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Fluid convection in a rotating porous layer under modulated temperature on the boundaries

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Abstract

The linear stability of thermal convection in a rotating horizontal layer of fluid-saturated porous medium, confined between two rigid boundaries, is studied for temperature modulation, using Brinkman’s model. In addition to a steady temperature difference between the walls of the porous layer, a time-dependent periodic perturbation is applied to the wall temperatures. Only infinitesimal disturbances are considered. The combined effect of rotation, permeability and modulation of walls’ temperature on the stability of flow through porous medium has been investigated using Galerkin method and Floquet theory. The critical Rayleigh number is calculated as function of amplitude and frequency of modulation, Taylor number, porous parameter and Prandtl number. It is found that both, rotation and permeability are having stabilizing influence on the onset of thermal instability. Further it is also found that it is possible to advance or delay the onset of convection by proper tuning of the frequency of modulation of the walls’ temperature.

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Abbreviations

a :

Horizontal wave number \(\left({a_x^2 +a_y^2}\right)^{1/2}\)

a c :

Critical wave number

d :

Depth of the porous layer

g :

Gravitational acceleration

k :

Permeability of the porous medium

κf :

Thermal conductivity of the fluid

κs :

Thermal conductivity of the solid

κ m :

δκf + (1 − δ)κs, effective thermal conductivity of porous media

p :

Pressure

P l :

Porous parameter, k/d 2

P r :

Prandtl number, ν /κ

R :

Thermal Rayleigh number, \(\frac{\alpha g\Delta Td^3}{\nu \kappa}\)

Ω:

Angular velocity vector (0, 0, Ω)

T :

Taylor number 4Ω2 d 4/ν2

R c :

Critical Rayleigh number

T :

Temperature

θ:

Perturbed temperature

ΔT :

Temperature difference between the walls

V :

Mean filter velocity, (u, v, w)

x, y, z :

Space coordinates

(ρ c p )f :

Heat capacity of the fluid

(ρ c p )s :

Heat capacity of the solid

(ρ c p ) m :

δ(ρ c p )f + (1 − δ)(ρ c p )s relative heat capacity of the porous medium

T S (z):

Steady temperature field

To(z, t):

Oscillating temperature field

Greek symbols :

 

ζ:

Z-component of vorticity

α:

Coefficient of thermal expansion

\(\varepsilon\) :

Amplitude of modulation

δ:

Porosity

γ:

Heat capacity ratio, (ρ c p ) m /(ρ c p )f

κ:

Effective thermal diffusivity, κ m /(ρ c p )f

μ:

Coefficient of viscosity

ν:

Kinematic viscosity, μ/ρ R

ρ:

Density

ω:

Modulation frequency

\(\phi\) :

Phase angle

Other symbols :

 

\(\nabla_1^2\) :

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

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

\(\nabla_1^2 +\frac{\partial^2}{\partial z^2}\)

D:

\(\frac{\partial} {\partial z}\)

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  1. Department of Mathematics and Statistics, Jai Narain Vyas University, Jodhpur, 342005, India

    B. S. Bhadauria

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Bhadauria, B.S. Fluid convection in a rotating porous layer under modulated temperature on the boundaries. Transp Porous Med 67, 297–315 (2007). https://doi.org/10.1007/s11242-006-9027-x

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