Transport in Porous Media

, Volume 126, Issue 2, pp 275–294 | Cite as

Linear Stability of Horizontal Throughflow in a Brinkman Porous Medium with Viscous Dissipation and Soret Effect

  • Rashmi DubeyEmail author
  • P. V. S. N. Murthy


The onset of double-diffusive convective instability of a horizontal throughflow induced by viscous dissipation in a fluid-saturated porous layer of high permeability is investigated. The porous layer is infinitely long along the horizontal direction and is bounded by two rigid surfaces maintained at constant, but different solute concentrations. The lower surface is thermally insulated, whereas the upper surface is considered to be isothermal. The Darcy–Brinkman model is adopted for deriving the equations governing flow in the medium, and the Soret effect is considered to persist in the flow. The instability in the base flow is considered to be induced by the non-negligible viscous heating. Disturbances in the base flow are assumed in the form of oblique rolls, where the longitudinal and the transverse rolls are at two extreme inclinations. The disturbance functions are assumed to be of O(1). It is considered that \(Ge\ll 1\) and \(|Pe|\gg 1\), where Ge is the Gebhart number and Pe is the Péclet number. The eigenvalue problem with coupled ordinary differential equations governing the disturbances in the flow is solved numerically using bvp4c in MATLAB. Results obtained depict that the flow is most stable in the Brinkman regime and the longitudinal rolls are the preferred mode of instability. The solute concentration gradient and the Soret parameter have both stabilizing and destabilizing effect on the flow in the medium, when the values of both are either positive or negative. However, they have either monotonically stabilizing or monotonically destabilizing effect on the flow, when the values of both have opposite signs.


Linear stability Throughflow Viscous dissipation Brinkman model Soret effect 


  1. Al-Hadhrami, A.K., Elliott, L., Ingham, D.B.: A new model for viscous dissipation in porous media across a range of permeability values. Transp. Porous Media 53, 117–122 (2003)CrossRefGoogle Scholar
  2. Altawallbeh, A.A., Bhadauria, B.S., Hashim, I.: Linear and nonlinear double-diffusive convection in a saturated anisotropic porous layer with Soret effect and internal heat source. Int. J. Heat Mass Transf. 59, 103–111 (2013)CrossRefGoogle Scholar
  3. Bahloul, A., Boutana, N., Vasseur, P.: Double-diffusive and Soret-induced convection in a shallow horizontal porous layer. J. Fluid Mech. 491, 325–352 (2003)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 Transf. 54, 1641–1648 (2011)CrossRefGoogle Scholar
  5. 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 Transf. 52, 337–344 (2009a)CrossRefGoogle Scholar
  6. Barletta, A., Celli, M., Rees, D.A.S.: Darcy–Forchheimer flow with viscous dissipation in a horizontal porous layer: onset of convective instabilities. Trans. ASME J. Heat Transf. 131, 072602 (2009b)CrossRefGoogle Scholar
  7. Barletta, A., Rossi di Schio, E., Celli, M.: Instability and viscous dissipation in the horizontal Brinkman flow through a porous medium. Transp. Porous Media 87, 105–119 (2011)CrossRefGoogle Scholar
  8. Breugem, W.P., Rees, D.A.S.: A derivation of the volume-averaged Boussinesq equations for flow in porous media with viscous dissipation. Transp. Porous Media 63, 1–12 (2006)CrossRefGoogle Scholar
  9. Brinkman, H.C.: A calculation of the viscous force exerted by a flowing fluid on a dense swarm of particles. Appl. Sci. Res. A1, 27–34 (1947)Google Scholar
  10. Dubey, R., Murthy, P.V.S.N.: The onset of convective instability of horizontal throughflow in a porous layer with inclined thermal and solutal gradients. Phys. Fluids 30, 074104 (2018)CrossRefGoogle Scholar
  11. 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, 3617–3635 (2009)CrossRefGoogle Scholar
  12. Gebhart, B.: Effects of viscous dissipation in natural convection. J. Fluid Mech. 14, 225–232 (1962)CrossRefGoogle Scholar
  13. Horton, C.W., Rogers, F.T.: Convection currents in a porous medium. J. Appl. Phys. 16, 367–370 (1945)CrossRefGoogle Scholar
  14. Kuznetsov, A.V., Xiong, M., Nield, D.A.: Thermally developing forced convection in a porous medium: circular duct with walls at constant temperature, with longitudinal conduction and viscous dissipation effects. Transp. Porous Media 53, 331–345 (2003)CrossRefGoogle Scholar
  15. Lapwood, E.R.: Convection of a fluid in a porous medium. Math. Proc. Camb. Philos. Soc. 44, 508–521 (1948)CrossRefGoogle Scholar
  16. Mojtabi, M.C., Elhajjar, B., Mojtabi, A.: Analytical and numerical stability analysis of Soret-driven convection in a horizontal porous layer. Phys. Fluids 19, 124104 (2007)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.: The boundary correction for the Rayleigh–Darcy problem: limitations of the Brinkman equation. J. Fluid Mech. 128, 37–46 (1983)CrossRefGoogle Scholar
  19. Nield, D.A.: The modeling of viscous dissipation in a saturated porous medium. Trans. ASME J. Heat Transf. 129, 1459–1463 (2007)CrossRefGoogle Scholar
  20. Nield, D.A., Kuznetsov, A.V., Xiong, M.: Thermally developing forced convection in a porous medium: parallel plate channel with walls at uniform temperature, with axial conduction and viscous dissipation effects. Int. J. Heat Mass Transf. 46, 643–651 (2003)CrossRefGoogle Scholar
  21. Rees, D.A.S.: The onset of Darcy–Brinkman convection in a porous layer: an asymptotic analysis. Int. J. Heat Mass Transf. 45, 2213–2220 (2002)CrossRefGoogle Scholar
  22. 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 Transf. 91, 700–710 (2015)CrossRefGoogle Scholar
  23. Storesletten, L., Barletta, A.: Linear instability of mixed convection of cold water in a porous layer induced by viscous dissipation. Int. J. Therm. Sci. 48, 655–664 (2009)CrossRefGoogle Scholar
  24. Taslim, M.E., Narusawa, U.: Binary fluid convection and double-diffusive convection in a porous medium. Trans. ASME J. Fluid Mech. 108(1), 221–224 (1986)Google Scholar
  25. Walker, K., Homsy, G.M.: A note on convective instabilities in Boussinesq fluids and porous media. Trans. ASME J. Heat Transf. 99, 338–339 (1977)CrossRefGoogle Scholar

Copyright information

© Springer Nature B.V. 2018

Authors and Affiliations

  1. 1.Department of MathematicsIndian Institute of Technology KharagpurKharagpurIndia

Personalised recommendations