Magnetohydrodynamic (MHD) Stability of Oscillating Fluid Cylinder with Magnetic Field

The classical of the capillary instability of a gas cylinder submerged into a liquid are given for first time by Chandrasekhar[1] for axisymmetric perturbation. Hasan [2], Elazab et al. [3] and Drazin and Reid [4] gave the dispersion relation valid for all axisymmetric and nonaxisymmetric modes. Cheng [5] discussed the instability of a gas jet in an incompressible liquid for all modes of perturbation. However, we have to mention here that the results given by Cheng [5]. Kindall [6] performed experiments with modern equipment to check the breaking up of that model. Moreover, he attracted the attention for the importance of the stability and discussions of that model for its application in many domains of science. Concerning the hydrodynamic stability of a hollow jet endowed with surface tension we may refer to Chauhan et al.[7], Abramowitz and Stegun [8], Chen and Lin [9], Cousin and Dumouchel [10], Lee and Wang [11,12], Mehring and Sirignano [13], Parthasarathy and Chiang [14], Shen and Li [15], Shi et al. [16], Shukudov and Sisoev [17], and Villermaux [18], Melachlan and Kelly [19,20]. Soon afterwards a lot of researchers treated with the magneto-dynamic stability of such model analytically [21-23] and numerically upon utilizing appropriate basic equation and boundary condition. Hamdy M Brakat and Khloud R Khater studied the Hydromagnetic self-gravitating stability of streaming fluid cylinder with longitudinal magnetic filed [24]. It all foregoing works the liquid may be at rest or unform streaming in the unperturbed state. Here we explain the hydromagnetic stability of oscillating fluid cylinder with longitudinal magnetic field.


Introduction
The classical of the capillary instability of a gas cylinder submerged into a liquid are given for first time by Chandrasekhar [1] for axisymmetric perturbation. Hasan [2], Elazab et al. [3] and Drazin and Reid [4] gave the dispersion relation valid for all axisymmetric and non-axisymmetric modes. Cheng [5] discussed the instability of a gas jet in an incompressible liquid for all modes of perturbation. However, we have to mention here that the results given by Cheng [5]. Kindall [6] performed experiments with modern equipment to check the breaking up of that model. Moreover, he attracted the attention for the importance of the stability and discussions of that model for its application in many domains of science. Concerning the hydrodynamic stability of a hollow jet endowed with surface tension we may refer to Chauhan et al. [7], Abramowitz and Stegun [8], Chen and Lin [9], Cousin and Dumouchel [10], Lee and Wang [11,12], Mehring and Sirignano [13], Parthasarathy and Chiang [14], Shen and Li [15], Shi et al. [16], Shukudov and Sisoev [17], and Villermaux [18], Melachlan and Kelly [19,20]. Soon afterwards a lot of researchers treated with the magneto-dynamic stability of such model analytically [21][22][23] and numerically upon utilizing appropriate basic equation and boundary condition. Hamdy M Brakat and Khloud R Khater studied the Hydromagnetic self-gravitating stability of streaming fluid cylinder with longitudinal magnetic filed [24]. It all foregoing works the liquid may be at rest or unform streaming in the unperturbed state. Here we explain the hydromagnetic stability of oscillating fluid cylinder with longitudinal magnetic field.

Formulation of the problem
We consider a fluid cylinder (with negligible motion) of radius R 0 surrounded by an oscillating liquid cylinder with velocity, Where U and Ω are the amplitude and oscillation frequency of the velocity.
The interior cylinder is being a gas with constant pressure 0 where H 0 is the intensity of the magnetic field and α is the parameter of 0 g H . The components of equations (1 -3) are considered along the cylindrical polar coordinates (r, Φ, z) with the z-axis is coinciding with the axis of the hollow jet. The model is acted by the inertia, pressure gradient, capillary and electromagnetic forces. The hydro-magnetic fundamental equations appropriate for studying the stability of the fluid model under consideration are the combination of the pure hydrodynamic equations and those of Maxwell concerning the electromagnetic theory.
These equations may be given as follows in the liquid region.  (8) In the region surrounding the gas cylinder, we have and along the gas-liquid interface the curvature pressure is, where H and g H are the magnetic field intensities in the gas and liquid regions, ρ is the mass density, u velocity vector, P kinetic pressure, and P s the curvature pressure due to the capillary force, T the surface tension coefficient, N is the unite outward normal vector to the gasliquid interface. In the unperturbed state equation (4) reduces to where Π is the total hydromagnetics pressure which is the sum of the liquid kinetic pressure and magnetic pressure, and equation (8) can be rewritten as using the equations (4) and (14) we get, The continuity of the total stress tensor across the gas-liquid interface at (r = R 0 ) yields, the unperturbed pressure P 0 of the liquid as Here 0 g P is the gas constant pressure in the initial state, is the contribution of capillary force, while is the net magnetic pressure due to the effect of electromagnetic force acting in the gas and liquid regions,

