Arabian Journal for Science and Engineering

, Volume 36, Issue 8, pp 1517–1528 | Cite as

Effect of Variable Viscosity on Vortex Instability of Mixed Convection Boundary Layer Flow Adjacent to a Non-isothermal Horizontal Surface in a Porous Medium

  • A. M. Elaiw
  • F. S. Ibrahim
  • A. A. Bakr
  • A. A. Salama
Research Article - Mathematics

Abstract

In this paper, we study the effect of variable viscosity on the vortex instability of horizontal mixed convection boundary layer flow in a saturated porous medium with variable wall temperature. The variation of viscosity is expressed as an exponential function of temperature. The analysis of the disturbance flow is based on linear stability theory. The entire mixed convection regime is divided into two regions. The first region covers the forced convection dominated regime, which is characterized by the parameter \({\xi_{f}=Ra_{x}/Pe_{x}^{3/2}}\) and the eigenvalue Peclet number. The second region covers the free convection dominated regime, which is characterized by the parameter \({\xi_{n}=Pe_{x}/Ra_{x}^{2/3}}\) and the eigenvalue Rayleigh number. The two solutions provide results that cover the entire mixed-convection regime from pure-forced to pure-free convection. The local Nusselt number, critical Peclet and Rayleigh numbers and the associated wave numbers at the onset of vortex instability are presented over a wide range of wall to ambient viscosity ratio parameters \({\mu^{\ast}=\mu_{w}/\mu_{\infty}}\). The variable viscosity effect is found to enhance the heat transfer rate and destabilize the flow for liquid heating, while the opposite trend is true for gas heating.

Keywords

Vortex instability Porous media Variable viscosity Mixed convection Finite difference method 

Mathematics Subject Classification (2000)

