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Double diffusive convection in a porous medium with modulated temperature on the boundaries

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Abstract

The effect of temperature modulation on the onset of double diffusive convection in a sparsely packed porous medium is studied by making linear stability analysis, and using Brinkman-Forchheimer extended Darcy model. The temperature field between the walls of the porous layer consists of a steady part and a time dependent periodic part that oscillates with time. Only infinitesimal disturbances are considered. The effect of permeability and thermal modulation on the onset of double diffusive convection has been studied using Galerkin method and Floquet theory. The critical Rayleigh number is calculated as a function of frequency and amplitude of modulation, Vadasz number, Darcy number, diffusivity ratio, and solute Rayleigh number. Stabilizing and destabilizing effects of modulation on the onset of double diffusive convection have been obtained. The effects of other parameters are also discussed on the stability of the system. Some results as the particular cases of the present study have also been obtained. Also the results corresponding to the Brinkman model and Darcy model have been compared.

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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

V :

Mean filter velocity, (u, v, w)

p :

Pressure

S :

Solute concentration

T :

Temperature

θ:

Perturbed temperature

ΔS :

Salinity difference between the walls

ΔT :

Temperature difference between the walls

D a :

Darcy number, k/d 2

P r :

Prandtl number, ν /κ T

R a :

Darcy Rayleigh number, \(\frac{\alpha g \Delta T k d}{\nu \kappa _T }\)

\(R_a^O \) :

Oscillatory Rayleigh number

R ac :

Critical Rayleigh number

R S :

Solute Rayleigh number, \(\frac{\beta \,g\,\Delta S\,k\,d}{\nu \kappa _S }\)

\(R_S^\ast \) :

Crossover solute Rayleigh number

x, y, z :

Space coordinates

Other symbols :

 

\(\nabla _1^2 \) :

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

∇ 2 :

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

D :

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

Greek symbols :

 

α:

Coefficient of thermal expansion

β:

Coefficient of solute expansion

\(\varepsilon\) :

Amplitude of modulation

γ:

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

δ:

Porosity

K f :

Thermal conductivity of the fluid

K SO :

Thermal conductivity of the solid

K m :

Effective thermal conductivity of Porous media, δK f + (1−δ )K SO

κ T :

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

κ S :

Solute diffusivity

μ:

coefficient of viscosity

ν:

Kinematic viscosity μ /ρR

ρ:

Density

ω:

Modulation frequency

\(\phi \) :

Phase angle

σ:

Growth rate (a complex number)

τ:

Diffusivity ratio, κ S /κ T

Subscripts :

 

b :

Basic state

c :

Critical

f :

Fluid

R :

Reference value

Superscript :

 

/:

Perturbed state

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Authors and Affiliations

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

    B. S. Bhadauria

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Bhadauria, B.S. Double diffusive convection in a porous medium with modulated temperature on the boundaries. Transp Porous Med 70, 191–211 (2007). https://doi.org/10.1007/s11242-006-9095-y

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