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Double-Diffusive Convection in a Saturated Anisotropic Porous Layer with Internal Heat Source

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Abstract

In the present study, double-diffusive convection in an anisotropic porous layer with an internal heat source, heated and salted from below, has been investigated. The generalized Darcy model is employed for the momentum equation. The fluid and solid phases are considered to be in equilibrium. Linear and nonlinear stability analyses have been performed. For linear theory normal mode technique has been used, while nonlinear analysis is based on a minimal representation of truncated Fourier series. Heat and mass transfers across the porous layer have been obtained in terms of Nusselt number Nu and Sherwood number Sh, respectively. The effects of internal Rayleigh number, anisotropy parameters, concentration Rayleigh number, and Vadasz number on stationary, oscillatory, and weak nonlinear convection are shown graphically. The transient behaviors of Nusselt number and Sherwood number have been investigated by solving the finite amplitude equations using a numerical method. Streamlines, isotherms, and isohalines are drawn for both steady and unsteady (time-dependent) cases. The results obtained, during the above analyses, have been presented graphically, and the effects of various parameters on heat and mass transfers have been discussed.

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Abbreviations

a :

Non-dimensionalized wave number

a c :

Critical wave number

d :

Depth of the porous layer

Da :

Darcy number, K z /d 2

K :

Permeability of the porous medium, \({K_{x}(\hat{i}\hat{i}+\hat{j}\hat{j})+K_{z}(\hat{k}\hat{k})}\)

p :

Pressure

g :

Gravitational acceleration (0, 0, −g)

q :

Velocity (u, v, w)

Pr :

Prandtl number, \({\nu/{\kappa_{T}}_{z}}\)

Ra T :

Thermal Rayleigh number \({Ra_{T}=(\alpha_{T}gK_{z}d(\Delta T)/\nu {\kappa_{T}}_{z})}\)

Ra S :

Concentration Rayleigh number \({Ra_{S}=(\alpha_{S}gK_{z}d(\Delta S)/\nu {\kappa_{S}})}\)

R i :

Internal Rayleigh number (\({Qd^{2}/{\kappa_{T}}_{z}}\))

Ra T, c :

Critical Rayleigh number

t :

Time

Va :

Vadasz number, δ Pr/Da

T :

Temperature

ΔT :

Temperature difference between the walls

S :

Concentration

ΔS :

Concentration difference between the walls

\({\bar{H}}\) :

Rate of heat transport per unit area

\({\bar{J}}\) :

Rate of mass transport per unit area

(x, y, z):

Space co-ordinate

α T :

Thermal expansion coefficient

α S :

Concentration expansion coefficient

κ T :

Thermal diffusivity of the fluid

κ S :

Concentration diffusivity of the fluid

τ :

Diffusivity ratio, \({{\kappa_{S}/\kappa_{T}}_{z}}\)

ξ :

Mechanical anisotropy parameter

δ :

Porosity

ρ :

Density

μ :

Dynamic viscosity

ν :

Kinematic viscosity, μ/ρ 0

σ :

Growth rate of fluid

η :

Thermal anisotropy parameter

α :

Rescaled wave number, a 2/π 2

ψ :

Stream function

b:

Basic state

c:

Critical

0:

Reference state

\({\hat{i}}\) :

Unit normal vector in x-direction

\({\hat{j}}\) :

Unit normal vector in y-direction

\({\hat{k}}\) :

Unit normal vector in z-direction

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

\({\dfrac{{\rm \partial}^{2}}{\partial x^{2}}+\dfrac{\partial^{2}}{\partial y^{2}}}\) , horizontal Laplacian

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

\({\nabla^{2}_{h} + \dfrac{\partial^{2}}{\partial z^{2}}}\)

D :

d/dz

i :

\({\sqrt{-1}}\)

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Authors and Affiliations

  1. Department of Applied Mathematics and Statistics, School for Physical Sciences, Babasaheb Bhimrao Ambedkar University, Lucknow, UP, 226025, India

    B. S. Bhadauria

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Bhadauria, B.S. Double-Diffusive Convection in a Saturated Anisotropic Porous Layer with Internal Heat Source. Transp Porous Med 92, 299–320 (2012). https://doi.org/10.1007/s11242-011-9903-x

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