars, Paris.Google Scholar
[12] Mikelić, A. & Tapiéro, R. (1995) Mathematical derivation of the power law describing polymer flow through a thin slab. RAIRO-Model. Math. Anal. Num. 29, 3–21.Google Scholar
[13] Nguestseng, G. (1989) A general convergence result for a functional related to the theory of homogenization. SIAM J. Math. Anal. 20, 608–623.Google Scholar
[14] Panasenko, G. P. (1981) Higher order asymptotics of solutions of problems on the contact of periodic structures. Math. U.S.S.R. Sbornik 38, 465–494.Google Scholar
[15] Pham Huy, H. & Sanchez-Palencia, E. (1974) Phénomènes de transmission à travers des couches minces de conductivité élevée. J. Math. Anal. Appl. 47, 284–309.Google Scholar
[16] Shah, S. N. (1982) Propant settling correlations for non-Newtonian fluids under static and dynamic conditions. Soc. Pet. Eng. J., 164–170.Google Scholar
[17] Tartar, L. 1980 Incompressible fluid flow in a porous medium convergence of the homogenization process. Appendix to Lecture Notes in Physics, 127, Springer-Velag, Berlin.Google Scholar
[18] Temam, R. 1983 Navier-Stokes equations and nonlinear functional analysis. CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 41, Society for Industrial and Applied Mathematics (SIAM), Philadelphia.Google Scholar
[19] Zhao, H. & Yao, Z. (2008) Homogenization of a non-stationary Stokes flow in porous medium including a layer. J. Math. Anal. Appl. 342, 108–124.Google Scholar
[20] Zhao, H. & Yao, Z. (2012) Effective models of the Navier–Stokes flow in porous media with a thin fissure. J. Math. Anal. Appl. 387, 542–555.Google Scholar