M A G N E T O - G R A V I T A T I O N A L I N S T A B I L I T Y AND

S U S P E N D E D P A R T I C L E S

R. C. SHARMA Dept. of Mathematics, Himachal Pradesh University, Simla, India

(Received 4 October; revised 22 November, 1975)

Abstract. The gravitational instability of an infinite homogeneous and infinitely conducting self- gravitating gas-particle medium in the presence of a vertical magnetic field and suspended particles is considered. It is found that in the presence of suspended particles and magnetic field, Jeans' criterion determines the gravitational instability.

1. Introduction

A comprehensive account of the gravitational instability of an infinite homogeneous self-gravitating medium, as discovered by Jeans (1902), has been given by Chandra- sekhar (1961). He has found that a uniform magnetic field do not alter Jeans' criterion for instability.

Sharma et al. (1976) have studied the effect of suspended particles on the onset of B6nard convection in hydromagnetics. They have found that the effect of suspended particles is to destabilize the layer and that the magnetic field has a stabilizing influence. In another study, Sharma (1975) studied the effect of suspended particles on the gravitational instability of an infinite homogeneous gas-particle medium.

The object of the present investigation is to study the gravitational instability of an infinite homogeneous and infinitely conducting self-gravitating gas-particle medium in the presence of magnetic field and suspended particles. This aspect forms the subject matter of the present paper. The presence of suspended (or dust) particles in the gas in astronomical contexts is more realistic and a reconsideration of the magneto- gravitational instability problem in the presence of suspended particles is certainly called for. This aspect forms the subject matter of the present paper.

2. Perturbation Equations

Let u(u, v, w), h(hxl h r, hz), JQ, 6p and JU denote respectively the perturbations in gas velocity, magnetic field H(0, 0, H), density Q, pressure p and gravitational potential U; N(2, t) and v(2, t) denote the number density and velocity field of the particles, K = 6~rQvtl, tl being the particle radius and v the kinematic viscosity of the gas, is a constant and 2 = (x, y, z). Let c and G denote the velocity of sound in the medium and the constant of gravitation. Then the linearized perturbation equations governing the motion of the gas-particle medium are

Astrophysics and Space Science 46 (1977) 255-259. All Rights Reserved Copyright �9 1977 by D. Reidel Publishing Company, Dordrecht-Holland

256 R.C. SHARMA

~u

~t 1 V6p + VSU + K N (v - u) + v[V2u + �89 + Q

1 + 7 - " (V x h) x H, (1)

~ - ~ + 1 v = u , (2)

5Q = -QV.u, (3)

3p = c 2 50, (4)

V.h = 0, (5)

~h - - = V x (n x H), (6) 8t

V 2 5U = - 4 n G 5~, (7)

where m N is the mass of particles per unit volume and �9 = m/K. In writing the linear- ized perturbation form (2) of the equation of motion for the particles, we have made use of the assumptions that the buoyancy force and the interparticle reactions are neglected (Scanlon and Segel, 1973). In writing Equation (I), use has been made of the Stokes' assumption that the bulk viscosity 2 + ~Qv = 0.

3. Dispersion Relation

Assume that the perturbations of all the quantities are of the form

exp i(lc~x + kzZ + at), (8)

where kx, kz are the wave-numbers of the perturbation along the x- and z-axes and a is the growth rate of the perturbation.

Using expression (8), Equations (1)-(7) give

[--za3 + it72(1 + ~vk2z + KNT) + (4vk2 + zkZV2)tT- ik2V2]u =

ikx z ~- - a(-~-s + iza)s, (9)

[--Z~73 + ia2(1 + ~V]C2Z d-K~ ~) + tT(4vk2 + zkZV2) - ik2V2]l~ = O,

(lO)

MAGNETO-GRAVITATIONAL INSTABILITY AND SUSPENDED PARTICLES 257

where

/kz 2 (11)

t~O V2 H 2 g'2~ = c 2 k 2 - 4nGo, s = - - and = �9

O 4z~Q

The divergence of Equation (1) with the use of Equations (2)-(7) and expression (8) gives

4 2 k x k Z V 2 ( l + i z a ) u + [izt74-F (1 q - ~ v k ~ + KN~)ty3 -

- ( ~ v k 2 + z O ] ) i a 2 - O ] a ] s - 0. (12)

