Skip to main content
Log in

Lower-Dimensional Nonlinear Brinkman’s Law for Non-Newtonian Flows in a Thin Porous Medium

  • Published:
Mediterranean Journal of Mathematics Aims and scope Submit manuscript


In this paper, we study the stationary incompressible power law fluid flow in a thin porous medium. The media under consideration is a bounded perforated 3D domain confined between two parallel plates, where the distance between the plates is very small. The perforation consists in an array solid cylinders, which connect the plates in perpendicular direction, distributed periodically with diameters of small size compared to the period. For a specific choice of the thickness of the domain, we found that the homogenization of the power law Stokes system results a lower-dimensional nonlinear Brinkman type law.

This is a preview of subscription content, log in via an institution to check access.

Access this article

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

Instant access to the full article PDF.

Fig. 1
Fig. 2
Fig. 3

Similar content being viewed by others


  1. Allaire, G.: Homogenization of the Navier–Stokes equations in open sets perforated with tiny holes I. Abstract framework, a volume distribution of holes. Arch. Ration. Mech. Anal. 113, 209–259 (1991)

  2. Allaire, G.: Homogenization of the Navier–Stokes equations in open sets perforated with tiny holes II: non-critical sizes of the holes for a volume distribution and a surface distribution of holes Arch. Ration. Mech. Anal. 113, 261–298 (1991)

    Article  Google Scholar 

  3. Allaire, G.: One-phase Newtonian flow. In: Homogenization and Porous Media, Interdisciplinary Applied Mathematics Series, vol. 6, pp. 77–94. Springer, New York (1997)

  4. Anguiano, M.: Darcy’s laws for non-stationary viscous fluid flow in a thin porous medium. Math. Methods Appl. Sci. 40(8), 2878–2895 (2017)

    Article  MathSciNet  Google Scholar 

  5. Anguiano, M.: On the non-stationary non-Newtonian flow through a thin porous medium. Z. Angew. Math. Mech. 97, 895–915 (2017)

    Article  MathSciNet  Google Scholar 

  6. Anguiano, M.: Derivation of a quasi-stationary coupled Darcy–Reynolds equation for incompressible viscous fluid flow through a thin porous medium with a fissure. Math. Methods Appl. Sci. 40, 4738–4757 (2017)

    MathSciNet  MATH  Google Scholar 

  7. Anguiano, M.: Homogenization of a non-stationary non-Newtonian flow in a porous medium containing a thin fissure. Eur. J. Appl. Math. 30(2), 248–277 (2019)

    Article  MathSciNet  Google Scholar 

  8. Anguiano, M, Bunoiu, R.: On the flow of a viscoplastic fluid in a thin periodic domain. In: Constanda, C., Harris, P. (eds.) Integral Methods in Science and Engineering, pp. 15–24. Birkhäuser, Cham (2019)

  9. Anguiano, M., Bunoiu, R.: Homogenization of Bingham flow in thin porous media. Netw. Heterog. Media 15(1), 87–110 (2020)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  11. 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)

    Article  MathSciNet  Google Scholar 

  12. 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)

    Article  MathSciNet  Google Scholar 

  13. Anguiano, M., Suárez-Grau, F.J.: Analysis of the effects of a fissure for a non-Newtonian fluid flow in a porous medium. Commun. Math. Sci. 16, 273–292 (2018)

    Article  MathSciNet  Google Scholar 

  14. Anguiano, M., Suárez-Grau, F.J.: Newtonian fluid flow in a thin porous medium with non-homogeneous slip boundary conditions. Netw. Heterog. Media 14, 289–316 (2019)

    Article  MathSciNet  Google Scholar 

  15. Anguiano, M., Suárez-Grau, F.J.: Nonlinear Reynolds equations for non-Newtonian thin-film fluid flows over a rough boundary. IMA J. Appl. Math. 84, 63–95 (2019)

    Article  MathSciNet  Google Scholar 

  16. Boukrouche, M., El Mir, R.: Asymptotic analysis of a non-Newtonian fluid in a thin domain with Tresca law. Nonlinear Anal. 59, 85–105 (2004)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  18. Bourgeat, A., Gipouloux, O., Maru\({\rm \check{s}}\)ić-Paloka, E.: Filtration law for polymer flow through porous media. Multiscale Model. Sim. 1, 432–457 (2003)

