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MHD stability of a self-gravitating rotating streaming perfectly conducting compressible fluid medium.

1. Introduction

The pure self-gravitating instability of an infinite homogeneous gas medium at rest has been investigated since long time ago. It was Jeans, [1], who first to study such model and wrote down about its application in astrophysics. It is founded that the model is unstable under the restriction

[k.sup.2][c.sup.2] - 4[pi]G[[rho].sub.o] < 0

called after Jeans by Jeans' criterion, where k is the wave number of the propagated wave, c is a sound speed in the fluid, of density [[rho].sub.o], and G is the self-gravitating constant. Such studies have been extended by Chandrasekhar and Fermi, [2], and later in a wide range by Chandrasekhar, [3]. The Jeans' model of pure self-gravitational medium has been elaborated with streams of variable velocity distribution in a vertical direction for general wave propagation by Sengar [4]. Recently A. E. Radwan and Elazab, [5], A. E. Radwan, H. Radwan and M. Hendi, [6], developed the magnetogravitational stability of variable streams pervaded by the constant magnetic field ([H.sub.o],0,0).

Here in the present work we study the MHD stability of a self-gravitating-rotating streaming inviscid fluid medium pervaded by general magnetic field. Such studies have a correlation with the formation of sunspots. Also they have relevance in describing the condensation within astronomical bodies cf. Chandrasekhar and Fermi, [2], and also Chandrasekhar, [3].

2. Basic State

We consider an infinite self-gravitating fluid medium. The fluid is assumed to be homogeneous and non-viscous. The model is acting upon the forces (i) the pressure gradient force (ii) electromagnetic force (iii) self-gravitating force and (iv) the forces due to rotating factors. We shall utilize the cartesian coordinates (x, y, z) for investigating such problem. The magnetogravitational basic equations for rotating fluid are

p [([partial derivative][u.bar] / [partial derivative]t + ([u.bar] x [nabla])[u.bar.) = -[nabla)P + [mu] ([nabla] [and] [H.bar]) [and] [H.bar] + p[nabla]V + 2p ([u.bar] [and] [[OMEGA].bar]) + 1/2 p ([[OEMGA].bar] [and] [r.bar]).sup.2] (1)

[partial derivative][H.bar] / [partial derivative]t = [nabla] [and] ([u.bar.] [and] [H.bar]) + [eta][nabla] [and] [H.bar] (2)

[partial derivative]p / [partial derivative]t + ([u.bar] x [nabla]) [rho] = -[rho] ([nabla] x [u.bar]) (3)

[nabla] x [H.bar] = 0 (4)

[[nabla].sup.2] V = -4[pi]G[rho] (5)

P = [K[rho].sup.[GAMMA]] (6)

Here [rho], [u.bar], and P are the fluid density, velocity vector and kinetic pressure, [mu], and [H.bar] are the magnetic field permeability and intensity, V and G are the self-gravitating potential and constant, [eta] is the coefficient resistivity, [[OMEGA].bar] is the angular velocity of rotation, K and [GAMMA] are constants where [GAMMA] is the polytropic exponent. Equation (1) is the vector equation of motion including the acting forces on the model. Equation (2) is the equation of magnetic field derived from the Maxwell's electrodynamic equations. Equation (3) is the continuity equation expressing the conservation of mass. Equation (4) expresses the conservation of magnetic flux. Equation (5) is the Poisson's equation satisfying the self-gravitating potential and equation (6) is the polytropic equation of state correlating the fluid density and kinetic pressure.

We assume that the medium: (i) rotates with the general uniform angular velocity

[[OMEGA].bar] = ([[OMEGA].sub.x], [[OMEGA].sub.y], [[OMEGA].sub.z]) (7)

(ii) be pervaded by the two dimensions homogeneous magnetic field

[[H.bar].sub.o] = (0, [H.sub.oy], [H.sub.oz]) (8)

and (iii) posses streams moving in the x-direction with speeds

[[u.bar].sub.o] = (U (z),0,0) (9)

varying along the z-direction of the cartesian coordinates (x, y, z).

