Advertisement

Transport in Porous Media

, Volume 79, Issue 2, pp 215–223 | Cite as

On the Domain of Validity of Brinkman’s Equation

  • Jean-Louis AuriaultEmail author
Article

Abstract

An increasing number of articles are adopting Brinkman’s equation in place of Darcy’s law for describing flow in porous media. That poses the question of the respective domains of validity of both laws, as well as the question of the value of the effective viscosity μ e which is present in Brinkman’s equation. These two topics are addressed in this article, mainly by a priori estimates and by recalling existing analyses. Three main classes of porous media can be distinguished: “classical” porous media with a connected solid structure where the pore surface S p is a function of the characteristic pore size l p (such as for cylindrical pores), swarms of low concentration fixed particles where the pore surface is a function of the characteristic particle size l s , and fiber-made porous media at low solid concentration where the pore surface is a function of the fiber diameter. If Brinkman’s 3D flow equation is valid to describe the flow of a Newtonian fluid through a swarm of fixed particles or fibrous media at low concentration under very precise conditions (Lévy 1983), then we show that it cannot apply to the flow of such a fluid through classical porous media.

Keywords

Darcy’s law Brinkman’s equation Effective viscosity Domain of validity 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  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 and 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)Google Scholar
  2. Auriault J.-L., Geindreau C., Boutin C.: Filtration law in porous media with poor separation of scales. Transp. Porous Med. 60, 89–108 (2005)CrossRefGoogle Scholar
  3. Brinkman H.C.: A calculation of the viscous force exerted by a flowing fluid on a dense swarm of particles. Appl. Sci. Res. A 1, 27–34 (1949)CrossRefGoogle Scholar
  4. Childress S.: Viscous flow past a random array of spheres. J. Chem. Phys 56(6), 2527–2539 (1972)CrossRefGoogle Scholar
  5. Darcy H.: Les Fontaines Publiques de la Ville de Dijon. Victor Darmon, Paris (1856)Google Scholar
  6. Durlovsky L., Brady J.F.: Analysis of the Brinkman equation as a model for flow in porous media. Phys. Fluids 30(11), 3329–3341 (1987)CrossRefGoogle Scholar
  7. Freed K.F., Muthukumar M: On the Stokes problem for a suspension of spheres at finite concentration. J. Chem. Phys 68(5), 2088–2096 (1978)CrossRefGoogle Scholar
  8. Givler R.C., Altobelli S.A.: A determination of the effective viscosity for Brinkman-Forchheimer flow model. J. Fluid Mech. 258, 355–370 (1994)CrossRefGoogle Scholar
  9. Hinch E.J.: An averaged-equation approach to particle interactions in a fluid suspension. J. Fluid Mech. 83, 695–720 (1977)CrossRefGoogle Scholar
  10. Happel J., Brenner H.: Low Reynolds Number Hydrodynamics. Noordhoff, The Netherlands (1973)Google Scholar
  11. Happel J., Epstein N.: Cubical assemblages of uniform spheres. Ind. Chem. Eng. 46(6), 1187–1194 (1954)CrossRefGoogle Scholar
  12. Howells I.D.: Drag due to the motion of a Newtonian fluid through a sparse random array of small fixed rigid objects. J. Fluid Mech. 64, 449–475 (1974)CrossRefGoogle Scholar
  13. Lévy T.: Fluid flow through an array of fixed particles. Int. J. Eng. Sci. 21(1), 11–23 (1983)CrossRefGoogle Scholar
  14. Muthukumar M., Freed K.F.: On the Stokes problem for a suspension of spheres at nonzero concentrations. II. Calculations for effective medium theory. J. Chem. Phys. 70(12), 5875–5887 (1979)CrossRefGoogle Scholar
  15. Rubinstein J.: Effective equation for flow in random porous media with a large number of scales. J. Fluid Mech. 170, 379–383 (1986)CrossRefGoogle Scholar
  16. Sanchez-Palencia E.: On the asymptotics of the fluid flow past an array of fixed particles. Int. J. Eng. Sci. 20(12), 1291–1301 (1982)CrossRefGoogle Scholar
  17. Sobera M.P., Kleijn C.R.: Hydraulic permeability of ordered and disordered single-layer arrays of cylinders. Phys. Rev. E 74(036301), 1–9 (2006)Google Scholar
  18. Tachie M.F., James D.F., Currie I.G.: Slow flow through a brush. Phys. Fluids 16(2), 445–451 (2004)CrossRefGoogle Scholar
  19. Tam C.K.W.: The drag on a cloud of spherical particles in low Reynolds number flow. J. Fluid Mech. 38, 537–546 (1969)CrossRefGoogle Scholar
  20. Vernescu B.: Asymptotic analysis for an incompressible flow in fractured porous media. Int. J. Eng. Sci. 28(9), 959–964 (1990)CrossRefGoogle Scholar
  21. Wang C.Y.: Stokes flow through a rectangular array of circular cylinders. Fluid Dyn. Res. 29, 65–80 (2001)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2008

Authors and Affiliations

  1. 1.Laboratoire Sols Solides Structures (3S)UJF, INPG, CNRS, Domaine UniversitaireGrenoble CedexFrance

Personalised recommendations