Applied Mathematics and Mechanics

, Volume 39, Issue 10, pp 1385–1410 | Cite as

Stability of triple diffusive convection in a viscoelastic fluid-saturated porous layer

  • K. R. Raghunatha
  • I. S. ShivakumaraEmail author


The triple diffusive convection in an Oldroyd-B fluid-saturated porous layer is investigated by performing linear and weakly nonlinear stability analyses. The condition for the onset of stationary and oscillatory is derived analytically. Contrary to the observed phenomenon in Newtonian fluids, the presence of viscoelasticity of the fluid is to degenerate the quasiperiodic bifurcation from the steady quiescent state. Under certain conditions, it is found that disconnected closed convex oscillatory neutral curves occur, indicating the requirement of three critical values of the thermal Darcy-Rayleigh number to identify the linear instability criteria instead of the usual single value, which is a novel result enunciated from the present study for an Oldroyd-B fluid saturating a porous medium. The similarities and differences of linear instability characteristics of Oldroyd-B, Maxwell, and Newtonian fluids are also highlighted. The stability of oscillatory finite amplitude convection is discussed by deriving a cubic Landau equation, and the convective heat and mass transfer are analyzed for different values of physical parameters.

Key words

Oldroyd-B fluid bifurcation instability perturbation method nonlinear stability heat and mass transfer 




depth of the porous layer


gravitational acceleration


permeability of the porous medium


unit vector in the vertical direction


ratio of heat capacities




Darcy-Prandtl number


velocity vector

RS i

solute Darcy-Rayleigh number of the ith-component


thermal Darcy-Rayleigh number



x, y, z

space coordinates

Greek symbols


horizontal wave number


thermal expansion coefficient

αS i

solute analogue of αT, i = 1, 2




thermal diffusivity

κS i

solute diffusivity, i = 1, 2


stress relaxation time


strain retardation time


stress relaxation parameter


strain retardation parameter


dynamic viscosity


kinematic viscosity


fluid density


growth term


ratio of diffusivities, i = 1, 2


stream function



basic state


lower boundary


upper boundary


dimensionless variable

Chinese Library Classification

O357.3 O175.8 

2010 Mathematics Subject Classification

76E06 76A10 76S05 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.



The authors thank reviewers for their constructive remarks and useful suggestions, which improve the work significantly.