For small departure from the initial state, every variable quantity Q may be expressed as

Q = [Q.sub.o] + [Q.sub.i], |[Q.sub.1]| << [Q.sub.o] (10)

where Q stands for [rho], u, P, H and V. Based on the expansion (10), the perturbation equations could be obtained from (1)-(6) in the form:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (11)

[[partial derivative][H.bar].sub.1] = [nabla] [and] ([[u.bar].sub.1] [and] [[H.bar].sub.o]) + [nabla] [and] ([[u.bar].sub.o] [and] [H.bar.1]) (12)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (13)

[nabla] x [[H.bar.].sub.1] = 0 (14)

[[nabla].sup.2][V.sub.1] = -4[pi]G[p.sub.1] (15)

[dP.sub.1] / dt = [c.sup.2] [dp.sub.1] / dt (16)

where c (= [square root of ([[TAU][rho].sub.o] / [[rho].sub.o])]) is a sound speed in the fluid.

By the use of the components of [[u.bar].sub.1] and [H.bar]1 viz.

[[u.bar].sub.1] = (u, v, w) (17)

[H.bar]1 = ([h.sub.x], [h.sub.y], [h.sub.z]) (18)

together with the postulates (7)-(9), the system (11)-(16) may be rewritten as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (19)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (20)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (21)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (22)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (23)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (24)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (25)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (26)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (27)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (28)

3. Eigenvalue Relation

We consider an acting sinsoidal wave along the fluid surface. Consequently, from the viewpoint of the stability approaches given by Chandrasekhar, [3], we assume that the space-time dependence of the wave propagation of the form

exp [i([k.sub.x]x + [k.sub.y]y + [k.sub.z]z + [[sigma].sub.t])] (29)

Here o is gyration frequency of the assuming wave. [k.sub.x], [k.sub.y] and [k.sub.z] are (any real values) the wave numbers in the (x, y, z) directions. By an appeal to the time-space dependence (29), the relevant perturbation equations (19)-(28) are given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (30)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (31)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (32)

[nh.sub.x] = i ([H.sub.oy][k.sub.y] + [H.sub.oz][k.sub.z])u + [h.sub.z][DU.sub.o] (33)

[nh.sub.y] = i ([H.sub.oy][k.sub.y] + [H.sub.oz][k.sub.z])v - i [H.sub.oy] ([k.sub.x]u + [k.sub.y]v + [k.sub.z]w) (34)

[nh.sub.z] = i ([H.sub.oy][k.sub.y] + [H.sub.oz][k.sub.z])w - i [H.sub.oz] ([K.sub.x]u + [k.sub.y]v + [k.sub.z]w) (35)

[np.sub.1] = i [[rho].sub.o] ([k.sub.x]u + [k.sub.y]u + [k.sup.y]v + [k.sub.z]w) (36)

i [k.sub.x][h.sub.x] + i [k.sub.y][h.sub.y] + i [k.sub.z][h.sub.z] = 0 (37)

[k.sup.2][V.sub.1] - 4[pi]G[p.sub.1] = 0 (38)

where

D = d / dz

[k.sup.2] = [k.sup.2.sub.x] + [k.sup.2.sub.y] + [k.sup.2.sub.z] (39)

n = i ([sigma] + [k.sub.x][U.sub.o]) (40)

The foregoing system equations (30)-(38) could be rewritten in the matrix form

[[a.sub.ij]][[b.sub.j]] = 0 (41)

where the elements [a.sub.ij] of the matrix are given in the appendix I while the elements of the column matrix [[b.sub.j]] are being u, v, w, [h.sub.x], [h.sub.y], [h.sub.z], [p.sub.1] and [V.sub.1], taking into account that equation (37) is identically satisfied.

For non-trivial solution of the equations (41), setting the determinant of the matrix [aij] equal to zero, we get the general eigenvalue relation of seven order in n in the form

[A.sup.7][n.sup.7] + [A.sub.6][n.sup.6] + [A.sub.5][n.sup.5] + [A.sub.4][n.sup.4] + [A.sub.3][n.sup.3] + [A.sub.2][n.sup.2] + [A.sub.1]n + [A.sub.o] = 0 (42)

where the compound coefficients [A.sub.i] (i = 1,2,3,4,5,6,7) are not given here but may be written down at once if we want to do that.

4. Discussions and Results

Equation (42) is a general MHD eigenvalue relation of a rotating self-gravitating fluid medium pervaded by magnetic field of two dimensions. Some previously publishing results may be obtained as limiting cases here. That confirms the present analysis.

