Skip to main content
SpringerLink
Log in

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

  • Published:
Applied Mathematics and Mechanics Aims and scope Submit manuscript

Abstract

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Abbreviations

d :

depth of the porous layer

g :

gravitational acceleration

K :

permeability of the porous medium

k :

unit vector in the vertical direction

M :

ratio of heat capacities

p :

pressure

Pr D :

Darcy-Prandtl number

q :

velocity vector

R S i :

solute Darcy-Rayleigh number of the ith-component

R T :

thermal Darcy-Rayleigh number

t :

time

x, y, z :

space coordinates

α :

horizontal wave number

α T :

thermal expansion coefficient

α S i :

solute analogue of αT, i = 1, 2

ε :

porosity

κ T :

thermal diffusivity

κ S i :

solute diffusivity, i = 1, 2

λ1 :

stress relaxation time

λ2 :

strain retardation time

Λ1 :

stress relaxation parameter

Λ2 :

strain retardation parameter

μ :

dynamic viscosity

ν :

kinematic viscosity

ρ :

fluid density

σ :

growth term

τ i :

ratio of diffusivities, i = 1, 2

ψ :

stream function

b:

basic state

L:

lower boundary

U:

upper boundary

*:

dimensionless variable

References

  1. HORTON, C.W. and ROGERS, F. T. Convection currents in a porous medium. Journal of Applied Physics, 16, 367–370 (1945)

    Article  MathSciNet  MATH  Google Scholar 

  2. LAPWOOD, E. R. Convection of a fluid in a porous medium. Mathematical Proceedings of the Cambridge Philosophical Society, 44, 508–512 (1948)

    Article  MathSciNet  MATH  Google Scholar 

  3. NIELD, D. A. and BEJAN, A. Convection in Porous Media, 5th ed., Springer, New York (2017)

    Book  MATH  Google Scholar 

  4. STRAUGHAN, B. Stability and Wave Motion in Porous Media, Springer, New York (2008)

    MATH  Google Scholar 

  5. STRAUGHAN, B. Convection with Local Thermal Non-Equilibrium and Microfluidic Effects, Springer, New York (2015)

    Book  MATH  Google Scholar 

  6. VAFAI, K. Handbook of Porous Media, Marcel Dekker, New York (2000)

    MATH  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  14. LARSON, R. G. Instabilities in viscoelastic flows. Rheological Acta, 31, 213–221 (1992)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  16. PERKINS, T. T., SMITH, D. E., and CHU, S. Single polymer dynamics in an elongational flow. Science, 276, 2016–2021 (1997)

    Article  Google Scholar 

  17. KOLODNER, P. Oscillatory convection in viscoelastic DNA suspensions. Journal of Non-Newtonian Fluid Mechanics, 75, 167–192 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  20. GRIFFITHS, R. W. The transport of multiple components through thermohaline diffusive inter-faces. Deep-Sea Research, 26, 383–397 (1979)

    Article  Google Scholar 

  21. TURNER, J. S. Multicomponent convection. Annual Review of Fluid Mechanics, 17, 11–44 (1985)

    Article  Google Scholar 

  22. MOROZ, I. M. Multiple instabilities in a triply diffusive system. Studies in Applied Mathematics, 80, 137–164 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  25. TERRONES, G. and PEARLSTEIN, A. J. The onset of convection in a multicomponent fluid layer. Physics of Fluids, 1, 845–853 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  26. TERRONES, G. Cross diffusion effects on the stability criteria in a triply diffusive system. Physics of Fluids, 5, 2172–2182 (1993)

    Article  MATH  Google Scholar 

  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)

    MATH  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  31. TRACEY, J. Multi-component convection-diffusion in a porous medium. Continuum Mechanics and Thermodynamics, 8, 361–381 (1996)

    Article  MATH  Google Scholar 

  32. RIONERO, S. Long-time behaviour of multi-component fluid mixtures in porous media. Interna-tional Journal of Engineering Science, 48, 1519–1533 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  33. RIONERO, S. Triple diffusive convection in porous media. Acta Mechanica, 224, 447–458 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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) https://doi.org/10.1007/s10483-018-2298-6

    Article  MATH  Google Scholar 

  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. 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)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  Google Scholar 

  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 

Download references

Acknowledgements

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

Author information

Authors and Affiliations

  1. Department of Mathematics, Bangalore University, Bangalore, 560056, India

    K. R. Raghunatha & I. S. Shivakumara

Authors
  1. K. R. Raghunatha
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. I. S. Shivakumara
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to I. S. Shivakumara.

Additional information

Citation: RAGHUNATHA, K. R. and SHIVAKUMARA, I. S. Stability of triple diffusive convection in a viscoelastic fluid-saturated porous layer. Applied Mathematics and Mechanics (English Edition), 39(10), 1385–1410 (2018) https://doi.org/10.1007/s10483-018-2376-8

Project supported by the Innovation in Science Pursuit for the Inspired Research (INSPIRE) Program (No. DST/INSPIRE Fellowship/[IF 150253])

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Raghunatha, K.R., Shivakumara, I.S. Stability of triple diffusive convection in a viscoelastic fluid-saturated porous layer. Appl. Math. Mech.-Engl. Ed. 39, 1385–1410 (2018). https://doi.org/10.1007/s10483-018-2376-8

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10483-018-2376-8

Key words

Chinese Library Classification

2010 Mathematics Subject Classification

Search

Navigation