Heat and Mass Transfer

, Volume 48, Issue 5, pp 863–874

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

  • Mahesha Narayana
  • P. Sibanda
  • S. S. Motsa
  • P. A. Lakshmi-Narayana
Original

DOI: 10.1007/s00231-011-0939-9

Cite this article as:
Narayana, M., Sibanda, P., Motsa, S.S. et al. Heat Mass Transfer (2012) 48: 863. doi:10.1007/s00231-011-0939-9
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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.

List of symbols

ABCDE

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/d2

Df

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

g

Acceleration due to gravity

K

Permeability constant

kk

Thermal and solutal diffusivity

k1, k2

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

RaD

Darcy–Rayleigh number, Ra Da

Rs

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

RsD

Solutal Darcy–Rayleigh number, LeNRaD

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

xz

Streamwise and normal coordinate axes

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

Dimensionless counterparts of xz

Greek symbols

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

ρf

Density at current state

σ

Growth rate

ω

Frequency

τ

Dimensionless time

ϕ

Non-dimensional concentration

\(\Upphi\)

Concentration function

Subscripts

b

Basic state

o

Reference state

1,2

Lower/upper wall value

Copyright information

© Springer-Verlag 2011

Authors and Affiliations

  • Mahesha Narayana
    • 1
  • P. Sibanda
    • 1
  • S. S. Motsa
    • 1
  • P. A. Lakshmi-Narayana
    • 1
  1. 1.School of Mathematical SciencesUniversity of KwaZulu-NatalPietermaritzburgSouth Africa