In absence of the rotating and electromagnetic forces i.e. [[OMEGA].bar] = 0 and [[H.bar].sub.o] = 0, equation (42) yields

[k.sup.2][n.sup.3] + [k.sup.2]([k.sup.2][c.sup.2] - 4[pi]G[[rho].sub.o])n - [k.sub.x][k.sub.z]([k.sup.2][c.sup.2] - 4[pi]G[[rho].sub.o])[DU.sub.o] = 0 (43)

This relation coincides with the dispersion relation, of a pure self-gravitating fluid medium streams with variable streams ([U.sub.o](z), 0, 0) derived by Sengar, [4]. The analytical discussions of the relation (43) reveal that there is must be at least one positive real root i.e.

[n.sub.1] > 0, say (44)

from which we deduce that the streaming medium is unstable. This shows that the streaming motion has a destabilizing influence.

If [[OMEGA].bar] = 0, [[H.bar].sub.o] = 0 and [U.sub.o] = 0, equation (42) reduces to

[n.sup.2] = -4[pi]G[[rho].sub.o] + [k.sup.2][c.sup.2] (45)

This gives the same results givenbyJeans, [1]. Formore details concerning the instability of this case we may refer to the discussions of Jeans, [1].

In absence of the magnetic field and we assume that the fluid medium is stationary i.e. [[H.bar].sub.o] = 0 and [U.sub.o] = 0, equation (42) degenerates to a somewhat complicated relation.

The purpose of the present part is to determine the influence of rotation on the Jeans's criterion (45) of a uniform streaming fluid. So in order to carry out and to facilate the present situation we may choose [[H.bar].sub.o] = 0, [[OMEGA].sub.x] = 0, [k.sub.x] = 0 and [k.sub.y] = 0 equation (42), at once, yields

[n.sup.4] + (4[pi]G[[rho].sub.o] - [c.sup.2][k.sup.2.sub.z] - [4[OMEGA].sup.2])[n.sup.2] + 4[[OMEGA].sup.2.subz] ([c.sup.2][k.sup.2.sub.z] - 4[pi]G[[rho].sub.o]) = 0 (46)

with

[[OMEGA].sup.2] = [[OMEGA].sup.2.sub.y] + [[OMEGA].sup.2.sub.z] (47)

Equation (46) indicates that there must be two modes in which a wave can be propagated in the medium. If the roots of (46) are supposed to be [n.sup.2.sub.1] and [n.sup.2.sub.2] then from the characteristics of the quadratic algebraic equation we have

[n.sup.2.sub.1] + [n.sup.2.sub.2] = [c.sup.2][k.sup.2.sub.z] + [4[OMEGA].sup.2] - 4[pi]G[[rho].sub.o] (48)

[n.sup.2.sub.1][n.sup.2.sub.2] = 4[[OMEGA].sup.2.sub.z] ([c.sup.2][k.sup.2.sub.z] - 4[pi]G[[rho].sub.o]) (49)

and we can see that both the roots [n.sup.2.sub.1] and [n.sup.2.sub.2] are real. The discussions of (40) indicate that if the Jeans's restriction

[c.sup.2][k.sup.2.sub.z] - 4[pi]G[[rho].sub.o] < 0 (50)

is valid, then one of the two roots [n.sup.2.sub.1] or [n.sup.2.sub.2] must be negative and consequently the model will be unstable. This means that under the Jeans's restriction (50), the self-gravitating rotating fluid medium is unstable. This shows that the Jeans's criterion for a self-gravitating medium is unaffected by the influence of the uniform rotation.

In order to examine the influence of the electromagnetic force on the instability of a gravitating streaming fluid medium we shall use the relation (42) with postulates [[OMEGA].bar] = 0, [k.sub.x] = 0 and [k.sub.y] = 0. In this case equation (42) degenerates to

[n.sup.4] + [C.sub.1][n.sup.2] + [C.sub.2] = 0 (51)

with

[C.sub.1] = 4[pi]G[[rho].sub.o] - [c.sup.2][k.sup.2.sub.2] - [mu][H.sup.2.sub.o][k.sup.2.sub.z] / [[rho].sub.o] (52)

