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


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.


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


temperature gradient along thex-direction


Darcy number,K/h′2


intensity of gravity


layer depth


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


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


thermal conductivity of fluid-saturated porous medium


permeability of the porous medium


integer (n π=vertical wave number)


constant heat flux


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


Rayleigh number for a fluid,R/Da


dimensionless time, α2=(1+H2)/Da


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


reference temperature at the origin of the coordinate system


characteristic temperature difference

u, v

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

x, y

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

Greek symbols


dimensionless parameter, α2=(1+H2)/Da


effective thermal diffusivity


coefficient of thermal expansion of fluid


horizontal wave length


dimensionless temperature


kinematic viscosity of fluid


dynamic viscosity of fluid




electrical conductivity


dimensionless stream function, Ψ′/α f


dimensional quantities


perturbation from the rest state



critical value at the onset of motion

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


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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  1. 1.
    Horton, C. W.;Rogers, F. T.: Convection currents in a porous medium. J. applied Physics 16 (1945) 367–370Google Scholar
  2. 2.
    Lapwood, E. R.: Convection of a fluid in a porous medium. Proc. Camb. Phil. Soc. 44 (1948) 508–521Google Scholar
  3. 3.
    Nield, D. A.; Bejan, A.: Convection In Porous Media, Springer Verlag 1992Google Scholar
  4. 4.
    Raptis, A.;Vlahos, J.: Unsteady hydromagnetic free convection flow through a porous medium. Letters in Heat and Mass Transfer 9 (1982) 59–64Google Scholar
  5. 5.
    Raptis, A.;Tzivanidis, G.: Hydromagnetic free convection flow through a porous medium between two parallel plates. Physics Letters 90A (1982) 288–289Google Scholar
  6. 6.
    Singh, A. K.;Dikshit, C. K.: Natural convection effects on hydromagnetic generalized Couette flow in a porous medium. Indian J. Theor. Phys. 35 (1987) 331–335Google Scholar
  7. 7.
    Sattar, A.;Hossain, M.: Unsteady hydromagnetic free convection flow with Hall current and mass transfer along an accelerated porous plate with time-dependent temperature and concentration. Zanadian J. Phys. 70 (1991) 369–374Google Scholar
  8. 8.
    Takhar, H. S.;Ram, P. C.: Effects of Hall current on hydromagnetic free-convective flow through a porous medium. Astrophysics and Space Science 192 (1992) 45–51Google Scholar
  9. 9.
    Rudraiah, N.: Universal stability of hydromagnetic convection flows in a porous medium. Indian J. of pure and Appl. Maths. 4 No. 3 (1972) 738–748Google Scholar
  10. 10.
    Rudraiah, N.;Vortmeyer, D.: Stability of finite-amplitude and overs table convection of a conducting fluid through fixed porous bed. Wärme- and Stoffübertragung 11 (1978) 241–254Google Scholar
  11. 11.
    Alchaar, S.; Vasseur, P.; Bilgen, E.: Hydromagnetic natural convection in a tilted rectangular porous enclosure. Numerical Heat Transfer, in press (1995)Google Scholar
  12. 12.
    Brinkman, H. C.: Calculation of the viscous force exerted by a flow in fluid on a dense swarm of particles. Appl Sci. Res. A1 (1947) 27–34Google Scholar
  13. 13.
    Walker, K. L.;Homsy, G. M.: A note of convective instability in Boussinesq fluid and porous media. J. Heat Transfer 99 (1977) 338–339Google Scholar
  14. 14.
    Vasseur, P.;Wang, C. H.;Sen, M.: The Brinkman model for natural convection in a shallow porous cavity with uniform heat flux. Numerical Heat Transfer 15 (1989) 221–242Google Scholar
  15. 15.
    Rudraiah N.;Veerappa, B.;Balachandra Rao, S.: Effects of nonuniform thermal gradient and adiabatic boundaries on convection in porous media. J. Heat Transfer 102 (1980) 254–260Google Scholar
  16. 16.
    Vasseur, P.;Robillard, L.: The Brinkman model for natural convection in a porous layer: effects of nonuniform thermal gradient. Int. J. Heat Mass Transfer 36 (1993) 4199–4206Google Scholar
  17. 17.
    Garandet, J. P.;Alboussiere, T.;Moreau, R.: Buoyancy driven convection in a rectangular enclosure with a transverse magnetic field. Int. J. Heat Mass Transfer 35 (1992) 741–748Google Scholar
  18. 18.
    Vasseur, P.;Wang, C. H.;Sen, S.: Thermal instability and natural convection in a fluid layer over a porous substrate. Wärme- und Stoffübertragung 24 (1989) 337–347Google Scholar
  19. 19.
    Nield, D. A.: Onset of thermohaline convection in a porous medium. Water Resour. Res. 4 (1968) 553–560Google Scholar
  20. 20.
    Nield, D. A.: The onset of transient convective instability. J. Fluid Mech. 71 (1975) 441–454Google Scholar
  21. 21.
    Kulacki, F. A.;Goldstein, R. J.: Hydrodynamic instability in fluid layers with uniform volumetric energy sources. Appl. Sci. Res. 31 (1975) 81–109Google Scholar
  22. 22.
    Sparrow, E. M.;Goldstein, R. J.;Jonsson, V. K.: Thermal instability in a horizontal fluid layer: effect of boundary conditions and nonlinear temperature profile. J. fluid Mech. 18 (1964) 513–528Google Scholar
  23. 23.
    Hurle, D. T. J.;Jakeman, E.;Pike, E. R.: On the solution of the Benard problem with boundaries of finite conductivity. Proc. Roy. Soc. Lond. A 296 (1967) 469–475Google Scholar
  24. 24.
    Chandrasekhar, S.: Hydrodynamic and Hydromagnetic Stability. Oxford University Press, London 1961Google Scholar
  25. 25.
    Vasseur, P.; Hasnaoui, M.; Bilgen E.; Robillard, L.: Natural convection in an clined fluid layer with a transverse magnetic field: Analogy with a porous medium. J. Heat Transfer, in press (1995)Google Scholar

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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