Advertisement

Transport in Porous Media

, Volume 122, Issue 3, pp 693–712 | Cite as

Linear Stability of the Double-Diffusive Convection in a Horizontal Porous Layer with Open Top: Soret and Viscous Dissipation Effects

  • Kamalika Roy
  • P. V. S. N. Murthy
Article
  • 112 Downloads

Abstract

The linear stability of the double-diffusive convection in a horizontal porous layer is studied considering the upper boundary to be open. A horizontal temperature gradient is applied along the upper boundary. It is assumed that the viscous dissipation and Soret effect are significant in the medium. The governing parameters are horizontal Rayleigh number (\(Ra_\mathrm{H}\)), solutal Rayleigh number (\(Ra_\mathrm{S}\)), Lewis number (Le), Gebhart number (Ge) and Soret parameter (Sr). The Rayleigh number (Ra) corresponding to the applied heat flux at the bottom boundary is considered as the eigenvalue. The influence of the solutal gradient caused due to the thermal diffusion on the double-diffusive instability is investigated by varying the Soret parameter. A horizontal basic flow is induced by the applied horizontal temperature gradient. The stability of this basic flow is analyzed by calculating the critical Rayleigh number (\(Ra_\mathrm{cr}\)) using the Runge–Kutta scheme accompanied by the Shooting method. The longitudinal rolls are more unstable except for some special cases. The Soret parameter has a significant effect on the stability of the flow when the upper boundary is at constant pressure. The critical Rayleigh number is decreasing in the presence of viscous dissipation except for some positive values of the Soret parameter. How a change in Soret parameter is attributing to the convective rolls is presented.