[C.sub.2] = ([mu][H.sup.2.sub.o][k.sup.2.sub.z] / [[rho].sub.o] ([c.sup.2][k.sup.2.sub.2] - 4[pi]G[[rho].sub.o]) (53) [[rho].sub.o]

where

[H.sup.2.sub.o] = [H.sup.2.sub.oz] + [H.sup.2.sub.oy] (54)

Again as in the previous case of rotation, we have here also two modes of wave propagation. If the [n.sup.2.sub.1] and [n.sup.2.sub.1] are the roots of the quadratic equation (51) in [n.sup.2], then from the theory of equations we have

[n.sup.2.sub.1] + [n.sup.2.sub.2] = [c.sup.2][k.sup.2.sub.z] - 4[pi]G[[rho].sub.o] + [mu][H.sup.2.sub.o][k.sup.2.sub.z] / [[rho].sub.o] (55)

[n.sup.2.sub.1] [n.sup.2.sub.2] = [mu][H.sup.2.sub.o][k.sup.2.sub.z] / [[rho].sub.o] ([c.sup.2][k.sup.2.sub.z] - 4[pi]G[[rho].sub.o]) (56)

By comparing (48) & (49) with (55) & (56) we may say that [4[OMEGA].sup.2] is replaced by [mu][H.sup.2.sub.o][k.sup.2.sub.z] / [[rho].sub.o].

Following the same analysis of the rotating case we finally find out that Jeans's self-gravitating instability restriction of a streaming fluid medium is not influenced by the effect of the electromagnetic force.

In order to determined the combined effect of the rotation and the electromagnetic force we may put [k.sub.x] = 0, [k.sub.y] = 0 for simplicity but [[OMEGA].bar] and [[H.bar].sub.o] are still as they are given in their general forms given by equations (7) and (8). In such a case the cumbersome dispersion relation (42) reduces to

[n.sup.6] - [E.sub.1][n.sup.4] + [E.sub.2][n.sup.2] - [E.sub.3] = 0 (57)

with

[E.sub.1] = [4[OMEGA].sup.2] + 2[mu][H.sup.2.sub.oz][k.sup.2.sub.z] / p + [H.sup.2.sub.oy][k.sup.2.sub.z] / p + [c.sup.2][k.sup.2] - 4[pi]G[[rho].sub.o] (58)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (59)

and

[E.sub.3] = [(2[mu][H.sup.2.sub.oz][k.sup.2.sub.z] / p).sup.2] ([c.sup.2][k.sup.2] - 4[pi]G[[rho].sub.o]) (60)

where

[[OMEGA].sup.2] = [[OMEGA].sup.2.sub.x] + [[OMEGA].sup.2.sub.y] + [[OMEGA].sup.2.sub.z] (61)

The stability criterion (57) is of sixth order algebraic equation in n. We expect that there will be three modes for which the proposing sinusoidal wave may be propagated in the medium. If we assume that the oscillation frequencies i.e. the roots of (57) are [n.sub.1], [n.sub.2] and [n.sub.3], then using the theory of equations we get

[n.sup.2.sub.1] + [n.sup.2.sub.2] + [n.sup.2.sub.3] = E1 (62)

[n.sup.2.sub.1] [n.sup.2.sub.2] [n.sup.2.sub.3] = E3

= [[mu].sup.2][H.sup.4.sub.oz][k.sup.4.sub.z] / [p.sup.2.sub.o] ([c.sup.2][k.sup.2] - 4[pi]G[[rho].sub.o]) (63)

By the use of equation of multiple roots [n.sub.1], [n.sub.2] and [n.sub.3] we see that if Jeans's criterion (cf. equation 50) is satisfied then one of the three roots is negative and consequently the model will be unstable with respect to one of the three modes.

We conclude that the Jeans's self-gravitating restriction of a streaming medium is not affected by the combined influence of the electromagnetic and rotational forces.