  19. Brillard, A.: Asymptotic analysis of incompressible and viscous fluid flow through porous media. Brinkman’s law via epi-convergence methods. In: Ann. Fac. Sci. Toulouse, Math. Toulouse: Université Paul Sabatier, Faculté des Sciences, vol. 8, pp. 225–252 (1986-1987)

  20. Cioranescu, D., Damlamian, A., Griso, G.: The Periodic Unfolding Method: Theory and Applications to Partial Differential Problems, Series in Contemporary Mathematics, vol. 3. Springer, Singapore (2018)

    Book  Google Scholar 

  21. Duvnjak, A.: Derivation of non-linear Reynolds-type problem for lubrication of a rotating shaft. Z. Angew. Math. Mech. 82, 317–333 (2002)

    Article  MathSciNet  Google Scholar 

  22. Fabricius, J., Hellstr\(\ddot{\rm o}\)m, J.G. I., Lundstr\(\ddot{\rm o}\)m, 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)

  23. Fabricius, J., Manjate, S., Wall, P.: On pressure-driven Hele-Shaw flow of power-law fluids. Appl. Anal. (2021).

  24. Fratrović, T., Maru\({\rm \check{s}}\)ić-Paloka, E.: Low-volume-fraction limit for polymer fluids. J. Math. Anal. Appl. 373, 399–409 (2011)

  25. Fratrović, T., Maru\({\rm \check{s}}\)ić-Paloka, E.: Nonlinear Brinkman-type law as a critical case in the polymer fluid filtration. Appl. Anal. 95, 562–583 (2016)

  26. Galdi, G.P.: An Introduction to the Mathematical Theory of the Navier–Stokes Equations. Springer, New York (1994)

    Book  Google Scholar 

  27. Jouybari, N.F., Lundstr\(\ddot{\rm o}\)m, T.S.: Investigation of Post-Darcy flow in thin porous media. Transp. Porous Media 138, 157–184 (2021)

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

    MATH  Google Scholar 

  29. Maru\({\rm \check{s}}\)ić-Paloka, E.: On the Stokes paradox for power-law fluids, Z. Angew. Math. Mech. 81, 31–36 (2001)

  30. Mikelić, A., Tapiero, R.: Mathematical derivation of power law describing polymer flow through a thin slab. RAIRO Model Math. Anal. Numer. 29, 3–21 (1995)

    Article  MathSciNet  Google Scholar 

  31. Mikelić, A.: Non-Newtonian flow. In: Homogenization and Porous Media, Interdisciplinary Applied Mathematics Series, vol. 6, pp. 45–68. Springer, New York (1997)

  32. Mikelić, A.: An introduction to the homogenization modeling of non-Newtonian and electrokinetic flows in porous media. In: Non-Newtonian Fluid Mechanics and Complex Flows. Lecture Notes in Mathematics, vol. 2212. Springer, New York (2018)

  33. Prat, M., Aga\({\rm \ddot{e}}\)sse, T.: Thin Porous Media. In: Vafai, K. (ed.) Handbook of Porous Media, pp. 89–112. CRC Press, Boca Raton (2015)

  34. Suárez-Grau, F.J.: Asymptotic behavior of a non-Newtonian flow in a thin domain with Navier law on a rough boundary. Nonlinear Anal. 117, 99–123 (2015)

    Article  MathSciNet  Google Scholar 

  35. Suárez-Grau, F.J.: Mathematical modeling of micropolar fluid flows through a thin porous medium. J. Eng. Math. 126, 7 (2021)

    Article  MathSciNet  Google Scholar 

  36. Yeghiazarian, L., Pillai, K., Rosati, R.: Thin porous media. Transp. Porous Med. 115, 407–410 (2016)

    Article  MathSciNet  Google Scholar 

  37. Zhengan, Y., Hongxing, Z.: Homogenization of a stationary Navier–Stokes flow in porous medium with thin film. Acta Math. Sci. 28, 963–974 (2008)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations


Corresponding author

Correspondence to Francisco J. Suárez-Grau.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Anguiano, M., Suárez-Grau, F.J. Lower-Dimensional Nonlinear Brinkman’s Law for Non-Newtonian Flows in a Thin Porous Medium. Mediterr. J. Math. 18, 175 (2021).

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI:


Mathematics Subject Classification