Mathematical modeling of micropolar fluid flows through a thin porous medium


We study the flow of a micropolar fluid in a thin domain with microstructure, i.e., a thin domain with thickness \(\varepsilon \) which is perforated by periodically distributed solid cylinders of size \(a_\varepsilon \). A main feature of this study is the dependence of the characteristic length of the micropolar fluid on the small parameters describing the geometry of the thin porous medium under consideration. Depending on the ratio of \(a_\varepsilon \) with respect to \(\varepsilon \), we derive three different generalized Darcy equations where the interaction between the velocity and the microrotation fields is preserved.

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

Fig. 1


  1. 1.

    Eringen AC (1964) Simple mocrofluids. Internat J. Eng Sci 2:205–207

    MathSciNet  Google Scholar 

  2. 2.

    Eringen AC (1966) Theory of micropolar fluids. J Math Mech 16:1–18

    MathSciNet  Google Scholar 

  3. 3.

    Frishfelds V, Lundström TS, Jakovics A (2011) Lattice gas analysis of liquid front in non-crimp fabrics. Transp Porous Med 84:75–93

    MathSciNet  Google Scholar 

  4. 4.

    Jeon W, Shin CB (2009) Design and simulation of passive mixing in microfluidic systems with geometric variations. Chem Eng J 152:575–582

    Google Scholar 

  5. 5.

    Lundström TS, Toll S, Hakanson JM (2002) Measurements of the permeability tensor of compressed fibre beds. Transp Porous Med 47:363–380

    Google Scholar 

  6. 6.

    Nordlund M, LundströmTS TS (2008) Effect of multi-scale porosity in local permeability modelling of non-crimp fabrics. Transp Porous Med 73:109–124

    Google Scholar 

  7. 7.

    Singh F, Stoeber B, Green SI (2015) Micro-PIV measurement of flow upstream of papermaking forming fabrics. Transp Porous Med 107:435–448

    Google Scholar 

  8. 8.

    Tan H, Pillai KM (2012) Multiscale modeling of unsaturated flow in dual-scale fiber preforms of liquid composite molding I: isothermal flows. Compos Part A Appl Sci Manuf 43:1–13

    Google Scholar 

  9. 9.

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

    MathSciNet  Google Scholar 

  10. 10.

    Anguiano M, Suárez-Grau FJ (2018) The transition between the Navier–Stokes equations to the Darcy equation in a thin porous medium. Mediterr J Math 15:45

    MathSciNet  MATH  Google Scholar 

  11. 11.

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

    MathSciNet  MATH  Google Scholar 

  12. 12.

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

    MathSciNet  MATH  Google Scholar 

  13. 13.

    Cioranescu D, Damlamian A, Griso G (2008) The periodic unfolding method in homogenization. SIAM J Math Anal 40:1585–1620

    MathSciNet  MATH  Google Scholar 

  14. 14.

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

    MathSciNet  MATH  Google Scholar 

  15. 15.

    Anguiano M, Bunoiu R (2019) On the flow of a viscoplastic fluid in a thin periodic domain. In: Constanda C, Harris P (eds) Integral methods in science and engineering. Birkauser, Cham

    Google Scholar 

  16. 16.

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

    MathSciNet  MATH  Google Scholar 

  17. 17.

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

    MathSciNet  MATH  Google Scholar 

  18. 18.

    Anguiano M (2017) 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

    MathSciNet  MATH  Google Scholar 

  19. 19.

    Anguiano M (2017) On the non-stationary non-Newtonian flow through a thin porous medium. ZAMM 97:895–915

    MathSciNet  Google Scholar 

  20. 20.

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

    MathSciNet  MATH  Google Scholar 

  21. 21.

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

    MathSciNet  MATH  Google Scholar 

  22. 22.

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

    MathSciNet  MATH  Google Scholar 

  23. 23.

    Anguiano M, Suárez-Grau FJ (2017) Derivation of a coupled Darcy–Reynolds equation for a fluid flow in a thin porous medium including a fissure. ZAMP 68:52

    MathSciNet  MATH  Google Scholar 

  24. 24.

    Abbas S, Malik MY, Nadeem S (2020) Derivation of a coupled Darcy–Reynolds equation for a fluid flow in a thin porous medium including a fissure. Comput Methods Programs Biomed 185:105136

    Google Scholar 

  25. 25.

    Abbas S, Nadeem S, Malik MY (2020) On extended version of Yamada–Ota and Xue models in micropolar fluid flow under the region of stagnation point. Physica A 542:123512

    MathSciNet  Google Scholar 

  26. 26.

    Abbas S, Nadeem S, Malik MY (2020) Theoretical study of micropolar hybrid nanofluid over Riga channel with slip conditions. Physica A 551:124083

    MathSciNet  Google Scholar 

  27. 27.