Appendix I

The elements [a.sub.ij] (i = 1,2,... ,8 and j = 1,2,... , 8) of the matrix [[a.sub.ij]] in equation (41) of the linear algebraic equations (30)-(38) are being

[a.sub.11] = n[[rho].sub.o], [a.sub.12] = [2[[rho].sub.o][[OMEGA].sub.z],

[a.sub.13] = -[2[[rho].sub.o][[OMEGA].sub.y] + [[rho].sub.o][DU.sub.o], [a.sub.14] = i[mu]([k.sub.y][H.sub.oy] + [k.sub.z][H.sub.oz]),

[a.sub.15] = i[mu][k.sub.x], [a.sub.16] = i [k.sub.x] [H.sub.oz], [a.sub.17] = i [k.sub.x][c.sup.2], [a.sub.18] = -i [[rho].sub.o][k.sub.x],

[a.sub.21] = - [2[rho].sub.o][[OMEGA].sub.z], [a.sub.22] = n[[rho].sub.o], [a.sub.23] = 2[[rho].sub.o][[OMEGA].sub.x], [a.sub.24] = 0,

[a.sub.25] = i[mu]([k.sub.y][H.sub.oy] + [k.sub.z][H.sub.oz]), [a.sub.26] = i[mu][k.sub.y][H.sub.oz,], [a.sub.27] = i [k.sub.y][c.sup.2], [a.sub.28] = -i [[rho].sub.o][k.sub.y],

[a.sub.31] = -2[[rho].sub.o][[OMEGA].sub.y], [a.sub.32] = 2[[rho].sub.o][[OMEGA].sub.x] , [a.sub.33] = n[[rho].sub.o], [a.sub.34] = 0,

[a.sub.35] = i[mu][k.sub.z][H.sub.oz], [a.sub.36] = i[mu]([2k.sub.z][H.sub.oz] + [k.sub.y][H.sub.oy]), [a.sub.37] = i [k.sub.z][c.sup.2], [a.sub.38] =- i [[rho].sub.o][k.sub.z],

[a.sub.41] = i([k.sub.y][H.sub.oy] + [k.sub.z][H.sub.oz]), [a.sub.42] = 0, [a.sub.43] = 0, [a.sub.44] =-n,

[a.sub.45] = 0, [a.sub.46] = i[k.sub.z][U.sub.o], [a.sub.47] = 0, [a.sub.48] = 0,

[a.sub.51] = i[k.sub.x][H.sub.oy], [a.sub.52] = i[k.sub.z][H.sub.oz], [a.sub.53] = ?i [k.sub.z][H.sub.oy],

[a.sub.54] = 0, [a.sub.55] = ?n, [a.sub.56] = 0, [a.sub.57] = 0, [a.sub.58] = 0,

[a.sub.61] = ?i[k.sub.x][H.sub.oz], [a.sub.62] = ?i [k.sub.y][H.sub.oz] , [a.sub.63] = i [k.sub.y][H.sub.oy], [a.sub.64] = 0,

[a.sub.65] = 0, [a.sub.66] = ?n, [a.sub.67] = 0, [a.sub.68] = 0,

[a.sub.71] = i [[rho].sub.o][k.sub.x], [a.sub.72] = i [[rho].sub.o][k.sub.y], [a.sub.73] = i [[rho].sub.o][k.sub.z], [a.sub.74] = 0,

[a.sub.75] = 0, [a.sub.76] = 0, [a.sub.77] = ?n, [a.sub.78] = 0,

[a.sub.81] = 0, [a.sub.82] = 0, [a.sub.83] = 0, [a.sub.84] = 0, [a.sub.85] = 0,

[a.sub.86] = 0, [a.sub.87] = 4[pi]G, [a.sub.88] = [k.sup.2]

with D = d / dz.

References

[1] Jeans, J. H., 1902, Philos. Trans. R. Soc., London 199, 1.

[2] Chandrasekhar, S. and Fermi, E., 1953 J. Astriphys, 118, 116.

[3] Chandrasekhar, S., Hydrodynamic and Hydromagnetic Stability, Dover Publ., New York, 1981.

[4] Sengar, R. S., 1981, Proc. Acad. Sci. India, 51A, 39.

[5] Radwan, A. E. and Elazab, S. S., Magnetogravitational stability of variable streams, 1988, J. Phys. Soc. Japan, 57, 461.

[6] Ahmed E. Radwan, Helmy A. Radwan and Mohamed Hendi, Magnetodynamic stability of cylindrical fluid jet, Chaos, Solitons and Fractals, 12:1729-1735, 2001.

Helmy A. Radwan

Maths. Dept., Faculty of Science, Omer El-Mokhtar University, El-Bidya, Lybia. Permanent Address: Maths. Dept., Faculty of Sc., Ain Shams Univ., Cairo, Egypt.
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Date:Jun 1, 2012
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