Equations (9)-(12) can be written in the form

-- rt73 -k it72 • 0 0 ikx~'22 k2 tr(l + irtr)

/ 4 - 2 KNz\

"4- tT(~3 vk 2 -1- rk 2 V 2) -

- ik 2 V21

kxk 2 V2(1 + ira)

-ro3 + ia2 x 0

x (l +~ vk2z +KNr) +

q- 0"(~ k'k 2 q- rk~ V 2) -

_ ilCz ~ v 2]

ik,~2~ -- ro'2 "q- io" X k 2 (1 + irtr)

x( l+~vkzr+KNoz)+

+-~vk 2]

0 [ira'+

+(1+~ vk2r+

KN'c\ 3 + Q ) a - -

-- (-~vk 2 + ~ ] ) •

IX ia 2 1 ~J~]

~ 0 . (13)

258 R.C. SHARMA

Equation (13) yields the dispersion relation

4 2 K~VT) ik~V2)

[ ( 4 I • iz2~r 6 + 2 z 1 + ~ v k 2 z + (7 5 - z(Tf2s z + k zV 2 + I r k 2)+

4 4 2 K~rT) X § (1 § K ~ z ) 2 } i a 4 - I ( 1 + ~ v k T ; + --

s vkZ~ z(f2~ k 2 V2)}o "3 + • (T~'2 2 § k2V 2 + ~ ./ § "q-

§ i-gy4vk2i4vk2~ § T.('2~) _{_ 4Tk4V2 v § ,,.262~ ~z rl:202~'~J -1- (k2V 2 + 02) •

' ?')I x I + -~ vk27: § i(C + {s 2 + 2zk~ V 2) +

+ -~k4V2v},:r - ik2:V2t2J] = O. (14)

4. Discussion

The first factor of Equation (14) gives

o" = 0, (15)

which is a neutrally stable mode. The second factor of Equation (14) can be written as

( 4 K N T ) w 2 Tk~V2)W + k2V 2 0, (16) "rW 3+ 1 +-~ vk2r + + (4vk2 + =

where we have written i a= W. Equation (16) does not allow a positive real root or a complex root whose real part is positive and so the system is stable.

The third factor of Equation (14), on putting i a= W, gives

(4 zW 2 + 1 q- ~ vk2z q- I~ + -~vk 2 = 0, (17)

which does not allow any root whose real part is positive, meaning thereby that the

sYstem is stable. The last factor of Equation (14) gives

MAGNETO-GRAVITATIONAL INSTABILITY AND SUSPENDED PARTICLES 259

( 4 2 KN'C)W5 (z( f f2~+ +-]vk2)+ z 2 W 6 + 2z 1 + -~ vk z + - + k 2 V 2

( 4 2 KN'c) 2} (~'(~t"2~ -k k 2 V2) q - + 1 + - ~ v k r + W 4 +

4 z . k 2 V z ) t w 3 + + ( 1 + - ~ v k z + ~T-)(~Vk2q-Zg22q-

+ ((kZVZ + g22)(1 + KNz + 4 ) "* z4 2 zg-22) ---d- 3 vk2~ + -svk (~vk + +

2 2 2 2} W2 (~i(~vk 2 + 2~k~V 2) + + 4a-'rk4V2v + z kzV s +

l:-2 ] ' . : 202 + ~k4VZv} W + '~z- ~ r = 0, (18)

where again we have put ia= W. It is clear that, when 122<0, Equation (18) has at least one positive root. This means that at least one value of ia is positive. Hence, the system is unstable.

We therefore conclude that the gravitational instability of an infinite homogeneous and infinitely conducting self-gravitating gas-particle medium is determined by Jeans' criterion in the presence of magnetic field and suspended particles.

References

Chandrasekhar, S.: 1961, Hydrodynamic and Hydromagnetic Stability (Clarendon Press, Oxford), Chap. 13.

Jeans, J. H.: 1902, Phil. Trans. Roy. Soc. (London) 199, 1. Scanlon, J. W. and Segel, L. A.." 1973, Phys. Fluids 16, 1573. Sharma, R. C.: 1975, Zeit. Appl. Math: Mech. 55, 615. Sharma, R. C., Prakash, K., and Dube, S. N.: 1976, Acta Physica Hungarica 40, 2.