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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
Article
  • 18 Downloads

Abstract

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.

Keywords

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

Mathematics Subject Classification

70K20 70K50 34A34 76E06 76R50 76A10 

Notes

Acknowledgements

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]).

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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

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