Skip to main content

Derivation of a coupled Darcy–Reynolds equation for a fluid flow in a thin porous medium including a fissure

Zeitschrift für angewandte Mathematik und Physik volume 68, Article number: 52 (2017) Cite this article

Abstract

We study the asymptotic behavior of a fluid flow in a thin porous medium of thickness \(\varepsilon \), which is characteristic size of the pores \(\varepsilon \) and contains a fissure of width \(\eta _\varepsilon \). We consider the limit when the size of the pores tends to zero, and we find a critical size \(\eta _\varepsilon \approx \varepsilon ^{2\over 3}\) in which the flow is described by a 2D Darcy law coupled with a 1D Reynolds problem. We also discuss the other cases.

This is a preview of subscription content, access via your institution.

Access options

Buy single article

Instant access to the full article PDF.

USD 39.95

Price includes VAT (USA)
Tax calculation will be finalised during checkout.

References

  1. Allaire, G.: Homogenization of the Stokes flow in a connected porous medium. Asymptot. Anal. 2, 203–222 (1989)

    MathSciNet  MATH  Google Scholar 

  2. Allaire, G.: Homogenization and two-scale convergence. SIAM J. Math. Anal. 23, 1482–1518 (1992)

    MathSciNet  Article  MATH  Google Scholar 

  3. Anguiano, M.: Darcy’s laws for non-stationary viscous fluid flow in a thin porous medium. Math. Methods Appl. Sci. (2016). doi:10.1002/mma.4204

    Google Scholar 

  4. Bourgeat, A., ElAmri, H., Tapiero, R.: Existence d’une taille critique pour une fissure dans un milieu poreux. Second Colloque Franco Chilien de Mathematiques Appliquées, Cepadués Edts, Tolouse, pp. 67–80 (1991)

  5. Bourgeat, A., Tapiero, R.: Homogenization in a perforated domain including a thin full interlayer. Int. Ser. Numer. Math. 114, 25–36 (1993)

    MathSciNet  MATH  Google Scholar 

  6. Bourgeat, A., Marušic-Paloka, E., Mikelić, A.: Effective fluid flow in a porous medium containing a thin fissure. Asymptot. Anal. 11, 241–262 (1995)

    MathSciNet  MATH  Google Scholar 

  7. Bourgeat, A., Mikelić, A.: Homogenization of a polymer flow through a porous medium. Nonlinear Anal. 26, 1221–1253 (1996)

    MathSciNet  Article  MATH  Google Scholar 

  8. Ciarlet, P.G., Ledret, H., Nzwenga, R.: Modélisation de la jonction entre un corps élastique tridimensionnel et une plaque. C. R. Acad. Sci. Paris Ser. I(305), 55–58 (1987)

    Google Scholar 

  9. Nguetseng, G.: A general convergence result for a functional related to the theory of homogenization. SIAM J. Math. Anal. 20, 608–623 (1989)

    MathSciNet  Article  MATH  Google Scholar 

  10. Panasenko, G.P.: Higher order asymptotics of solutions of problems on the contact of periodic structures. Math. U.S.S.R. Sbornik 38, 465–494 (1981)

    Article  MATH  Google Scholar 

  11. Sanchez-Palencia, E.: Non-homogeneous media and vibration theory. In: Ehlers, J., Hepp, K., Kippenhahn, R., Weidenmüller, H.A., Zittartz, J. (eds.) Lecture Notes in Physics. Springer, Berlin/Heidelberg/New York (1980)

  12. Tartar, L.: Incompressible fluid flow in a porous medium convergence of the homogenization process. In: Ehlers, J., Hepp, K., Kippenhahn, R., Weidenmüller, H.A., Zittartz, J. (eds.) Appendix to Lecture Notes in Physics, vol. 127. Springer, Berlin (1980)

  13. Temam, R.: Navier–Stokes equations and nonlinear functional analysis. In: CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 41. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (1983)

  14. Zhao, H., Yao, Z.: Effective models of the Navier–Stokes flow in porous media with a thin fissure. J. Math. Anal. Appl. 387, 542–555 (2012)

    MathSciNet  Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

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

    María Anguiano

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

    Francisco Javier Suárez-Grau

Authors
  1. María Anguiano
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Francisco Javier Suárez-Grau
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to María Anguiano.

Additional information

María Anguiano has been supported by Junta de Andalucía (Spain), Proyecto de Excelencia P12-FQM-2466, and in part by European Commission, Excellent Science-European Research Council (ERC) H2020-EU.1.1.-639227. Francisco Javier Suárez-Grau has been supported by Ministerio de Economía y Competitividad (Spain), Proyecto Excelencia MTM2014-53309-P.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Anguiano, M., Suárez-Grau, F.J. Derivation of a coupled Darcy–Reynolds equation for a fluid flow in a thin porous medium including a fissure. Z. Angew. Math. Phys. 68, 52 (2017). https://doi.org/10.1007/s00033-017-0797-5

Download citation

  • Received:

  • Revised:

  • Published:

  • DOI: https://doi.org/10.1007/s00033-017-0797-5

Mathematics Subject Classification

  • 75A05
  • 76A20
  • 76M50
  • 35B27

Keywords

  • Stokes equation
  • Darcy’s law
  • Reynolds equation
  • Thin porous medium
  • Fissure