76S05 76R05 76R10 76M20 76E20 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Ali M.E.: The effect of variable viscosity on mixed convection heat transfer along a vertical moving surface. Int. J. Thermal Sci. 45, 60–66 (2006)CrossRefGoogle Scholar
  2. 2.
    Aldoss T.K., Chen T.S., Armaly B.F.: Nonsimilarity solutions for mixed convection from horizontal surfaces in a porous medium-variable wall temperature. Int. J. Heat Mass Transf. 36, 471–477 (1993)MATHCrossRefGoogle Scholar
  3. 3.
    Chung T.J., Park J.H., Choi C.K., Yoon D.Y.: The onset of vortex instability in a laminar forced convection flow through a horizontal porous channel. Int. J. Heat Mass Transf. 45, 3061–3064 (2002)MATHCrossRefGoogle Scholar
  4. 4.
    Chin K.E., Nazar R., Arifin N.M., Pop I.: Effect of variable viscosity on mixed convection boundary layer flow over a vertical surface embedded in a porous medium. Int. Commun. Heat Mass Transf. 34, 464–473 (2007)CrossRefGoogle Scholar
  5. 5.
    Elaiw A.M., Ibrahim F.S.: Vortex instability of mixed convection boundary layer flow adjacent to a non isothermal horizontal surface in a porous medium with variable permeability. J. Porous Media. 11, 305–321 (2007)CrossRefGoogle Scholar
  6. 6.
    Elaiw A.M., Ibrahim F.S., Bakr A.A.: The influence of variable permeability on vortex instability of a horizontal combined free and mixed convection flow in a saturated porous medium. Z. Angew. Math. Mech. 87, 528–536 (2007)MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Elaiw, A.M.: Vortex instability of mixed convection boundary layer flow adjacent to a non-isothermal inclined surface in a porous medium with variable permeability. Z. Angew. Math. Mech. 88, 121–128 (2008)Google Scholar
  8. 8.
    Gray, J., Kassoy, D.R., Tadjeran, H., Zebib, A.: The effect of significant viscosity variation on convective heat transport in water saturated porous media. J. Fluid Mech. 117, 233–249 (1982)Google Scholar
  9. 9.
    Hsu, C.T., Cheng, P., Homsy, G.M.: Instability of free convection flow over a horizontal impermeable surface in a porous medium. Int. J. Heat Mass Transf. 21, 1221–1228 (1978)Google Scholar
  10. 10.
    Hsu, C.T., Cheng, P.: Vortex instability of mixed convection flow in a semi-infinite porous medium bounded by horizontal surface. Int. J. Heat Mass Transf. 23, 789–798 (1980)Google Scholar
  11. 11.
    Hassanien, I.A., Essawy, A.H., Moursy, N.M.: Variable viscosity and thermal conductivity effects on combined heat and mass transfer in mixed convection over a UHF/UMF wedge in porous media: the entire regime. Appl. Math. Comput. 145, 667–682 (2003)Google Scholar
  12. 12.
    Hassanien, I.A., Salama, A.A., Elaiw, A.M.: Variable permeability effect on vortex instability of mixed convection flow in a semi-infinite porous medium bounded by a horizontal surface. Appl. Math. Comput. 146, 829–847 (2003)Google Scholar
  13. 13.
    Hassanien I.A., Salama A.A., Elaiw A.M.: Variable permeability effect on vortex instability of a horizontal natural convection flow in a saturated porous medium with variable wall temperature. Z. Angew. Math. Mech. 84, 39–47 (2004)MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Hamad, M.A.A., Bashir, M.A.: Similarty solutions of the effect of variable viscosity on unsteady mixed convection bundary layer flow over a vertical surface embedded in a porous medium via HAMAD formulations. World Appl. Sci. J. 12(4), 519–530 (2011)Google Scholar
  15. 15.
    IMSL: References Manual. IMSL, Houston (1990)Google Scholar
  16. 16.
    Jang, J.Y., Leu, J.S.: Variable viscosity effects on the vortex instability of free convection boundary layer flow over a horizontal surface in porous medium. Int. J. Heat Mass Transf. 36, 1287–1294 (1993)Google Scholar
  17. 17.
    Jayanthi, S., Kumari, M.: Effect of variable viscosity on non-Darcy free or mixed flow on a vertical surfae in a non-Newtonian fluid saturated porous medium. Appl. Math. Comput. 182(2), 1643–1659 (2007)Google Scholar
  18. 18.
    Kassoy D.R., Zebib A.: Variable viscosity effects on the onset of convection in porous media. Phys. Fluids 18, 1649–1651 (1975)CrossRefGoogle Scholar
  19. 19.
    Kumari, M.: Effect of variable viscosity on non-darcy free or mixed convection flow on a horizontal surface in a saturated porous medium. Int. Commun. Heat Mass Transf. 28, 723–732 (2001)Google Scholar
  20. 20.
    Lee, D.H., Yoon, D.Y., Choi, C.K.: The onset of vortex instability in laminar convection flow over an inclined plate embedded in a porous medium. Int. J. Heat Mass Transf. 43, 2895–2908 (2000)Google Scholar
  21. 21.
    Lai, F.C., Kulacki, F.A.: The effect of variable viscosity on convective heat transfer along a vertical surface in a saturated porous medium. Int. J. Heat Mass Transf. 33, 1028–1031 (1990)Google Scholar
  22. 22.
    Ling, J.X., Dybbs, A.: The effect of variable viscosity on forced convection over a flat plate submersed in a porous medium. Trans. ASME J. Heat Transf. 114, 1063–1065 (1992)Google Scholar
  23. 23.
    Nield D.A., Bejan A.: Convection in Porous Media. 3rd edn. Springer, New York (2006)Google Scholar
  24. 24.
    Nakayama, A., Pop, I.: A unified similarity transformation for free, forced and mixed convection in Darcy and non-Darcy porous media. Int. J. Heat Mass Transf. 34, 357–367 (1991)Google Scholar
  25. 25.
    Nakayama, A.: PC-Aided Numerical Heat Transfer and Convective Flow. CRC Press, Boca Raton (1995)Google Scholar
  26. 26.
    Sparrow, E.M., Quack, H., Boerner, C.T.: Local nonsimilarity boundary layer solutions. AIAA J. 8, 1936–1942 (1970)Google Scholar
  27. 27.
    Usmani R.A.: Some new finite difference methods for computing eigenvalues of two-point boundary value problems. Comput. Math. Appl. 9, 903–909 (1985)MathSciNetCrossRefGoogle Scholar

Copyright information

© King Fahd University of Petroleum and Minerals 2011

Authors and Affiliations

  • A. M. Elaiw
    • 1
    • 2
  • F. S. Ibrahim
    • 3
  • A. A. Bakr
    • 2
  • A. A. Salama
    • 3
  1. 1.Department of Mathematics, Faculty of ScienceKing Abdulaziz UniversityJeddahSaudi Arabia
  2. 2.Department of Mathematics, Faculty of ScienceAssiut UniversityAssiutEgypt
  3. 3.Department of Mathematics, Faculty of ScienceAl-Azhar UniversityAssiutEgypt

Personalised recommendations