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Heat and Mass Transfer

, Volume 30, Issue 4, pp 259–267 | Cite as

Effects of a magnetic field on the onset of convection in a porous medium

  • S. Alchaar
  • P. Vasseur
  • E. Bilgen
Originals

Abstract

The stability of a conducting fluid saturating a porous medium, in the presence of a uniform magnetic field, is investigated using the Brinkman model. In the first part of the paper constant-flux thermal boundary conditions are considered for which the onset of convection is known to correspond to a vanishingly small wave number. The external magnetic field is assumed to be aligned with gravity. Closed form solutions are obtained, based on a parallel flow assumption, for a porous layer with either rigid-rigid, rigid-free or free-free boundaries. In the second part of the paper, the linear stability of a porous layer, heated isothermally from below, is investigated using the normal mode technique. The external magnetic field is applied either vertically or horizontally. Solutions are obtained for the case of a porous layer with free boundaries. Results for a pure viscous fluid and a Darcy (densely packed) porous medium emerge from the present analysis as limiting cases.

Keywords

Convection Porous Medium External Magnetic Field Free Boundary Closed Form Solution 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

List of symbols

\(\vec B\)

applied magnetic field

C

temperature gradient along thex-direction

Da

Darcy number,K/h′2

g

intensity of gravity

h′

layer depth

H

Hartmann number for a porous medium,B(Kσ/ξ)1/2

Ha

Hartmann number for a fluid,\(H/\sqrt {Da} \)

k

thermal conductivity of fluid-saturated porous medium

K

permeability of the porous medium

n

integer (n π=vertical wave number)

q′

constant heat flux

R

Rayleigh number for a porous medium (K g βΔT'h')/(k afv)

Ra

Rayleigh number for a fluid,R/Da

t

dimensionless time, α2=(1+H2)/Da

T

dimensionless temperature (T′−T′0)/ΔT′

T′0

reference temperature at the origin of the coordinate system

ΔT′

characteristic temperature difference

u, v

dimensionless velocities inx andy directions (u′, v′)h′f

x, y

dimensionless Cartesian coordinates (x′, y′)/h′

Greek symbols

α

dimensionless parameter, α2=(1+H2)/Da

αf

effective thermal diffusivity

β

coefficient of thermal expansion of fluid

λ

horizontal wave length

θ

dimensionless temperature

ν

kinematic viscosity of fluid

μ

dynamic viscosity of fluid

ρ

density

σ

electrical conductivity

ψ

dimensionless stream function, Ψ′/α f

Superscript

dimensional quantities

*

perturbation from the rest state

3|Subscript

C

critical value at the onset of motion

Einfluß eines Magnetfeldes auf das Einsetzen der Konvektion in einem porösen Medium

Zusammenfassung

Mit Hilfe des Brinkman-Modells wird die Stabilität eines, ein poröses Medium tränkenden, leitfähigen Fluids untersucht, auf das ein homogenes Magnetfeld einwirkt. In einer ersten Teilbetrachtung steht das System unter der Randbedingung zweiter Art (konstanter Wärmefluß), für die der Konvektionsbeginn bekanntermaßen mit einer verschwindend kleinen Wellenzahl korrespondiert. Magnet- und Gravitationsfeld wirken gleichsinnig. Parallelströmungen vorausgesetzt, lassen sich geschlossene Lösungen für eine poröse Schicht finden, deren Berandungen entweder als starr/starr, oder starr/frei, oder frei/frei angenommen werden können. Im zweiten Teil der Arbeit wird die lineare Stabilität einer porösen, von unten isotherm beheizten Schicht mit Hilfe der Methode der Normalmode untersucht. Das äußere Magnetfeld kann sowohl vertikal als auch horizontal einwirken. Die gewonnenen Lösungen beziehen sich auf eine poröse Schicht mit freien Rändern. Als Grenzfälle dieser Untersuchung folgen die Lösungen für das sehr dicht gepackte poröse Darcy-Medium und für das reine zähe Fluid.

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Copyright information

© Springer-Verlag 1995

Authors and Affiliations

  • S. Alchaar
    • 1
  • P. Vasseur
    • 1
  • E. Bilgen
    • 1
  1. 1.Department of Mechanical Engineering Ecole PolytechniqueMontreal UniversityMontrealCanada

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