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Weakly nonlinear stability analysis of triple diffusive convection in a Maxwell fluid saturated porous layer

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Abstract

The weakly nonlinear stability of the triple diffusive convection in a Maxwell fluid saturated porous layer is investigated. In some cases, disconnected oscillatory neutral curves are found to exist, indicating that three critical thermal Darcy-Rayleigh numbers are required to specify the linear instability criteria. However, another distinguishing feature predicted from that of Newtonian fluids is the impossibility of quasi-periodic bifurcation from the rest state. Besides, the co-dimensional two bifurcation points are located in the Darcy-Prandtl number and the stress relaxation parameter plane. It is observed that the value of the stress relaxation parameter defining the crossover between stationary and oscillatory bifurcations decreases when the Darcy-Prandtl number increases. A cubic Landau equation is derived based on the weakly nonlinear stability analysis. It is found that the bifurcating oscillatory solution is either supercritical or subcritical, depending on the choice of the physical parameters. Heat and mass transfers are estimated in terms of time and area-averaged Nusselt numbers.

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Abbreviations

d :

depth of the porous layer

Pr D :

Darcy-Prandtl number

g :

gravitational acceleration

q :

velocity vector

K :

permeability of the porous medium

R Si :

solute Darcy-Rayleigh number of the ith component

\(\hat k\) :

unit vector in the vertical direction

R T :

thermal Darcy-Rayleigh number

M :

ratio of heat capacities

t :

time

p :

pressure

x :

y, z, space coordinates.

α :

horizontal wave number

Λ1 :

stress relaxation parameter

α T :

thermal expansion coefficient

μ :

dynamic viscosity

α Si :

solute analog of αT (i = 1, 2)

ν :

kinematic viscosity

ϵ :

porosity

ρ :

fluid density

κ T :

thermal diffusivity

σ :

growth term

κ Si :

solute diffusivity (i = 1, 2)

τ i :

ratio of diffusivity (i = 1, 2)

λ1 :

stress relaxation time

ψ :

stream function.

b:

basic state

U:

upper boundary

L:

lower boundary

*:

dimensionless variable.

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Acknowledgements

One of the authors (K. R. RAGHUNATHA) wishes to thank the 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]). The authors wish to thank the reviewers for their helpful suggestions.

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Authors and Affiliations

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

    K. R. Raghunatha & I. S. Shivakumara

  2. Department of Mathematics, PES University, Bangalore, 560085, India

    B. M. Shankar

Authors
  1. K. R. Raghunatha
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  2. I. S. Shivakumara
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  3. B. M. Shankar
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Correspondence to I. S. Shivakumara.

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Raghunatha, K.R., Shivakumara, I.S. & Shankar, B.M. Weakly nonlinear stability analysis of triple diffusive convection in a Maxwell fluid saturated porous layer. Appl. Math. Mech.-Engl. Ed. 39, 153–168 (2018). https://doi.org/10.1007/s10483-018-2298-6

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