Korean Journal of Chemical Engineering

, Volume 35, Issue 6, pp 1247–1256 | Cite as

Effect of vertically varying permeability on the onset of convection in a porous medium

  • Won Sun Ryoo
  • Min Chan Kim
Transport Phenomena


Considering the vertically varying permeability of a porous medium, we conducted theoretical and numerical analyses on the onset of buoyancy-driven instability in an initially quiescent, fluid-saturated, horizontal porous layer. Darcy’s law was employed to explain the fluid flow through a porous medium and linear and nonlinear analyses are conducted. In the semi-infinite domain, the growth of disturbance and the onset of convection were analyzed with and without the quasi-steady state approximation. The present analysis of initial growth rate shows that the system is initially unconditionally stable regardless of a vertical heterogeneity parameter. The onset conditions of buoyancy-driven instabilities were investigated as a function of the Darcy-Rayleigh number and the heterogeneity parameter. To find the effect of a vertical heterogeneity on the flow after the onset of convection, nonlinear numerical simulations also were conducted using the result of the linear analysis as a starting point. Nonlinear numerical simulations show that the finger-like instability motion is not readily observable at a critical time and it becomes visible approximately when a mass transfer rate substantially increases.


Buoyancy-driven Convection Onset Condition Vertically Heterogeneous Porous Medium Linear Stability Analysis Direct Numerical Simulation 


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  1. 1.
    C. W. Horton and F. T. Rogers, J. Appl. Phys., 6, 367 (1945).CrossRefGoogle Scholar
  2. 2.
    E. R. Lapwood, Proc. Camb. Philos. Soc., 44, 508 (1948).CrossRefGoogle Scholar
  3. 3.
    D. A. Nield and A. Bejan, Convection in Porous Media, 4th Ed., Springer (2013).CrossRefGoogle Scholar
  4. 4.
    J.-P. Caltagirone, Q. J. Mech. Appl. Math., 33, 47 (1980).Google Scholar
  5. 5.
    J. Ennis-King, I. Preston and L. Paterson, Phys. Fluids, 17, 084107 (2005).CrossRefGoogle Scholar
  6. 6.
    X. Xu, S. Chen and D. Zhang, Adv. Water Resour., 29, 397 (2006).CrossRefGoogle Scholar
  7. 7.
    H. Hassanzadeh, M. Pooladi-Darvish and D.W. Keith, Trans. Porous Med., 65, 193 (2006).CrossRefGoogle Scholar
  8. 8.
    S. Rapaka, S. Chen, R. J. Pawar, P. H. Stauffer and D. Zhang, J. Fluid Mech., 609, 285 (2008).CrossRefGoogle Scholar
  9. 9.
    A. Riaz, M. Hesse, H.A. Tchelepi and F. M. Orr, J. Fluid Mech., 548, 87 (2006).CrossRefGoogle Scholar
  10. 10.
    A. Selem and D. A. S. Rees, J. Porous Med., 10, 1 (2007).CrossRefGoogle Scholar
  11. 11.
    D. Wessel-Berg, SIAM J. Appl. Math., 70, 1219 (2009).CrossRefGoogle Scholar
  12. 12.
    M.C. Kim and C.K. Choi, Phys. Fluids, 19, 088103 (2007).CrossRefGoogle Scholar
  13. 13.
    A. Riaz and Y. Cinar, J. Petrol. Sci. Eng., 124, 367 (2014).CrossRefGoogle Scholar
  14. 14.
    S. Rapaka, R. J. Pawar, P. H. Stauffer, D. Zhang and S. Chen, J. Fluid Mech., 641, 227 (2009).CrossRefGoogle Scholar
  15. 15.
    D.A. Nield and A.V. Kuznetsov, Transp. Porous Med., 85, 691 (2010).CrossRefGoogle Scholar
  16. 16.
    P. Ranganathan, R. Farajzadeh, H. Bruining and P. L. J. Zitha, Transp. Porous Med., 95, 25 (2012).CrossRefGoogle Scholar
  17. 17.
    X.-Z. Kong and M.O. Saar, Int. J. Greenhouse Gas Control, 19, 160 (2013).CrossRefGoogle Scholar
  18. 18.
    A.A. Hill and M.R. Morad, Proc. R. Soc. A, 470, 20140373 (2014).CrossRefGoogle Scholar
  19. 19.
    M.C. Kim and C. K. Choi, Korean J. Chem. Eng., 32, 2400 (2015).CrossRefGoogle Scholar
  20. 20.
    S. Chandrasekhar, Hydrodynamic and Hydromagnetic Stability, Oxford U. P. (1961).Google Scholar
  21. 21.
    A. J. Harfash and A.A. Hill, Int. J. Heat Mass Transfer, 72, 609 (2014).CrossRefGoogle Scholar
  22. 22.
    A. J. Harfash, Transp. Porous Med., 102, 43 (2014).CrossRefGoogle Scholar
  23. 23.
    Y. Ben, E. A. Demekhin and H.-C. Chang, Phys. Fluids, 14, 999 (2002).CrossRefGoogle Scholar
  24. 24.
    M.C. Kim and C. K. Choi, Phys. Fluids, 24, 044102 (2012).CrossRefGoogle Scholar
  25. 25.
    A.C. Slim and T. S. Ramakrishnan, Phys. Fluids, 22, 124103 (2010).CrossRefGoogle Scholar
  26. 26.
    M.C. Kim, Korean J. Chem. Eng., 35(2), 364 (2017).CrossRefGoogle Scholar
  27. 27.
    W. Lick, J. Fluid Mech., 21, 565 (1965).CrossRefGoogle Scholar
  28. 28.
    C.T. Tan and G.M. Homsy, Phys. Fluids, 29, 3549 (1986).CrossRefGoogle Scholar
  29. 29.
    K. Ghesmat, H. Hassanzadeh and J. Abedi, J. Fluid Mech., 673, 480 (2011).CrossRefGoogle Scholar
  30. 30.
    C.T. Tan and G.M. Homsy, Phys. Fluids, 31, 1330 (1988).CrossRefGoogle Scholar
  31. 31.
    W.B. Zimmerman and G.M. Homsy, Phys. Fluids A, 4, 2348 (1992).CrossRefGoogle Scholar
  32. 32.
    O. Manickam and G. M. Homsy, J. Fluid Mech., 288, 75 (1995).CrossRefGoogle Scholar

Copyright information

© Korean Institute of Chemical Engineers, Seoul, Korea 2018

Authors and Affiliations

  1. 1.Department of Chemical EngineeringHongik UniversitySeoulKorea
  2. 2.Department of Chemical EngineeringJeju National UniversityJejuKorea

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