  1. [1]
    HORTON, C.W. and ROGERS, F. T. Convection currents in a porous medium. Journal of Applied Physics, 16, 367–370 (1945)MathSciNetCrossRefzbMATHGoogle Scholar
  2. [2]
    LAPWOOD, E. R. Convection of a fluid in a porous medium. Mathematical Proceedings of the Cambridge Philosophical Society, 44, 508–512 (1948)MathSciNetCrossRefzbMATHGoogle Scholar
  3. [3]
    NIELD, D. A. and BEJAN, A. Convection in Porous Media, 5th ed., Springer, New York (2017)CrossRefzbMATHGoogle Scholar
  4. [4]
    STRAUGHAN, B. Stability and Wave Motion in Porous Media, Springer, New York (2008)zbMATHGoogle Scholar
  5. [5]
    STRAUGHAN, B. Convection with Local Thermal Non-Equilibrium and Microfluidic Effects, Springer, New York (2015)CrossRefzbMATHGoogle Scholar
  6. [6]
    VAFAI, K. Handbook of Porous Media, Marcel Dekker, New York (2000)zbMATHGoogle Scholar
  7. [7]
    RUDRAIAH, N., SRIMANI, P. K., and FRIEDRICH, R. Finite amplitude convection in a two-component fluid saturated porous layer. International Journal of Heat and Mass Transfer, 25, 715–722 (1982)CrossRefzbMATHGoogle Scholar
  8. [8]
    KIM, M. C., LEE, S. B., KIM, S., and CHUNG, B. J. Thermal instability of viscoelastic fluids in porous media. International Journal of Heat and Mass Transfer, 46, 5065–5072 (2003)CrossRefzbMATHGoogle Scholar
  9. [9]
    SHIVAKUMARA, I. S. and SURESHKUMAR, S. Convective instabilities in a viscoelastic-fluid-saturated porous medium with through flow. Journal of Geophysics and Engineering, 4, 104–115 (2007)CrossRefGoogle Scholar
  10. [10]
    BERTOLA, V. and CAFARO, E. Thermal instability of viscoelastic fluids in horizontal porous layers as initial problem. International Journal of Heat and Mass Transfer, 4, 4003–4012 (2006)CrossRefzbMATHGoogle Scholar
  11. [11]
    WANG, S. and TAN, W. Stability analysis of soret-driven double-diffusive convection of Maxwell fluid in a saturated porous medium. International Journal of Heat and Fluid Flow, 32, 88–94 (2011)CrossRefGoogle Scholar
  12. [12]
    MALASHETTY, M. S., TAN, W. C., and SWAMY, M. The onset of double diffusive convection in a binary viscoelastic fluid saturated anisotropic porous layer. Physics of Fluids, 21, 084101–084111 (2009)CrossRefzbMATHGoogle Scholar
  13. [13]
    AWAD, F. G., SIBANDA, P., and MOTSA, S. S. On the linear stability analysis of a Maxwell fluid with double-diffusive convection. Applied Mathematical Modeling, 34, 3509–3517 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  14. [14]
    LARSON, R. G. Instabilities in viscoelastic flows. Rheological Acta, 31, 213–221 (1992)CrossRefGoogle Scholar
  15. [15]
    PERKINS, T. T., QUAKE, S. R., SMITH, D. E., and CHU, S. Relaxation of a single DNA molecule observed by optical microscopy. Science, 264, 822–826 (1994)CrossRefGoogle Scholar
  16. [16]
    PERKINS, T. T., SMITH, D. E., and CHU, S. Single polymer dynamics in an elongational flow. Science, 276, 2016–2021 (1997)CrossRefGoogle Scholar
  17. [17]
    KOLODNER, P. Oscillatory convection in viscoelastic DNA suspensions. Journal of Non-Newtonian Fluid Mechanics, 75, 167–192 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  18. [18]
    CELIA, M. A., KINDRED, J. S., and HERRERA, I. Contaminant transport and biodegrasdation I: a numerical model for reactive transport in porous media. Water Resources Research, 25, 1141–1148 (1989)CrossRefGoogle Scholar
  19. [19]
    CHEN, B., CUNNINGHAM, A., EWING, R., PERALTA, R., and VISSER, E. Two-dimensional modelling of micro scale transport and biotransformation in porous media. Numerical Methods Partial Differential Equations, 10, 65–83 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  20. [20]
    GRIFFITHS, R. W. The transport of multiple components through thermohaline diffusive inter-faces. Deep-Sea Research, 26, 383–397 (1979)CrossRefGoogle Scholar
  21. [21]
    TURNER, J. S. Multicomponent convection. Annual Review of Fluid Mechanics, 17, 11–44 (1985)CrossRefGoogle Scholar
  22. [22]
    MOROZ, I. M. Multiple instabilities in a triply diffusive system. Studies in Applied Mathematics, 80, 137–164 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  23. [23]
