Effect of cross-diffusion on the stability of a triple-diffusive Oldroyd-B fluid layer

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


The onset and stability of a triple cross-diffusive viscoelastic fluid layer is investigated. The rheology of viscoelastic fluid is approximated by the nonlinear Oldroyd-B constitutive equation which encompasses Maxwell and Newtonian fluid models as special cases. By performing the linear instability analysis, analytical expression for the occurrence of stationary and oscillatory convection is obtained. The numerical results show that the elasticity and cross-diffusion effects reinforce together in displaying complex dynamical behavior on the system. The presence of cross-diffusion is found to either stabilize or destabilize the system depending on the strength of species concentration as well as elasticity of the fluid and also alters the nature of convective instability. The disconnected closed oscillatory neutral curve lying well below the stationary neutral curve is observed to be convex in its shape in contrast to quasiperiodic bifurcation from the quiescent basic state noted in the case of Newtonian fluids. This striking feature is attributed to the viscoelasticity of the fluid. By performing a weakly nonlinear stability analysis, the stability of bifurcating solution is discussed. It is worth reporting that the viscoelastic parameters significantly influence the stability of stationary bifurcation though the stationary onset is unaffected by viscoelasticity. Besides, subcritical instability is occurs and the critical Rayleigh number at which such an instability is possible decreases in the presence of cross-diffusion terms. The results of Maxwell and Newtonian fluids are delineated as particular cases from the present study.


Instability Cross-diffusion terms Triple-diffusive convection Nonlinear stability Bifurcation Perturbation method 

Mathematics Subject Classification

70K20 70K50 34A34 76E06 76R50 76A10 



The authors thank the reviewer for the constructive comments and useful suggestions which helped in improving the paper considerably. One of the authors K. R. Raghunatha (SRF) wishes to thank the Department of Science and Technology, New Delhi, for Granting him a fellowship under the Innovation in Science Pursuit for the Inspired Research (INSPIRE) Program (No. DST/INSPIRE Fellowship/[IF 150253]).


