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Hydromagnetic thermal convection between non-uniformly heated plates

Hydromagnetische, thermische Konvektion zwischen zwei ungleichförmig beheizten Platten

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Summary

The effect of the vertical magnetic field of uniform strength on the stability of natural convection in a thin horizontal fluid layer subject to horizontal as well as vertical temperature gradient has been studied on the basis of linear theory. The boundaries are taken to be rigid, perfectly thermally and electrically conducting having prescribed temperatures and the horizontal temperature gradient is assumed to be small. The analysis is restricted to the case when Prandtl number is greater than magnetic Prandtl number which is met by a large margin under most terristrial conditions. It has been found that the preferred mode of disturbance is stationary and will be a transverse roll or a longitudinal roll depending on three parameters namely Prandtl number, Chandrasekhar number and the ratio of Prandtl number to magnetic Prandtl number. The critical Rayleigh numbers for various values of the above three parameters have been reported.

Zusammenfassung

Es wird der Einfluß eines vertikalen, homogenen Magnetfeldes auf die Stabilität der natürlichen Konvektion in einer dünnen, horizontalen Fluidschicht, welche sowohl einem horizontalen als auch einem vertikalen Temperaturgradienten ausgesetzt ist, auf der Basis einer linearen Theorie untersucht. Die Begrenzungen sind dabei starr, vollkommen thermisch und elektrisch leitend mit vorgeschriebenen Temperaturen angenommen, wobei der horizontale Temperaturgradient als klein vorausgesetzt wird. Die Behandlung wird auf den Fall einer Prandtl-Zahl größer als die magnetische Prandt-Zahl, wie er mit großem Spielraum unter den meisten terrestrischen Bedingungen auftritt, beschränkt. Es wurde gefunden, daß der bevorzugte Mode der Störung stationär ist und eine transversale oder longitudinale Walze in Abhängigkeit dreier Parameter, hauptsächlich der Prandtl-Zahl, der Chandrasekhar-Zahl und des Verhältnisses der Prandtl-Zahl zur magnetischen Prandtl-Zahl darstellt. Die kritischen Rayleigh-Zahlen werden für verschiedene Werte der vorgenannten drei Parameter angegeben.

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Abbreviations

a :

Overall wave number

A :

variational parameter (see Table 1)

d :

depth of the layer

D :

differential operator\(\frac{d}{{dx_3 }}\)

g :

acceleration of gravity

h i (i+1, 2, 3):

perturbed magnetic field components

H :

Strength of uniform magnetic field alongx 3 direction

H i (i+1, 2, 3):

basic magnetic field components (H x(x3),O, H)

k :

Unit vector alongx 3 direction

k t :

thermal diffusivity

k, m :

wave numbers inx 1 andx 2 directions

ox 1 x 2 x 3 :

coordinate system

p :

pressure

P(x 1,x 3):

basic flow pressure

P r :

Prandtl number\(\frac{v}{{k_t }}\)

P m :

magnetic Prandtl number\(\frac{v}{\eta }\)

Q m :

Chandrasekhar number (magnetic field parameter)\(\frac{{\mu _1 H^2 d^2 }}{{4\pi \varrho _0 \eta v}}\)

R :

Rayleigh number\(\frac{{g\alpha \Delta {\rm T}d^3 }}{{k_t v}}\)

t :

time

T 0 :

standard temperature

T :

temperature

ΔT :

temperature difference between lower and upper plates

u i (i+1, 2, 3):

velocity components

U(x 3):

basic flow velocity

V 2 :

Laplacian operator\(\frac{{\partial ^2 }}{{\partial x_1^2 }} + \frac{{\partial ^2 }}{{\partial x_2^2 }} + \frac{{\partial ^2 }}{{\partial x_3^2 }}\)

α:

coefficient of volume expansion

β:

horizontal temperature gradient

ν:

kinematic viscosity

ϱ:

density

ϱ0 :

standard density

θ:

temperature

σ:

amplification factor of disturbance σr i+σi

μ 1 :

magnetic permeability

σ e :

electrical conductivity

η:

ressistivity\(\frac{1}{{4\pi \mu _1 \sigma _e }}\)

τ:

ratio of Prandtl number of magnetic Prandtl number\(\frac{{P_r }}{{P_m }}\)

References

  1. Chandrasekhar, S.: Hydrodynamic and hydromagnetic stability, pp. 146–195. Oxford: University Press 1961.

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  2. Platten, J. K., Rasse, D.: Formulation variationnelle pour la stabilité hydromagnétique. Entropie45, 7–16 (1972).

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  3. Weber, J. E.: On thermal convection between nonuniformly heated planes. Int. J. Heat Mass Transfer16, 961–970 (1973).

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  4. Bhattacharyya, S. P., Nadoor, S.: Stability of thermal convection between nonuniformly heated plates. Appl. Sci. Res.32, 555–570 (1976).

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Authors and Affiliations

  1. Mission Operations and Planning Division, ISRO Satellite Centre Peenya, 560058, Bangalore, India

    S. Nadoor

  2. Department of Mathematics, Indian Institute of Techology, 400076, Bombay, India

    S. P. Bhattacharyya

Authors
  1. S. Nadoor
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  2. S. P. Bhattacharyya
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Rest of the symbols are defined whenever they appear in the discussion.

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Nadoor, S., Bhattacharyya, S.P. Hydromagnetic thermal convection between non-uniformly heated plates. Acta Mechanica 41, 265–282 (1981). https://doi.org/10.1007/BF01177352

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  • DOI: https://doi.org/10.1007/BF01177352

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