    COX, S. M. and MOROZ, I. M. Multiple bifurcations in triple convection with non-ideal boundary conditions. Physica D: Nonlinear Phenomena, 93, 1–22 (1996)CrossRefzbMATHGoogle Scholar
  24. [24]
    PEARLSTEIN, A. J., HARRIS, R. M., and TERRONES, G. The onset of convective instability in a triply diffusive of fluid layer. Journal of Fluid Mechanics, 202, 443–465 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  25. [25]
    TERRONES, G. and PEARLSTEIN, A. J. The onset of convection in a multicomponent fluid layer. Physics of Fluids, 1, 845–853 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  26. [26]
    TERRONES, G. Cross diffusion effects on the stability criteria in a triply diffusive system. Physics of Fluids, 5, 2172–2182 (1993)CrossRefzbMATHGoogle Scholar
  27. [27]
    LOPEZ, A. R., ROMERO, L. A., and PEARLSTEIN, A. J. Effect of rigid boundaries on the onset of convective instability in a triply diffusive fluid layer. Physics of Fluids, 2, 896–902 (1990)zbMATHGoogle Scholar
  28. [28]
    SHIVAKUMARA, I. S. and NAVEEN-KUMAR, S. B. Linear and weakly nonlinear triple diffusive convection in a couple stress fluid layer. International Journal of Heat and Mass Transfer, 68, 542–553 (2015)CrossRefGoogle Scholar
  29. [29]
    RUDRAIAH, N. and VORTMEYER, D. Influence of permeability and of a third diffusing compo-nent upon the onset of convection in a porous medium. International Journal of Heat and Mass Transfer, 25, 457–464 (1982)CrossRefzbMATHGoogle Scholar
  30. [30]
    POULIKAKOS, D. Effect of a third diffusing component on the onset of convection in a horizontal layer. Physics of Fluids, 28, 3172–3174 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  31. [31]
    TRACEY, J. Multi-component convection-diffusion in a porous medium. Continuum Mechanics and Thermodynamics, 8, 361–381 (1996)CrossRefzbMATHGoogle Scholar
  32. [32]
    RIONERO, S. Long-time behaviour of multi-component fluid mixtures in porous media. Interna-tional Journal of Engineering Science, 48, 1519–1533 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  33. [33]
    RIONERO, S. Triple diffusive convection in porous media. Acta Mechanica, 224, 447–458 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  34. [34]
    ZHAO, M., WANG, S., and ZHANG, Q. Onset of triply diffusive convection in a Maxwell fluid saturated porous layer. Applied Mathematical Modeling, 38, 2345–2352 (2014)MathSciNetCrossRefGoogle Scholar
  35. [35]
    RAGHUNATHA, K. R., SHIVAKUMARA, I. S., and SHANKAR, B. M. Weakly nonlinear stabil-ity analysis of triple diffusive convection in a Maxwell fluid saturated porous layer. Applied Mathe-matics and Mechanics (English Edition), 39, 153–168 (2018) CrossRefzbMATHGoogle Scholar
  36. [36]
    ALISHAEV, M. G. and MIRZADJANZADE, A. K. For the calculation of delay phenomenon in filtration theory. Izvestya Vuzov Neft′i Gaz, 6, 71–77 (1975)Google Scholar
  37. [37]
    KHUZHAVOROV, B., AURIAULT, J. L., and ROVER, P. Derivation of macroscopic filtration law for transient linear viscoelastic fluid flow in porous media. International Journal of Engineering Science, 38, 487–504 (2000)MathSciNetCrossRefGoogle Scholar
  38. [38]
    TAN, W. and MASUOKA, T. Stability analysis of a Maxwell fluid in a porous medium heated from below. Physics Letters A, 360, 454–460 (2007)CrossRefzbMATHGoogle Scholar
  39. [39]
    VADASZ, P. Coriolis effect on gravity-driven convection in a rotating porous layer heated from below. Journal of Fluid Mechanics, 376, 351–375 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  40. [40]
    PLUTCHOK, G. J. and JOZEF, L. K. Predicting steady and oscillatory shear rheological proper-ties of CMC and guar gum blends from concentration and molecular weight data. Journal of Food Science, 51, 1284–1288 (1986)CrossRefGoogle Scholar
  41. [41]
    HAO, W. and FRIEDMAN, A. The LDL-HDL profile determines the risk of atherosclerosis: a mathematical model. PLoS One, 9, 1–15 (2014)Google Scholar

Copyright information

© Shanghai University and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsBangalore UniversityBangaloreIndia

Personalised recommendations