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Coriolis Effect on the Stability of Centrifugally Driven Convection in a Rotating Anisotropic Porous Layer Subjected to Gravity

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Abstract

We investigate natural convection in a fluid saturated rotating anisotropic porous layer subjected to centrifugal gravitational and Coriolis body forces. The Darcy model (including the centrifugal, gravitational and Coriolis terms; and permeability anisotropy effects) and a modified energy equation (including the effects of thermal anisotropy) is used in the current analysis. The linear stability theory is used to evaluate the critical Rayleigh number for the onset of convection in the presence of thermal and mechanical anisotropy. It is shown that the preferred solution comprises roll cells aligned parallel to the vertical z-axis. As a result, it is found that the Coriolis acceleration (or Taylor number) and the gravitational term play no role in the stability of convection.

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Abbreviations

Latin Symbols :

 

a k :

Coefficients in Galerkin expansion

g * :

Acceleration due to gravity

H * :

Height of the porous layer

\(\boldsymbol{\hat {e}}_x\) :

Unit vector in the x-direction

\(\boldsymbol{\hat {e}}_y\) :

Unit vector in the y-direction

\(\boldsymbol{\hat {e}}_z\) :

Unit vector in the z-direction

k :

Permeability tensor

k * x , k * y , k * z :

Characteristic permeabilities in the x-, y- and z- directions

L * :

Length of the porous layer

p :

Reduced pressure

R g :

Scaled gravitational Rayleigh number, Ra g / π2.

R ω :

Scaled centrifugal Rayleigh number, Ra ω/ π2.

Ra g :

Gravitational Rayleigh number, β* ΔTk * x g * L */ κ* x ν* ..

Ra ω :

Centrifugal Rayleigh number, \(\beta_{\ast} \Delta Tk_{\ast x} \omega_{\ast} ^{2} L_{\ast}^{2} \left/ \kappa_{\ast x} \nu _{\ast}\right.\).

s :

Wave number, \(\sqrt {s_y^2 +s_z^2} \)

s y :

y-component of wavenumber

s z :

z-component of wavenumber

t :

Time

T :

Dimensionless temperature, (T * −T C)/ (T H −T C) .

T C :

Temperature at cold wall

T H :

Temperature at hot wall

Ta :

Taylor number, defined as (2 ω* k * x / ν* ϕ *.)

u :

Horizontal x-component of the filtration velocity

v :

Horizontal y-component of filtration velocity

w :

Vertical component of filtration velocity

W * :

Width of porous layer

V :

Dimensionless filtration velocity vector, \(u\hat {e}_x +v\hat {e}_y+w\hat {e}_z\)

X :

Space vector, \(x\hat {e}_x +y\hat {e}_y +z\hat {e}_z\)

x :

Horizontal length co-ordinate

y :

Horizontal width co-ordinate

z :

Vertical co-ordinate

Greek Symbols :

 

α:

Scaled wavenumber, ξ s 2/ π2.

β* :

thermal expansion coefficient

η:

κ* y / κ * x .=κ * z / κ* x .

ϕ* :

Porosity

κ* :

Fluid thermal diffusivity

φ:

Parameter, η/ ξ .

μ* :

Fluid dynamic viscosity

ν* :

Fluid kinematic viscosity

θ:

Galerkin function

ρ* :

Fluid density

ω* :

Angular velocity

ξ:

k * y / k * x .=k * z / k * x .

Subscripts :

 

*:

Dimensional quantities

B :

Basic flow quantities

C:

Associated with cold wall

H:

Associated with hot wall

cr:

Critical values

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Author information

Authors and Affiliations

  1. School of Mechanical Engineering, University of Kwa-Zulu Natal, King George V Avenue, Durban, 4041, South Africa

    Saneshan Govender

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  1. Saneshan Govender
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Correspondence to Saneshan Govender.

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Govender, S. Coriolis Effect on the Stability of Centrifugally Driven Convection in a Rotating Anisotropic Porous Layer Subjected to Gravity. Transp Porous Med 67, 219–227 (2007). https://doi.org/10.1007/s11242-006-9003-5

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  • DOI: https://doi.org/10.1007/s11242-006-9003-5

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