Exact solutions of the Navier-Stokes equations generalized for flow in porous media

  • Edoardo Daly
  • Hossein Basser
  • Murray Rudman
Regular Article


Flow of Newtonian fluids in porous media is often modelled using a generalized version of the full non-linear Navier-Stokes equations that include additional terms describing the resistance to flow due to the porous matrix. Because this formulation is becoming increasingly popular in numerical models, exact solutions are required as a benchmark of numerical codes. The contribution of this study is to provide a number of non-trivial exact solutions of the generalized form of the Navier-Stokes equations for parallel flow in porous media. Steady-state solutions are derived in the case of flows in a medium with constant permeability along the main direction of flow and a constant cross-stream velocity in the case of both linear and non-linear drag. Solutions are also presented for cases in which the permeability changes in the direction normal to the main flow. An unsteady solution for a flow with velocity driven by a time-periodic pressure gradient is also derived. These solutions form a basis for validating computational models across a wide range of Reynolds and Darcy numbers.


  1. 1.
    J. Bear, Dynamics of Fluids in Porous Media (Elsevier, Amsterdam, 1972)Google Scholar
  2. 2.
    P.Y. Polubarinova-Kochina, Theory of Groundwater Movement (Princeton University Press, Princeton, N.J., 1962)Google Scholar
  3. 3.
    D.A. Nield, A. Bejan, Convection in Porous Media (Springer, Cham, 2017)Google Scholar
  4. 4.
    K.G. Boggs, R.W. Van Kirk, G.S. Johnson, J.P. Fairley, P.S. Porter, J. Am. Water Resour. Assoc. 46, 1116 (2010)ADSCrossRefGoogle Scholar
  5. 5.
    Peter Troch, Emiel van Loon, Arno Hilberts, Adv. Water Resour. 25, 637 (2002)ADSCrossRefGoogle Scholar
  6. 6.
    E. Daly, A. Porporato, Phys. Rev. E 70, 056303 (2004)ADSCrossRefGoogle Scholar
  7. 7.
    E. Daly, A. Porporato, Water Resour. Res. 40, W016011 (2004)CrossRefGoogle Scholar
  8. 8.
    M.S. Bartlett, A. Porporato, Water Resour. Res. 54, 767 (2018)ADSCrossRefGoogle Scholar
  9. 9.
    M.H. Hamdan, Appl. Math. Comput. 62, 203 (1994)MathSciNetGoogle Scholar
  10. 10.
    H.C. Brinkman, Appl. Sci. Res. A1, 27 (1947)Google Scholar
  11. 11.
    B. Goyeau, D. Lhuillier, D. Gobin, M.G. Velarde, Int. J. Heat Mass Transfer 46, 4071 (2003)CrossRefGoogle Scholar
  12. 12.
    F.J. Valds-Parada, C.G. Aguilar-Madera, J.A. Ochoa-Tapia, B. Goyeau, Adv. Water Resour. 62, 327 (2013)ADSCrossRefGoogle Scholar
  13. 13.
    H. Akbari, M.M. Namin, Coast. Eng. 74, 59 (2013)CrossRefGoogle Scholar
  14. 14.
    H. Akbari, Coast. Eng. 89, 1 (2014)CrossRefGoogle Scholar
  15. 15.
    H. Basser, M. Rudman, E. Daly, Adv. Water Resour. 108, 15 (2017)ADSCrossRefGoogle Scholar
  16. 16.
    J.J. Monaghan, Annu. Rev. Fluid Mech. 44, 323 (2012)ADSCrossRefGoogle Scholar
  17. 17.
    P.G. Drazin, N. Riley, The Navier-Stokes equations: A classification of flows and exact solutions, in London Mathematical Society Lecture Notes Series (Cambridge University Press, Cambridge, 2006)Google Scholar
  18. 18.
    C.Y. Wang, Appl. Mech. Rev. 42, S269 (1989)ADSCrossRefGoogle Scholar
  19. 19.
    C.Y. Wang, Annu. Rev. Fluid Mech. 23, 159 (1991)ADSCrossRefGoogle Scholar
  20. 20.
    A. Bourchtein, Int. J. Numer. Methods Fluids 39, 1053 (2002)ADSCrossRefGoogle Scholar
  21. 21.
    W. Khan, F. Yousafzai, M. Ikhlaq Chohan, A. Zeb, G. Zaman, I.H. Jung, Int. J. Pure Appl. Math. 96, 235 (2014)CrossRefGoogle Scholar
  22. 22.
    C. Beckermann, R. Viskanta, S. Ramadhyani, J. Fluid Mech. 186, 257 (2006)ADSCrossRefGoogle Scholar
  23. 23.
    C. Beckermann, S. Ramadhyani, R. Viskanta, J. Heat Transf. 109, 363 (1987)CrossRefGoogle Scholar
  24. 24.
    D.D. Joseph, D.A. Nield, G. Papanicolaou, Water Resour. Res. 18, 1049 (1982)ADSCrossRefGoogle Scholar
  25. 25.
    D.A. Nield, Transp. Porous Media 78, 537 (2009)CrossRefGoogle Scholar
  26. 26.
    I. Federico, S. Marrone, A. Colagrossi, F. Aristodemo, M. Antuono, Eur. J. Mech. B/Fluids 34, 35 (2012)ADSMathSciNetCrossRefGoogle Scholar
  27. 27.
    W.F. Ames, Nonlinear Ordinary Differential Equations in Transport Processes (Academic Press, New York, 1968)Google Scholar
  28. 28.
    M. Abramowitz, I.A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965)Google Scholar
  29. 29.
    C.Y. Wang, J. Appl. Mech. 38, 553 (1971)ADSCrossRefGoogle Scholar
  30. 30.
    D.A. Nield, Int. J. Heat Mass Transfer 46, 4351 (2003)CrossRefGoogle Scholar
  31. 31.
    P.G. Drazin, W.G. Reid, Hydrodynamic Stability, 2nd ed. (Cambridge University Press, Cambridge, 2004)Google Scholar
  32. 32.
    Antony A. Hill, Brian Straughan, Stability of Poiseuille flow in a porous medium (Springer, Berlin, Heidelberg, 2010) pp. 287--293Google Scholar
  33. 33.
    B. Straughan, Stability and wave motion in porous media, in Applied Mathematical Sciences (Springer, New York, 2008)Google Scholar
  34. 34.
    A.D. Polyanin, V.F. Zaitsev, Handbook of Exact Solutions for Ordinary Differential Equations (CRC Press, Boca Raton, 2003)Google Scholar

Copyright information

© Società Italiana di Fisica and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Civil EngineeringMonash UniversityMelbourneAustralia
  2. 2.Department of Mechanical and Aerospace EngineeringMonash UniversityMelbourneAustralia

Personalised recommendations