Keywords

Soret effect Viscous dissipation Open boundary Linear stability 

References

  1. Barletta, A., Celli, M., Nield, D.A.: Unstably stratified Darcy flow with impressed horizontal temperature gradient, viscous dissipation and asymmetric thermal boundary conditions. Int. J. Heat Mass Transfer 53(9), 1621–1627 (2010)CrossRefGoogle Scholar
  2. Barletta, A., Celli, M., Rees, D.A.S.: The onset of convection in a porous layer induced by viscous dissipation: a linear stability analysis. Int. J. Heat Mass Transfer 52(1), 337–344 (2009)CrossRefGoogle Scholar
  3. Barletta, A., Nield, D.A.: Instability of hadley–prats flow with viscous heating in a horizontal porous layer. Transp. Porous Media 84(2), 241–256 (2010)CrossRefGoogle Scholar
  4. Barletta, A., Nield, D.A.: Thermosolutal convective instability and viscous dissipation effect in a fluid-saturated porous medium. Int. J. Heat Mass Transfer 54(7), 1641–1648 (2011)CrossRefGoogle Scholar
  5. Barletta, A., Storesletten, L.: Stability of flow with viscous dissipation in a horizontal porous layer with an open boundary having a prescribed temperature gradient. Transp. Porous Media 85(3), 723–741 (2010)CrossRefGoogle Scholar
  6. Brand, H., Steinberg, V.: Convective instabilities in binary mixtures in a porous medium. Phys. A Stat. Mech. Appl. 119(1–2), 327–338 (1983)CrossRefGoogle Scholar
  7. Charrier-Mojtabi, M.C., Elhajjar, B., Mojtabi, A.: Analytical and numerical stability analysis of Soret-driven convection in a horizontal porous layer. Phys. Fluids 19(12), 124104 (2007)CrossRefGoogle Scholar
  8. Deepika, N.: Linear and nonlinear stability of double-diffusive convection with the Soret effect. Transp. Porous Media 121(1), 93–108 (2018)CrossRefGoogle Scholar
  9. Gaikwad, S.N., Malashetty, M.S., Rama Prasad, K.: An analytical study of linear and nonlinear double diffusive convection in a fluid saturated anisotropic porous layer with Soret effect. Appl. Math. Model. 33(9), 3617–3635 (2009)CrossRefGoogle Scholar
  10. Gebhart, B.: Effects of viscous dissipation in natural convection. J. Fluid Mech. 14(02), 225–232 (1962)CrossRefGoogle Scholar
  11. Gebhart, B., Mollendorf, J.: Viscous dissipation in external natural convection flows. J. Fluid Mech. 38(01), 97–107 (1969)CrossRefGoogle Scholar
  12. Horton, C.W., Rogers, F.T.: Convection currents in a porous medium. J. Appl. Phys. 16(6), 367–370 (1945)CrossRefGoogle Scholar
  13. Hurle, D.T.J., Jakeman, E.: Soret-driven thermosolutal convection. J. Fluid Mech. 47(04), 667–687 (1971)CrossRefGoogle Scholar
  14. Lapwood, E.R.: Convection of a fluid in a porous medium. In: Mathematical Proceedings of the Cambridge Philosophical Society, vol. 44, pp. 508–521. Cambridge University Press, Cambridge (1948)Google Scholar
  15. Lu, N., Zhang, Y., Ross, B.: Onset of gas convection in a moist porous layer with the top boundary open to the atmosphere. Int. Commun. Heat Mass Transfer 26(1), 33–44 (1999)CrossRefGoogle Scholar
  16. Manole, D.M., Lage, J.L., Nield, D.A.: Convection induced by inclined thermal and solutal gradients, with horizontal mass flow, in a shallow horizontal layer of a porous medium. Int. J. Heat Mass Transfer 37(14), 2047–2057 (1994)CrossRefGoogle Scholar
  17. Narayana, P.A.L., Murthy, P.V.S.N., Reddy Gorla, R.S.: Soret-driven thermosolutal convection induced by inclined thermal and solutal gradients in a shallow horizontal layer of a porous medium. J. Fluid Mech. 612, 1–9 (2008)CrossRefGoogle Scholar
  18. Nield, D.A.: Onset of thermohaline convection in a porous medium. Water Resour. Res. 4(3), 553–560 (1968)CrossRefGoogle Scholar
  19. Nield, D.A.: Convection in a porous medium with inclined temperature gradient. Int. J. Heat Mass Transfer 34(1), 87–92 (1991)CrossRefGoogle Scholar
  20. Nield, D.A.: The modeling of viscous dissipation in a saturated porous medium. Trans ASME J. Heat Transfer 129(10), 1459–1463 (2007)CrossRefGoogle Scholar
  21. Nield, D.A., Bejan, A.: Convection in Porous Media, 4th edn. Springer, Berlin (2013)CrossRefGoogle Scholar
  22. Nield, D.A., Manole, D.M., Lage, J.L.: Convection induced by inclined thermal and solutal gradients in a shallow horizontal layer of a porous medium. J. Fluid Mech. 257(1), 559–574 (1993)CrossRefGoogle Scholar
  23. Patil, P.R., Rudriah, N.: Linear convective stability and thermal diffusion of a horizontal quiescent layer of a two component fluid in a porous medium. Int. J. Eng. Sci. 18(8), 1055–1059 (1980)CrossRefGoogle Scholar
  24. Roy, K., Murthy, P.V.S.N.: Soret effect on the double diffusive convection instability due to viscous dissipation in a horizontal porous channel. Int. J. Heat Mass Transfer 91, 700–710 (2015)CrossRefGoogle Scholar
  25. Taunton, J.W., Lightfoot, E.N., Green, T.: Thermohaline instability and salt fingers in a porous medium. Phys. Fluids 15(5), 748–753 (1972)CrossRefGoogle Scholar
  26. Turcotte, D.L., Hsui, A.T., Torrance, K.E., Schubert, G.: Influence of viscous dissipation on bénard convection. J. Fluid Mech. 64(02), 369–374 (1974)CrossRefGoogle Scholar
  27. Weber, J.E.: Convection in a porous medium with horizontal and vertical temperature gradients. Int. J. Heat Mass Transfer 17(2), 241–248 (1974)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V., part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsIndian Institute of TechnologyKharagpurIndia

Personalised recommendations