    Ahmad S, Nadeem S (2020) Application of CNT-based micropolar hybrid nanofluid flow in the presence of Newtonian heating. Appl Nanosci 10:5265–5277

  28. 28.

    Ahmad S, Nadeem S, Muhammad N, Kahn MN (2020) Cattaneo–Christov heat flux model for stagnation point flow of micropolar nanofluid toward a nonlinear stretching surface with slip effects. J Therm Anal Calorim

  29. 29.

    Khan MN, Nadeem S, Muhammad N (2020) Micropolar fluid flow with temperature-dependent transport properties. Heat Transf 49:2375–2389

    Google Scholar 

  30. 30.

    Nadeem S, Malik MY, Abbas N (2020) Heat transfer of three-dimensional micropolar fluid on a Riga plate. Can J Phys 98:32–38

    Google Scholar 

  31. 31.

    Nadeem S, Abbas N, Elmasry Y, Malik MY (2020) Numerical analysis of water based CNTs flow of micropolar fluid through rotating frame. Comput Methods Program Biomed 186:105194

    Google Scholar 

  32. 32.

    Nadeem S, Kiani MN, Saleem A, Issakhov A (2020) Microvascular blood flow with heat transfer in a wavy channel having electroosmotic effects. Electrophoresis 41:1198–1205

    Google Scholar 

  33. 33.

    Lukaszewicz G (1999) Micropolar fluids, theory and applications. Modeling and simulation in science, engineering and technology. Birkhauser, Basel

  34. 34.

    Johnston GJ, Wayte R, Spikes HA (1991) The measurement and study of very thin lubricant films in concentrated contacts. Tribol Trans 34:187–194

    Google Scholar 

  35. 35.

    Luo JB, Huang P, Wen SZ (1996) Thin film lubrication part I: study on the transition between EHL and thin film lubrication using relative optical interference intensity technique. Wear 194:107–115

    Google Scholar 

  36. 36.

    Luo JB, Huang P, Wen SZ, Lawrence L (1999) Characteristics of fluid lubricant films at nano-scale. J Tribol 121:872–878

    Google Scholar 

  37. 37.

    Bonnivard M, Pazanin I, Suárez-Grau FJ (2018) Effects of rough boundary and nonzero boundary conditions on the lubrication process with micropolar fluid. Eur J Mech B Fluids 72:501–518

    MathSciNet  MATH  Google Scholar 

  38. 38.

    Boukrouche M, Paoli L (2012) Asymptotic analysis of a micropolar fluid flow in a thin domain with a free and rough boundary. SIAM J Math Anal 44:1211–1256

    MathSciNet  MATH  Google Scholar 

  39. 39.

    Dupuy D, Panasenko G, Stavre R (2008) Asymptotic solution for a micropolar flow in a curvilinear channel. ZAMM Z Angew Math Mech 88:793–807

    MathSciNet  MATH  Google Scholar 

  40. 40.

    Pazanin I, Suárez-Grau FJ (2014) Analysis of the thin film flow in a rough thin domain filled with micropolar fluid. Comput Math Appl 68:1915–1932

    MathSciNet  MATH  Google Scholar 

  41. 41.

    Bayada G, Lukaszewicz G (1996) On micropolar fluids in the theory of lubrication. Rigorous derivation of an analogue of the Reynolds equation. Internat J Eng Sci 34:1477–1490

    MathSciNet  MATH  Google Scholar 

  42. 42.

    Bayada G, Chambat M, Gamouana SR (2001) About thin film micropolar asymptotic equations. Quart Appl Math 59:413–439

    MathSciNet  MATH  Google Scholar 

  43. 43.

    Duvaut G, Lions JL (1972) Les inequations en mechanique et en physique [The inequations in mechanics and physics]. Dunod, Paris

    Google Scholar 

  44. 44.

    Tartar L (1980) Incompressible fluid flow in a porous medium convergence of the homogenization process, vol 127. Appendix to lecture notes in physics, Springer-Velag, Berlin

    Google Scholar 

  45. 45.

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

    MathSciNet  MATH  Google Scholar 

  46. 46.

    Suárez-Grau FJ (2020) Analysis of the roughness regimes for micropolar fluids via homogenization. Malaysian Math Sci Soc Bull

  47. 47.

    Anguiano M (2020) Existence, uniqueness and homogenization of nonlinear parabolic problems with dynamical boundary conditions in perforated media. Mediterr J Math 17:18

    MathSciNet  MATH  Google Scholar 

  48. 48.

    Anguiano M (2020) Homogenization of parabolic problems with dynamical boundary conditions of reactive-diffusive type in perforated media. ZAMM 100:e202000088

Download references

Author information



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

Verify currency and authenticity via CrossMark

Cite this article

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

Download citation


  • Darcy’s law
  • Homogenization
  • Micropolar fluid flow
  • Thin-film fluid
  • Thin porous medium