Perturbation analysis
For small departures from the equilibrium state due to an infinitesimal perturbation every perturbed quantity Q(r, q, z, t) can be expanded as; where the quantities with subscript 0 are those of equilibrium while those with 1 are small increments due to the perturbation. The Q(r, ϕ, z, t) stands for The amplitude ε of the perturbation is given by Where ε 0 is the initial amplitude at (t=0) and σ is the temporal amplification of instability. Consider a sinusoidal wave along the gasliquid interface, using a single fourier term, the perturbed cylinder radial distance of the gas jet is described by Is the elevation of the surface wave measured from unperturbed position. Here K (any real number) is the longitudinal wave number, and m an integer is the azimuthal wave number. In view of the foregoing expansions (18 -21), the relevant perturbation equations are given as follows. For the interior of the cylindrical jet and by using equations (6), (7) and (8) we get, In the gas region, as along the gas-liquid interface, Based on the equations (18 -21) every fluctuating quantity Q 1 (r, 0, z, t) could be written as 1 1 ( , 0, , ) = ( ) exp( ). Q r z t Q r t ikz σ + By the aid of this expansion, equations (22) and (24) are combined to give magnetic field intensity in the form, Since the liquid is assumed to be non-dissipative and irrotational, the velocity 1 u could be derived from a scalar function Φ 1 (r, 0, z, t) such that, Combining equations (24) and (32) we get, Combining equations (28) and (34) we get, . So by using equations (24) we get,

The magneto-dynamic condition
This condition is being "the normal component of the magnetic field must be continuous across the gas-liquid interface at (r = R 0 )",mathematically this condition is given by

Stresses condition
This condition is being" The jump of the normal component of the stresses in the gas and liquid regions must be discontinuous by the surface pressure P 1s across the cylindrical gas-liquid interface at (r = R 0 ) ". This condition is being From this equation we get the dispersion relation

General Discussion
The relation (56) is the dispersion equation of a fluid cylinder acting upon the electromagnetic force. It relates the growth rate σ with the modified Bessel function I 0 (x) and K 0 (x) and their derivative, the wave number m and x, the amplitude U of the streaming velocity, Ω the oscillation frequency of the oscillating streaming, α the parameter of the magnetic field in the gas cylinder and with the parameters T, ρ, R 0 , µ and H 0 of the problem. One has to mention here that the relation (56) contains as unit of (time) 2 and this fact is vital in formulating (56) in dimensionless form. Relation (56) is a simple linear combination of eigenvalue relations of gas-liquid cylinder subject to the electromagnetic force only.It is remarkable that in the axisymmetric mode at m = 0 for a fluid cylindrical jet pervaded by constant axial field, subject to its own attraction and lorentz force. This character of a simple linear combination is also true if the acting force are the capillary and electromagnetic forces. Whether the model are a full liquid-gas cylinder. The physical interpretation of such phenomenon is given explicitly in this paper reference. Since the relation (56) is somewhat more general, several stability criteria can be obtained as limiting cases from the eigenvalue of relation (60). The discussion of this equation reveals that the uniform streaming of the liquid has destabilizing effect, and that effect is valid not only in the axisymmetric mode m = 0.
If we assume that U = 0, then the eigenvalues relation(56) will be = 0. ∇ Π (37) Now,we may see that the system of perturbed equations (22) and (24 -28) could be solved, as Laplace,s equations (33), (35) and (37) are solved for this task, we may write the expansion 1 1 0 for Φ 1 (r, 0, z, t), Ψ 1 (r, 0, z, t) and 1 (r, 0, z, t). Substituting equation (38) into equations (33), (35) and (37), we obtain the total second order differential equation of Bessel. For the problem under consideration, the nonsingular solutions of (33), (35) and (37),are given by Where C 1 ,C 2 and C 3 are constant of integration to be determined, while K 0 and I 0 are Bessel functions of the first and second kind order m = 0. Along the gas-liquid cylinder fluid interface, the surface pressure in the perturbed state due to the capillary force, in view of equations (22) and (29), given by Where (x = kR 0 ) is the longitudinal dimensionless wave number.

Boundary Condition
The solution of the unperturbed and perturbed sates equations (39),(41) must satisfy appropriate boundary condition across the fluid interface at (r = R 0 ), these conditions are given as follows

Kinematic condition
The first condition states that "the normal component of the velocity u of the liquid must be compatible with the perturbed boundary gasliquid at (r = R 0 ). This condition is given, The second boundary condition could be described and utilized as follows, by using equations (24) and (31)

Conclusion
From the foregoing discussions, we may conclude the following: (1) The axial magnetic field pervaded interior the fluid cylinder has stabilizing effect.
(2) The stability behavior of the model comes after the destabilizing behavior of the model when it be reduced and suppressed.
(3) The capillary force is destabilizing only in a small axisymmetric domains and all domains of non-axisymmetric perturbations.
(4) The present model is purely stabilizing under the acting electromagnetic in gas and liquid regions.
(5) For very high intensity of magnetic field the destabilizing character of the model could be suppressed completely for all value of wave lengths.