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Effect of Thermal/Gravity Modulation on the Onset of Convection in a Maxwell Fluid Saturated Porous Layer

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Abstract

The effect of thermal/gravity modulation on the onset of convection in a Maxwell fluid saturated porous layer is investigated by a linear stability analysis. Modified Darcy–Maxwell model is used to describe the fluid motion. The regular perturbation method based on the small amplitude of modulation is employed to compute the critical Rayleigh number and the corresponding wavenumber. The stability of the system characterized by a correction Rayleigh number is calculated as a function of the viscoelastic parameter, Darcy–Prandtl number, normalized porosity, and the frequency of modulation. It is found that the low frequency symmetric thermal modulation is destabilizing while moderate and high frequency symmetric modulation is always stabilizing. The asymmetric modulation and lower wall temperature modulations are, in general, stabilizing while the system becomes unstable for large values of Darcy–Prandtl number and for small frequencies. It is shown that in general the gravity modulation produces a stabilizing effect on the onset of convection for moderate and high frequency. The small frequency gravity modulation is found to have destabilizing effect on the stability of the system.

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Abbreviations

a :

Wavenumber

c :

Specific heat of solid

c p :

Specific heat of fluid

d :

Height of the porous layer

Da :

Darcy number, k/d 2

g :

Gravitational acceleration (0, 0, −g)

k :

Permeability of the porous layer

l, m :

Horizontal wavenumbers

Pr :

Prandtl number, ν/κ

Pr D :

Darcy–Prandtl number, \({Pr_D = {\phi ^{2}Pr}/{Da}}\)

p :

Pressure

q :

Velocity vector (u, v, w)

Ra :

Rayleigh number, β gΔTdk/νκ

t :

Time

T :

Temperature

ΔT :

Temperature difference between the walls

x, y, z :

Space coordinates

β :

Thermal expansion coefficient

\({\phi}\) :

Porosity

ε :

Amplitude of modulation

Ω:

Frequency of modulation

φ :

Phase angle

γ :

Ratio of specific heat (ρ c)m /(ρ c p )f

κ :

Thermal diffusivity

\({\bar{{\lambda}}}\) :

Stress relaxation parameter

Γ:

Deborah number, \({{\bar{{\lambda }}\kappa }/{\phi d^{2}}}\)

χ :

Normalized porosity, \({\phi /\gamma}\)

μ :

Dynamic viscosity

ν :

Kinematic viscosity

ρ :

Density

ω :

Dimensionless frequency of modulation, Ω d 2 γ/κ

\({\nabla _1^2}\) :

\({\frac{\partial ^{2}}{\partial x^{2}}+\frac{\partial ^{2}}{\partial y^{2}}}\)

\({\nabla ^{2}}\) :

\({\nabla _1^2 +\frac{\partial ^{2}}{\partial z^{2}}}\)

b:

Basic state

c:

Critical

f:

Fluid

m:

Porous medium

0:

Reference value

s:

Solid

*:

Dimensionless quantity

′:

Perturbed quantity

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  1. Department of Mathematics, Gulbarga University, Gulbarga, 585106, India

    M. S. Malashetty & Irfana Begum

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  1. M. S. Malashetty
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  2. Irfana Begum
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Correspondence to M. S. Malashetty.

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Malashetty, M.S., Begum, I. Effect of Thermal/Gravity Modulation on the Onset of Convection in a Maxwell Fluid Saturated Porous Layer. Transp Porous Med 90, 889–909 (2011). https://doi.org/10.1007/s11242-011-9822-x

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