Abstract
The stability of infinitestimal steady and oscillatory motions and finite amplitude steady motions of a conducting fluid through porous media with free boundaries which is heated from below and cooled from above is investigated in the presence of a uniform magnetic field. Infinitesimal steady motions are investigated using Liapunov method and its is shown that the principle of exchange of stability is valid only when Pm/Pr≤1 with a restricted value of the Hartmann number. It is shown that overstable motions are due to the zonal current induced by the magnetic field. Finite amplitude steady motions are investigated using Veronis [1] analysis and it is shown that for a restricted range of Hartmann numbers and porous parameter Pl, steady finite-amplitude motions can exist for values of the Rayleigh number smaller than that value corresponding to oscillatory motions. Since the Busse number is greater than the wave number the horizontal scale of the steady finite-amplitude motions is larger than that of the overstable motions.
Zusammenfassung
Die Stabilität eines Fluids mit thermischer und elektrischer Leitfähigkeit wird in einem von unten beheizten und von oben gekühlten Medium behandelt.
Bei überlagertem magnetischen Feld können sich oszillatorische Instabilitäten ausbilden, die sich auf zonale, vom Magnetfeld induzierte Ströme zurückführen lassen.
Andere Formen der Instabilität treten unter anderen Bedingungen auf. Maßgebend dafür sind die Werte der Hartmann- und Rayleighzahlen.
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Abbreviations
- (x, y, z):
-
the cartesian Co-ordinates
- t:
-
the time
- d:
-
the depth of the porous media
- \(\vec q\) :
-
(u, v, w) the velocity field
- \(\vec H\) :
-
(Hx, Hy, Hz) the magnetic field
- P:
-
the pressure
- ρ:
-
the density
- T:
-
the temperature
- ψ:
-
the velocity stream function
- φ:
-
the magnetic stream function
- ν:
-
the kinematic viscosity
- K:
-
the thermal diffusivity
- νm:
-
the magnetic viscosity
- μ:
-
the magnetic permeability
- k:
-
the permeability of porous media
- Pl=d2/k:
-
the porous parameter
- Pr=ν/K:
-
the Prandtl number
- Pm=ν/νm:
-
the magnetic Prandtl number
- S=K/νm :
-
the Busse number
- R=αgΔTd3/Kν:
-
the Rayleigh number
- \(M = H_0 d\sqrt {\tfrac{\mu }{{\rho _0 ^{\nu \nu } m}}} \) :
-
the Hartmann number
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Rudraiah, N., Vortmeyer, D. Stability of finite-amplitude and overstable convection of a conducting fluid through fixed porous bed. Warme- und Stoffubertragung 11, 241–254 (1978). https://doi.org/10.1007/BF02587788
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DOI: https://doi.org/10.1007/BF02587788