Skip to main content
SpringerLink
Log in

The effect of mechanical and thermal anisotropy on the stability of gravity driven convection in rotating porous media in the presence of thermal non-equilibrium

  • Original Paper
  • Published:
Transport in Porous Media Aims and scope Submit manuscript

Abstract

We investigate Rayleigh–Benard convection in a porous layer subjected to gravitational and Coriolis body forces, when the fluid and solid phases are not in local thermodynamic equilibrium. The Darcy model (extended to include Coriolis effects and anisotropic permeability) is used to describe the flow, whilst the two-equation model is used for the energy equation (for the solid and fluid phases separately). The linear stability theory is used to evaluate the critical Rayleigh number for the onset of convection and the effect of both thermal and mechanical anisotropy on the critical Rayleigh number is discussed.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Abbreviations

g * :

Acceleration due to gravity

H * :

Height of the mushy layer

ê z :

Unit vector in the z-direction

k x ,k y ,k z :

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

L :

Length of the mushy layer

p :

Reduced pressure

Pr :

Prandtl number, ν*/κ *

R :

Scaled Rayleigh number, Ra/π 2

Ra :

Rayleigh number, \({\rho _{\ast 0} g_\ast \beta _\ast \Delta Tk_{\ast z} H_\ast \rho _{\ast f} c_{\ast f}}/({\kappa _{\ast fz} \mu _\ast})\)

s x :

x-component of wavenumber

s y :

y-component of wavenumber

t :

Time

T f :

Dimensionless temperature,(T f * −T C )(T H −T C )

T s :

Dimensionless temperature,(T s * −T C)(T H −T C)

u :

Horizontal x-component of the filtration velocity

V :

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

v :

Horizontal y-component of filtration velocity

w :

Vertical component of filtration velocity

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

α :

Scaled wavenumber, s 2/π2

β :

Thermal expansion coefficient

η :

thermal conductivity ratio, \({\kappa _{\ast (.)x}}/{\kappa _{\ast (.)z}} {\kappa _{\ast (.)y}}/{\kappa _{\ast (.)z}}\)

\(\phi\) :

Porosity

κ :

Thermal conductivity

λ :

Parameter akin to the Nield number

μ :

Dynamic viscosity of the fluid

ν :

Kinematic viscosity

ρ :

Fluid density

ω :

Angular velocity

ξ :

Anisotropy ratio,\({k_{\ast x}}/{k_{\ast z}}{{k_{\ast y}} /{k_{\ast z}}}\)

*:

Dimensional quantities

o :

Rescaled parameters

B :

Basic flow quantities

cr:

Critical values

f x , f y , f z :

Fluid conditions in the x-, y- and z- directions

s x , s y , s z :

Solid conditions in the x-, y- and z- directions

References

  • Alex S.M., Patil P.R. (2000) Thermal instability in an anisotropic rotating porous medium. J. Fluid Mech. 252, 79–98

    Google Scholar 

  • Banu N., Rees D.A.S. (2002) Onset of Darcy–Benard convection using a thermal non-equilibrium model. Int. J. Heat Mass Transfer. 45, 2221–2228

    Article  Google Scholar 

  • Bejan A. Convection Heat Transfer, 2nd edn. Wiley (1995)

  • Epherre J.F. (1975) Critère d’apparition de la convection naturelle dans une couche poreuse anisotrope. Rev. Gen. Therm. 168, 949–950

    Google Scholar 

  • Govender, S.: On the effect of anisotropy on the stability of convection in rotating porous media, Trans Porous Media 64, 413–422 (2006a) (In Press)

  • Govender, S.: Coriolis effect on the stability of centrifugally driven convection in a rotating anisotropic porous layer subjected to gravity, Trans Porous Media (TiPM 264) (2006b) (In Press)

  • Kuznetsov, A.V.: Thermal nonequilibrium forced convection in porous media, Transport Phenomena in Porous Media, pp. 103–129. Pergamon (An imprint of Elsevier Science), Netherlands

