Transport in Porous Media

, 81:73 | Cite as

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

Article

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.

Keywords

Anisotropy Rotation Porous medium Variable viscosity Thermal convection 

List of symbols

a

Horizontal wave number

ac

Critical wave number

BrD

Brinkman–Darcy number, ΛDa

c

Specific heat

cp

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

(kh ,kh ,kv)

Permeability along x, y and z-direction

l, m

Wave numbers

p

Pressure

p*

Hydrostatic pressure

pH

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

RD

Darcy–Rayleigh number,RDa

t

Time

T

Temperature field

Ta

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

Tb

Basic state temperature

TR

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

VaD

(Taylor–Vadasz number), TaDa2

x

Horizontal coordinate

x*

Dimensionless horizontal coordinate

z

Vertical coordinate

z*

Dimensionless vertical coordinate

(x, y, z)

Cartesian coordinates with z-axis vertically upward

Greek symbols

α

Coefficient of thermal expansion

Th , χTv)

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, χThTv

γ

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

τ

Scaled dimensionless time, tDa−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

μp

Effective viscosity

μf

Viscosity of the fluid

ν

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

ρ

Density

ρb

Basic state density

ρR

Density of the liquid at reference temperature T = Tm

ψ

Stream function

σ

Growth rate of perturbation

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

Angular velocity of rotation

ω

Scaled frequency of oscillation

ζ

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

Subscripts

b

Basic state

c

Critical quantity

f

Fluid

Superscripts

Dimensional quantities

*

Dimensionless quantities

o

Oscillatory

s

Stationary

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Copyright information

© Springer Science+Business Media B.V. 2009

Authors and Affiliations

  1. 1.Maharani’s Science college for WomenBangaloreIndia
  2. 2.Department of Mathematics, Central College CampusBangalore UniversityBangaloreIndia

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