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Linear and nonlinear stability analysis of binary Maxwell fluid convection in a porous medium

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Abstract

The stability analysis of the quiescent state in a Maxwell fluid-saturated densely packed porous medium subject to vertical concentration and temperature gradients is presented. A single phase model with local thermal equilibrium between the porous matrix and the Maxwell fluid is assumed. The critical Darcy–Rayleigh numbers and the corresponding wave numbers for the onset of stationary and oscillatory convection are determined. A Lorenz like system is obtained for weakly nonlinear stability analysis.

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Abbreviations

A, B, C, D, E :

Time dependent amplitudes

\(\bar{A},\bar{B},\bar{C},\bar{D},\bar{E}\) :

Scaled time dependent amplitudes

d :

Depth of porous medium

Da :

Darcy number, K/d 2

D f :

Dufour parameter, \(k_1\Updelta S/k^{\prime}\Updelta T\)

g :

Acceleration due to gravity

K :

Permeability constant

k, k′:

Thermal and solutal diffusivity

k 1, k 2 :

Heat and mass flux parameters

Le :

Lewis number, k/k′

M :

Heat capacity ratio, (ρc) f /(ρc) p

N :

Buoyancy ratio, \(\alpha^{\prime}\Updelta S/\alpha\Updelta T\)

Ra :

Thermal Rayleigh number, \(g\alpha d^{3}\Updelta T/\nu k\)

Ra D :

Darcy–Rayleigh number, Ra Da

Rs :

Solutal Rayleigh number, \(g\alpha^{\prime} d^{3}\Updelta S/\nu k\)

Rs D :

Solutal Darcy–Rayleigh number, LeNRa D

S :

Solute concentration

Sr :

Soret number, \(k_2\Updelta T/k\Updelta S\)

t :

Dimensional time

T :

Temperature

\(t^{\ast}\) :

Dimensionless time

v :

Darcian velocity

w :

z-Component of Darcian velocity

\(w^{\ast}\) :

Non-dimensional velocity

W :

Velocity function

x, z :

Streamwise and normal coordinate axes

\(x^{\ast},z^{\ast}\) :

Dimensionless counterparts of x, z

α:

Coefficient of thermal expansion

α′:

Coefficient of solutal expansion

\(\epsilon\) :

Scaled normalized porosity, \(\bar{\epsilon}M\)

\(\bar{\epsilon}\) :

Normalized porosity

θ:

Non-dimensional temperature

\(\Uptheta\) :

Temperature function

λ:

Dimensionless relaxation time, \(\bar{\lambda}k/d^{2}\)

\(\bar{\lambda}\) :

Stress relaxation

μ:

Dynamic viscosity

ν:

Kinematic viscosity

ρ:

Fluid density

ρ f :

Density at current state

σ:

Growth rate

ω:

Frequency

τ:

Dimensionless time

ϕ:

Non-dimensional concentration

\(\Upphi\) :

Concentration function

b :

Basic state

o:

Reference state

1,2:

Lower/upper wall value

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Acknowledgments

The authors wish to acknowledgement the many fruitful discussions with Prof PG Siddheshwar, Department of Mathematics, Bangalore University, Central College Campus, Bangalore during the preparation of this paper.

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Authors and Affiliations

  1. School of Mathematical Sciences, University of KwaZulu-Natal, P Bag X01, Scottsville, Pietermaritzburg, 3209, South Africa

    Mahesha Narayana, P. Sibanda, S. S. Motsa & P. A. Lakshmi-Narayana

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  1. Mahesha Narayana
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  2. P. Sibanda
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  3. S. S. Motsa
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  4. P. A. Lakshmi-Narayana
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Correspondence to P. Sibanda.

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Narayana, M., Sibanda, P., Motsa, S.S. et al. Linear and nonlinear stability analysis of binary Maxwell fluid convection in a porous medium. Heat Mass Transfer 48, 863–874 (2012). https://doi.org/10.1007/s00231-011-0939-9

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