1. MAGNETOHYDRODYNAMIC INSTABILITY OF
A COMPRESSIBLE ANNULAR MGNETIZED FLUID
Abstract
The capillary stability of a compressible magnetized fluid jet surrounding very dense cylinder is investigated. The object of the present work is to investigate the magneto-hyrodynamic (MHD) instability of a compressible annular fluid jet coaxial with very dense fluid cylinder of negligible motion and surrounded by tenuous medium of negligible inertia. This model is acted by the electromagnetic (with varying magnetic field) and the curvature pressure gradient forces. The derived dispersion relation is valid for all axisymmetric m = 0 and non-axisymmetric m ≠ 0 modes of perturbations where m is an integer called the azimuthal wave number . Some reported works are recovered from the present general result as limiting cases. Such a study has applications in the planetary physics

2. 1. Introduction
1.1 The Concept of Stability
Here in we follow the concept of stability which is given by Chandrasekhar
[1]. Suppose that we have a hydrodynamic system which, in accordance with its
governing equations, is in a stationary state, i.e. in a state in which none of the variables
describing it is a function of time. Let
be a set of parameters which
define the system. These parameters will include geometrical parameters
such as the dimensions of the system; parameters characterizing the forces which
may be acting on the system, such as pressure gradient, temperature gradient,
magnetic fields, rotation,….. etc.

3. In considering the stability of such a system (with a given set of parameters
we essentially seek the reaction of the system to small disturbance especially, we ask: if the system is disturbed, will the disturbance gradually die down, or will the disturbance grow in amplitude in such a way that the system progressively departs from the initial state and never reverts to it ? In the former case, we say that the system is stable with respect to the particular disturbance and in the latter case, we say that it is unstable.
1.2 Stability techniques
There are several methods for solving the stability problems [2].
We may mention here some of them, e.g., the variation principle method,
the energy method and the method of the normal mode analysis. However,
every method has its advantages and disadvantages. In the present work
we have used the normal mode technique for the perturbation analysis [3]-[4].

4. The dispersion relation of a full liquid jet endowed with surface tension is derived by Chandrasekhar [1] as
where σ, S, ρ, R, k, m and are the growth rate, the surface tension
coefficient ,fluid density , radius of the cylinder ,the longitudinal wave
number , the azimuthal wave number and modified Bessel function of
first kind [5]. We deduce from this equation that the liquid jet is stable
for all purely non-axisymmetric deformations; but it is unstable for
axisymmetric varicose deformations with wave lengths exceeding the
circumference of the cylinder. This last result is due to Plateau-Rayleigh [1]. In the present work we make an extension to the pervious equation taking under consideration the effect of magnetic field and the curvature pressure gradient forces on a compressible fluid. The field of MHD has been initiated by Alfvén and others [6], for which he received the Nobel Prize in The set of equations which describe MHD are a combination of the Navier –Stokes equations of fluid dynamics and Maxwell’s equations of electromagnetism. The study of Magnetohydrodynamics stability is important in Geophysics, Astrophysics and other fields [7].

5. Sketch of MHD compressible annular fluid cylinder
2 . Formulation of the Problem
fluid
tar
tenuous medium
Figure (0)
Sketch of MHD compressible annular fluid cylinder

7. (16)

8. 4. Perturbation Analysis
For small departures from the equilibrium state, the different variable
quantities could be expressed as its unperturbed part plus a fluctuation part
(17)
where 0 indicates the steady state value and 1 the perturbation
stands for each of
is an infinitesimal amplitude of the perturbation at any instant of time t, given by
(18)

9. where
is the initial amplitude and
is the temporal amplification of the
perturbation ( growth rate ). If
is the imaginary then
is the oscillation frequency
Consider a sinusoidal wave along the gas – tenuous interface, using
expansion (17) the perturbed cylinder radial distance of the gas jet
is described by
(19)
where
(20)
is the elevation of the surface wave measured from the unperturbed position,
Here k (a real number ) is the longitudinal wave number
and m (an integer)
is the azimuthal
wave number.

11. (31)

12. (39)
(40)
(41)
5. Boundary Conditions

13. longitudinal compressible wave number
(42)
(43)
where
=qkRo
while
is the dimensionless
longitudinal compressible
wave number
(44)
where
is the ordinary dimensionless longitudinal
wave number
The following eigenvalue relation is obtained

14. (45)
where
and
6. Numerical Calculations

22. References
[1] Chandrasekhar, S., "Hydrodynamic and Hydromagnetic Stability", (Dover Publ., New York, U. S. A.) (1981).
[2] Nayfeh, A. H.," Perturbation Methods", (John Willey & Sons, Inc.) (1973).
[3] Rosensweig, R.E., "Ferrohydrodynamics", (Cambridge, London U.K.) (1985).
[4] Drazin, P. G. and Reid, W. H., "Hydrodynamic Stability", (Cambridge Univ. Press, London, U. K.) (1980).
[5] Abramowitz M. and Stegun I., "Handbook of Mathematical Functions", (Dover Publ., New York, U. S. A.) (1970).
[6] Alfvén, H., Cosmical Electrodynamics, International Series of Monographs on physics, Oxford, England, (1950).
[7] David p. Stern, NASA, “The Sun’s Magnetic Cycle” (2006).
[8] Radwan, A. E. and Elmahdy, E. E., "Capillary Instability of a streaming fluid –core liquid jet under their self gravitating forces" J. Mech. and Mech. Eng vol. 8, No.1. (2005) pp
[9] Emad E. Elmahdy and Mohamed O. Shaker, “Magnetohydrodynamics Stability of a Compressible Fluid Layer below A Vacuum Medium” J. F. C vol. 33, May(2008) pp