The Transition Between the Navier–Stokes Equations to the Darcy Equation in a Thin Porous Medium

Abstract

We consider a Newtonian flow in a thin porous medium \(\Omega _{\varepsilon }\) of thickness \(\varepsilon \) which is perforated by periodically distributed solid cylinders of size \(a_\varepsilon \). Generalizing (Anguiano and Suárez-Grau, ZAMP J Appl Math Phys 68:45, 2017), the fluid is described by the 3D incompressible Navier–Stokes system where the external force takes values in the space \(H^{-1}\), and the porous medium considered has one of the most commonly used distribution of cylinders: hexagonal distribution. By an adaptation of the unfolding method, three different Darcy’s laws are rigorously derived from this model depending on the magnitude \(a_{\varepsilon }\) with respect to \(\varepsilon \).

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

References

  1. 1.

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

    MathSciNet  Article  MATH  Google Scholar 

  2. 2.

    Anguiano, M., Suárez-Grau, F.J.: Homogenization of an incompressible non-Newtonian flow through a thin porous medium. ZAMP J. Appl. Math. Phys. 68, 45 (2017)

    MathSciNet  MATH  Google Scholar 

  3. 3.

    Arbogast, T., Douglas, J.R., Hornung, U.: Derivation of the double porosity model of single phase flow via homogenization theory. SIAM J. Math. Anal. 21, 823–836 (1990)

    MathSciNet  Article  MATH  Google Scholar 

  4. 4.

    Cioranescu, D., Damlamian, A., Griso, G.: Periodic unfolding and homogenization. C.R. Acad. Sci. Paris Ser. I. 335, 99–104 (2002)

    MathSciNet  Article  MATH  Google Scholar 

  5. 5.

    Fabricius, J., Hellström, J.G.I., Lundström, T.S., Miroshnikova, E., Wall, P.: Darcy’s law for flow in a periodic thin porous medium confined between two parallel plates. Transp. Porous Med. 115, 473–493 (2016)

    MathSciNet  Article  Google Scholar 

  6. 6.

    Lions, J.L.: Quelques méthodes de résolution des problèmes aux limites non linéaires. Dunod, Gauthier-Villars, Paris (1969)

    Google Scholar 

  7. 7.

    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 

  8. 8.

    Snyder, W.A., Qi, H., Sander, W.: Coordinate system for hexagonal pixels. In: Proc. SPIE 3661, Medical Imaging 1999: Image Processing (1999)

  9. 9.

    Tartar, L.: Incompressible Fluid Flow in a Porous Medium Convergence of the Homogenization Process. Appendix to Lecture Notes in Physics, vol. 127. Springer, Berlin (1980)

    Google Scholar 

  10. 10.

    Temam, R.: Navier–Stokes Equations. North Holland, Amsterdam (1977)

    Google Scholar 

  11. 11.

    Zhengan, Y., Hongxing, Z.: Homogenization of the Navier–Stokes flow in porous medium with thin film. Acta Math. Sci. 28, 863–974 (2008)

    MathSciNet  Article  MATH  Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Francisco Javier Suárez-Grau.

Additional information

María Anguiano has been supported by Junta de Andalucía (Spain), Proyecto de Excelencia P12-FQM-2466. 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. The Transition Between the Navier–Stokes Equations to the Darcy Equation in a Thin Porous Medium. Mediterr. J. Math. 15, 45 (2018). https://doi.org/10.1007/s00009-018-1086-z

Download citation

Mathematics Subject Classification

  • 76A20
  • 76M50
  • 35B27
  • 35Q30

Keywords

  • Homogenization
  • Navier–Stokes equations
  • Darcy’s law
  • porous medium
  • thin-film fluids