  • Malashetty M.S., Shivakumara I.S., Kulkarni S. (2005) The onset of convection in an anisotropic porous layer using a thermal non-equilibrium model. Trans. Porous Media 60, 199–215

    Article  Google Scholar 

  • McKibbin R. (1986) Thermal convection in a porous layer: Effects of anisotropy and surface boundary conditions. Trans. in Porous Media 1, 271–292

    Article  Google Scholar 

  • Nield D.A., Bejan, A.: Convection in Porous Media. Springer (1992)

  • Storesletten L. (1993) Natural convection in a horizontal porous layer with anisotropic thermal diffusivity. Trans. Porous Media 12, 19–29

    Article  Google Scholar 

  • Vadasz P. (1992) Natural convection in porous media induced by the centrifugal body force: the solution for small aspect ratio. ASME J. Energy Res. Tech. 114, 250–254

    Google Scholar 

  • Vadasz P. (1993a) Fluid flow through heterogenous porous media in a rotating square channel. Trans. Porous Media 12, 43–54

    Article  Google Scholar 

  • Vadasz P. (1993b) Three-dimensional free convection in a long rotating porous box: analytical solution. ASME J. Heat Transfer 115, 639–644

    Article  Google Scholar 

  • Vadasz P. (1994a) Stability of free convection in a narrow porous layer subject to rotation. Int. Comm. Heat Mass Transfer 21, 881–890

    Article  Google Scholar 

  • Vadasz P. (1994b) Centrifugally generated free convection in a rotating porous box. Int. J. Heat Mass Transfer. 37, 2399–2404

    Article  Google Scholar 

  • Vadasz P. (1995) Coriolis effect on free convection in a long rotating porous box subject to uniform heat generation. Int. J. Heat Mass Transfer 38, 2011–2018

    Article  Google Scholar 

  • Vadasz P. (1996a) Stability of free convection in a rotating porous layer distant from the axis of rotation. Trans. Porous Media 23, 153–173

    Article  Google Scholar 

  • Vadasz P. (1996b) Convection and stability in a rotating porous layer with alternating direction of the centrifugal body force. Int. J. Heat Mass Transfer 39, 1639–1647

    Article  Google Scholar 

  • Vadasz P. (1998) Coriolis effect on gravity driven convection in a rotating porous layer heated from below. J. Fluid. Mech. 376, 351–375

    Article  Google Scholar 

  • Vadasz P., Govender S. (1998) Two-dimensional convection induced by gravity and centrifugal forces in a rotating porous layer far away from the axis of rotation. Int. J. Rot. Mach. 4, 73–90

    Article  Google Scholar 

  • Vadasz P., Heerah A. (1998) Experimental confirmation and analytical results of centrifugally-driven free convection in a rotating porous media. J. Porous Media 1, 261–272

    Google Scholar 

  • Vadasz P., Olek S. (1998) Transitions and chaos for free convection in a rotating porous layer. Int. J. Heat Mass Transfer 41, 1417–1435

    Article  Google Scholar 

  • Vadasz P. (2005) Explicit conditions for local thermal equilibrium in porous media heat conduction. Trans Porous Media 59(3): 341–355

    Article  Google Scholar 

Download references

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

  2. Department of Mechanical Engineering, Northern Arizona University, P.O. Box 15600, Flagstaff, AZ, 86011-5600, USA

    Peter Vadasz

Authors
  1. Saneshan Govender
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Peter Vadasz
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Saneshan Govender.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Govender, S., Vadasz, P. The effect of mechanical and thermal anisotropy on the stability of gravity driven convection in rotating porous media in the presence of thermal non-equilibrium. Transp Porous Med 69, 55–66 (2007). https://doi.org/10.1007/s11242-006-9063-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11242-006-9063-6

Keywords

Search

Navigation