  1. 1.
    Onsager, L., Fuoss, R.: Irreversible processes in electrolytes, diffusion, conductance and viscous flow in arbitrary mixtures of strong electrolytes. J. Phys. Chem. 36, 2689–2778 (1932)CrossRefGoogle Scholar
  2. 2.
    Baldwin, R.L., Dunlop, P.J., Gosting, L.J.: Interacting flows in liquid diffusion: equations for evaluation of the diffusion coefficients from moments of the refractive index gradient curves. J. Am. Chem. Soc. 77, 5235–5238 (1955)CrossRefGoogle Scholar
  3. 3.
    Mimura, M., Kawasaki, K.: Spatial segregation in competitive interaction diffusion equations. J. Math. Biol. 9, 49–64 (1980)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Turner, J.: Double-diffusive phenomena. Annu. Rev. Fluid Mech. 6, 37–54 (1974)CrossRefGoogle Scholar
  5. 5.
    Huppert, H.E., Turner, J.S.: Double-diffusive convection. J. Fluid Mech. 106, 299–329 (1981)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Platten, J.K., Legros, J.C.: Convection in Liquids. Springer, Berlin (1984)CrossRefGoogle Scholar
  7. 7.
    Garaud, P.: Double-diffusive convection at low Prandtl number. Annu. Rev. Fluid Mech. 50, 275–298 (2018)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Turner, J.S.: Multicomponent convection. Annu. Rev. Fluid Mech. 17, 11–44 (1985)CrossRefGoogle Scholar
  9. 9.
    Corriel, S.R., McFadden, G.B., Voorhees, P.W., Sekerka, R.F.: Stability of a planar interface solidification of a multicomponent system. J. Crystal Growth 82, 300–313 (1987)Google Scholar
  10. 10.
    Pearlstein, A.J., Harris, R.M., Terrones, G.: The onset of convective instability in a triply diffusive of fluid layer. J. Fluid Mech. 202, 443–465 (1989)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Cox, S.M., Moroz, I.M.: Multiple bifurcations in triple convection with non-ideal boundary conditions. Physica D 93, 122 (1996)CrossRefGoogle Scholar
  12. 12.
    Vanag, V.K., Epstein, I.R.: Cross-diffusion and pattern formation in reaction–diffusion systems. Phys. Chem. Chem. Phys. 11, 897–912 (2009)CrossRefGoogle Scholar
  13. 13.
    Annunziata, O., Vergara, A., Paduano, L., Sartorio, R., Miller, D.G., Albright, J.G.: Quaternary diffusion coefficients in a protein- polymer- salt-water system determined by Rayleigh interferometry. J. Phys. Chem. 113, 13446–13453 (2009)CrossRefGoogle Scholar
  14. 14.
    Griffiths, R.W.: The influence of a third diffusing component upon the onset of convection. J. Fluid Mech. 92, 659–670 (1979)CrossRefGoogle Scholar
  15. 15.
    Griffiths, R.W.: A note on the formulation of “salt-finger” and “diffusive” interfaces in three-component systems. Int. J. Heat Mass Transf. 22, 1687–1693 (1979)CrossRefGoogle Scholar
  16. 16.
    Terrones, G., Pearlstein, A.J.: The onset of convection in a multicomponent fluid layer. Phys. Fluids A 1, 845–853 (1989)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Lopez, A.R., Romero, L.A., Pearlstein, A.J.: Effects of rigid boundaries on the onset of convective instability in a triply diffusive fluid layer. Phys. Fluids 2, 896–902 (1990)CrossRefGoogle Scholar
  18. 18.
    Terrones, G.: Cross diffusion effects on the stability criteria in a triply diffusive system. Phys. Fluids A 5, 2172–2182 (1993)CrossRefGoogle Scholar
  19. 19.
    Ryzhkov, I.I., Shevtsova, V.M.: On thermal diffusion and convection in multicomponent mixtures with application to the thermo gravitational column. Phys. Fluids 19, 1–17 (2007)CrossRefGoogle Scholar
  20. 20.
    Shivakumara, I.S., Naveen Kumar, S.B.: Linear and weakly nonlinear triple diffusive convection in a couple stress fluid layer. Int. J. Heat Mass Transf. 68, 542–553 (2014)CrossRefGoogle Scholar
  21. 21.
    Rosenblat, S.: Thermal convection in a viscoelastic liquid. J. NonNewton. Fluid Mech. 21, 201–223 (1986)CrossRefGoogle Scholar
  22. 22.
    Li, Z., Khayat, R.E.: Finite-amplitude Rayleigh–Benard convection and pattern selection for viscoelastic fluids. J. Fluid Mech. 529, 221–251 (2005)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Park, H.M., Lee, H.S.: Hopf bifurcations of viscoelastic fluids heated from below. J. NonNewton. Fluid Mech. 66, 1–34 (1996)CrossRefGoogle Scholar
  24. 24.
    Khayat, R.E.: Non-linear over stability in the thermal convection of viscoelastic fluids. J. NonNewton. Fluid. 58, 331–356 (1995)CrossRefGoogle Scholar
  25. 25.
    Kolodner, P.: Oscillatory convection in viscoelastic DNA suspensions. J. NonNewton. Fluid Mech. 75, 167–192 (1998)CrossRefGoogle Scholar
  26. 26.
    Martinez-Mardones, J., Tiemann, R., Walgraef, D.: Rayleigh–Bénard convection in binary viscoelastic fluid. Physica A 283, 233–236 (2000)CrossRefGoogle Scholar
  27. 27.
    Malashetty, M.S., Swamy, M.: The onset of double diffusive convection in a viscoelastic fluid layer. J. NonNewton. Fluid. 165, 1129–1138 (2010)CrossRefGoogle Scholar
  28. 28.
    Awad, F.G., Sibanda, P., Motsa, S.S.: On the linear stability analysis of a Maxwell fluid with double- diffusive convection. Appl. Math. Model. 34, 3509–3517 (2010)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Raghunatha, K.R., Shivakumara, I.S.: Double-diffusive convection in an Oldroyd-B fluid layer-stability of bifurcating equilibrium solutions. J. Appl. Fluid Mech. 12, 85–94 (2019)CrossRefGoogle Scholar
  30. 30.
    Noulty, R.A., Leaist, D.G.: Quaternary diffusion in aqueous KCl–K2PO4–H3PO4 mixtures. J. Phys. Chem. 91, 1655–1658 (1987)CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • K. R. Raghunatha
    • 1
    • 2
  • I. S. Shivakumara
    • 1
    Email author
  • M. S. Swamy
    • 1
    • 3
  1. 1.Department of MathematicsBangalore UniversityBangaloreIndia
  2. 2.Department of MathematicsDavangere UniversityDavangereIndia
  3. 3.Government First Grade CollegeGulbargaIndia

Personalised recommendations