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Rayleigh-Taylor instability of compressible fluids in the presence of a vertical magnetic field

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Abstract

The influence of a vertical magnetic field on the stability of a compressible, inviscid fluid of variable density is investigated. The solution is characterized by a variational principle. Based on it an approximate solution for the fluid of exponentially varying density, confined between two planes is obtained. The magnetic field is found to have a stabilizing influence on the unstable arrangement. Also the effect of a magnetic field on the angular frequency of oscillations of the waves generated in stable arrangement is considered.

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Abbreviations

A :

constant, denoting a characteristic vertical component of velocity

c :

velocity of sound in the medium

c 0 :

velocity of sound at z=0

d :

depth of the fluid layer

D:

d/dz

g :

acceleration due to gravity (0, 0, −g)

h :

perturbation in magnetic field with components h x , h y and h z

h′ :

value of h corresponding to the characteristic value n′ of n

H :

uniform magnetic field pervading the fluid configuration, (0, 0, H)

I 1, I 2, I 3, I 4 :

integrals given by equations (23)–(26)

i:

√ − 1

k :

wave number vector, (k x , k y , 0)

l :

πs/d

n :

the growth rate of departure of equilibrium from its initial state

p :

pressure at any point at time t

Q :

dimensionless measure of magnetic field, (V 2/c 2) πs

Q crit :

critical value of Q for which the system is thoroughly stabilized, T/R

R :

πs

s :

any integer

t :

time

T :

dimensionless measure of buoyancy forces, (d 2/c 2)(gβ+g 2/c 2)

T 1 :

−T

u :

velocity at any general point P, (u, v, w)

V g :

group velocity of the waves

V w :

wave velocity

w′ :

the value of w corresponding to the characteristic value n′ of n

x*:

(T/QR−1)−1/2

β :

constant, measuring scale height in the fluid

ρ :

density of fluid at point P at time t

ρ 0 :

density at z=0

δρ :

perturbation in density

δp :

perturbation in pressure

μ :

magnetic permeability assumed constant

References

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  3. Mitchner, M. and R. K. M. Landshoff, Phys. of Fluids 7 (1962) 862.

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  4. Ariel, P. D., Ind. J. Pure Appl. Math. 2 (1971) 135.

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  5. Hide, R., Proc. Roy. Soc. London A 233 (1955) 376.

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Authors and Affiliations

  1. Dept. of Mathematics, Malaviya Regional Engineering College, Jaipur, India

    P. D. Ariel

Authors
  1. P. D. Ariel
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Ariel, P.D. Rayleigh-Taylor instability of compressible fluids in the presence of a vertical magnetic field. Appl. Sci. Res. 24, 294–304 (1971). https://doi.org/10.1007/BF00411720

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  • DOI: https://doi.org/10.1007/BF00411720

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