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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. Shivakumara
Article
  • 412 Downloads

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.

Key words

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

Abbreviations

Nomenclature

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

PrD

Darcy-Prandtl number

q

velocity vector

RS i

solute Darcy-Rayleigh number of the ith-component

RT

thermal Darcy-Rayleigh number

t

time

x, y, z

space coordinates

Greek symbols

α

horizontal wave number

αT

thermal expansion coefficient

αS i

solute analogue of αT, i = 1, 2

ε

porosity

κT

thermal diffusivity

κS 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

τi

ratio of diffusivities, i = 1, 2

ψ

stream function

Subscripts/superscripts

b

basic state

L

lower boundary

U

upper boundary

*

dimensionless variable

Chinese Library Classification

O357.3 O175.8 

2010 Mathematics Subject Classification

76E06 76A10 76S05 

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Notes

Acknowledgements

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

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Copyright information

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

Authors and Affiliations

  1. 1.Department of MathematicsBangalore UniversityBangaloreIndia

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