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Effect of Rotation on Thermal Convection in an Anisotropic Porous Medium with Temperature-dependent Viscosity

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Abstract

A linear stability analysis is performed for mono-diffusive convection in an anisotropic rotating porous medium with temperature-dependent viscosity. The Galerkin variant of the weighted residual technique is used to obtain the eigen value of the problem. The effect of Taylor–Vadasz number and the other parameters of the problem are considered for stationary convection in the absence or presence of rotation. Oscillatory convection seems highly improbable. Some new results on the parameters’ influence on convection in the presence of rotation, for both high and low rotation rates, are presented.

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Abbreviations

a :

Horizontal wave number

a c :

Critical wave number

Br D :

Brinkman–Darcy number, ΛDa

c :

Specific heat

c p :

Specific heat at a constant pressure

Da −1 :

Inverse Darcy number (porous parameter), \({\frac{d^{2}}{k_v }}\)

d :

Height of the porous layer

\({\vec {g}}\) :

Gravitational acceleration (0, 0, −g)

k :

Permeability

k :

Permeability tensor, \({\frac{1}{k_h }\hat{{i}}\hat{{i}}+\frac{1}{k_h }\hat{{j}}\hat{{j}}+\frac{1}{k_v }\hat{{k}}\hat{{k}}}\)

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

Unit vector in the vertical direction

(k h ,k h ,k v ):

Permeability along x, y and z-direction

l, m :

Wave numbers

p :

Pressure

p * :

Hydrostatic pressure

p H :

Basic state pressure

Pr:

Prandtl number, \({\frac{\nu \Phi }{\chi_{Tv}}}\)

\({\vec {q}}\) :

(u, v, w), velocity vector

\({{\vec {q}}'}\) :

Velocity of the perturbed state

R :

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

R D :

Darcy–Rayleigh number,RDa

t :

Time

T :

Temperature field

Ta :

Taylor number, \({\frac{4\Omega^{2}d^{4}}{\Phi^{2}\nu^{2}}}\)

T b :

Basic state temperature

T R :

Reference temperature

u, v, w :

Dimensional horizontal and vertical velocity components

u *,v *,w * :

Dimensionless velocity components

V :

Linear variable viscosity parameter, Γ ΔT

Va :

Vadasz or Prandtl–Darcy number, PrDa

Va D :

(Taylor–Vadasz number), Ta Da 2

x :

Horizontal coordinate

x * :

Dimensionless horizontal coordinate

z :

Vertical coordinate

z * :

Dimensionless vertical coordinate

(x, y, z):

Cartesian coordinates with z-axis vertically upward

α :

Coefficient of thermal expansion

(χ Th , χ Tv ):

Thermal conductivities in x- and z-directions

ε :

Mechanical anisotropy parameter, \({\frac{k_h }{k_v }}\)

Φ:

Porosity of the media

η :

Thermal anisotropy parameter, χ Th /χ Tv

γ :

Heat capacity ratio, \({{(\rho_R c_p )_m}/{(\rho_R c_p )_f }}\)

τ :

Scaled dimensionless time, t Da −1

ΔT :

Temperature gradient

▽:

\({\hat{i}\frac{\partial}{\partial x}+\hat{j}\frac{\partial }{\partial y}+\hat{k}\frac{\partial }{\partial z}}\) (vector differential operator)

▽2 :

\({\frac{\partial^{2}}{\partial x^{2}} +\frac{\partial^{2}}{\partial y^{2}}+\frac{\partial^{2}}{\partial z^{2}}}\) (three-dimensional Laplacian operator)

\({\nabla_1^2}\) :

\({\frac{\partial^{2}}{\partial x^{2}} +\frac{\partial^{2}}{\partial y^{2}}}\) (two-dimensional Laplacian operator)

Λ:

Brinkman number, \({\frac{\mu_p }{\mu_f }}\)

μ :

Dynamic viscosity

μ p :

Effective viscosity

μ f :

Viscosity of the fluid

ν :

Kinematic viscosity, \({\frac{\mu_f }{\rho_R }}\)

ρ :

Density

ρ b :

Basic state density

ρ R :

Density of the liquid at reference temperature T = T m

ψ :

Stream function

σ :

Growth rate of perturbation

\({\vec {\Omega }}\) :

Angular velocity of rotation

ω :

Scaled frequency of oscillation

ζ:

z- component of vorticity, \({\left( {\frac{\partial v}{\partial x}-\frac{\partial u}{\partial y}}\right)}\)

b :

Basic state

c :

Critical quantity

f :

Fluid

′:

Dimensional quantities

*:

Dimensionless quantities

o :

Oscillatory

s :

Stationary

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Authors and Affiliations

  1. Maharani’s Science college for Women, Bangalore, 560 001, India

    R. K. Vanishree

  2. Department of Mathematics, Central College Campus, Bangalore University, Bangalore, 560 001, India

    P. G. Siddheshwar

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  1. R. K. Vanishree
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  2. P. G. Siddheshwar
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Correspondence to R. K. Vanishree.

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Vanishree, R.K., Siddheshwar, P.G. Effect of Rotation on Thermal Convection in an Anisotropic Porous Medium with Temperature-dependent Viscosity. Transp Porous Med 81, 73–87 (2010). https://doi.org/10.1007/s11242-009-9385-2

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