Stability of Compressible Hollow Jet Pervaded by a Transverse Varying Magnetic Field

1 Department of Mathematics, Women’s University College, Ain Shams University, Asmaa Fahmy Street, Heliopolis, Cairo 11566, Egypt 2 Department of Engineering Physics and Mathematics, Faculty of Engineering, Ain Shams University, Cairo 11566, Egypt 3 Department of Basic and Applied Sciences, College of Engineering and Technology, Arab Academy for Science & Technology and Maritime Transport (AAST), P.O. Box 2033, Elhorria, Cairo 11361, Egypt 4 Department of Engineering Physics and Mathematics, Faculty of Engineering (Mataria), Helwan University, Cairo 11321, Egypt


Introduction
The stability of a fluid cylinder subjected to different external forces has been published and reported by Rayleigh 1 and Chandrasekhar 2 . The instability of the mirror case of a gas cylinder surrounded by an ideal fluid endowed with surface tension is envised by 2 ISRN Mathematical Physics Chandrasekhar 2 for axisymmetric perturbation. See also Drazin and Reid 3 and Cheng 4 . Kendall 5 made very neat experiments to check the breaking-up of this model due to its instability as the inertia force of the exterior liquid is predominant over that of the gas cylinder. Moreover, he did write about its applications in several domains of science and attracted the attention of researchers for elaborating the stability of this model. One has to infer here that the analytical results given by Cheng 4 , in 2.4 and 2.5 , are incorrect in the third term. In fact, the term 1−s 2 −k 2 R 2 0 must be in the numerator as it is clear from 2.3 . See also 5.1 -6.1 in the present work and Drazin and Reid's result 3, page 16 , and also Chandrasekhar's dispersion relation 2, page 538 and page 540 , 147 and 155 there . For other works extending some of the previous results, we may refer to  In all these foregoing studies, the fluids are considered to be incompressible and nonviscous. See also Chen and Lin 10 . In reality, in the model of a gas jet injected into a liquid, we have to consider the fluid is compressible. In 11,12 ,Radwan et al. tried to investigate the instability of that model pervaded by constant magnetic field.
The stability of different cylindrical models under the action of selfgravitating force in addition to other forces has been elaborated by Radwan and Hasan 13,14 . Hasan 15 has discussed the stability of oscillating streaming fluid cylinder subjected to combined effect of the capillary, selfgravitating, and electrodynamic forces for all axisymmetric and nonaxisymmetric perturbation modes.
There are many applications of magnetohydrodynamic stability in several fields of science such as the following.
i Geophysics: the fluid of the core of the Earth and other theorized to be a huge MHD dynamo that generates the Earth's magnetic field due to the motion of the liquid iron.
ii Astrophysics: MHD applies quite well to astrophysics since 99% of baryonic matter content of the universe is made of plasma, including stars, the interplanetary medium, nebulae and jets, stability of spiral arm of galaxy, and so forth. Many astrophysical systems are not in local thermal equilibrium and, therefore, require an additional kinematic treatment to describe all phenomena within the system.
iii Engineering applications: there are many forms in engineering sciences include oil and gas extraction process if it is surrounded by electric field or magnetic field, gas and steam turbines, MHD power generation systems, magneto-flow meters, and so forth.
Here, we present a complete analysis of the stability of compressible hollow cylinder pervaded by a transverse varying magnetic field for all axisymmetric and nonaxisymmetric modes of perturbation where the fluid velocity is not solenoidal any more in the present analysis.

Formulation of the Problem
Consider a gas cylinder of radius R 0 surrounded by a non-viscous, compressible and perfectly conducting liquid. The liquid is pervaded by the uniform magnetic field H 0 0, 0, H 0 while the gas cylinder is penetrated by the transverse varying magnetic field H g 0 0, βH 0 r/R 0 , 0 , where β is a parameter satisfying certain restrictions and H 0 is the intensity of the magnetic field in the liquid region, see Figure 1. We shall use the cylindrical coordinates r, ϕ, z with the z-axis coinciding with the axis of the gas cylinder; the components of H 0 and H g 0 are considered along the coordinates r, ϕ, z . The model is acting upon the electromagnetic, capillary, inertia and pressure gradient forces such that the liquid inertia force is paramount over that of the gas cylinder. The fundamental equations for such a study are the combination of the ordinary fluid dynamics, Maxwell electromagnetic equation and those of the perfect gas. Under the present circumstances these equations are given as follows.
Equations of motion: where ρΠ is the total magnetohydrodynamic pressure which is the sum of kinetic and magnetic pressures, given by Evaluation equations of a magnetic field derived from the electromagnetic Maxwell equation in a perfectly conducting fluid:

2.10
In the gas cylinder region, surrounded by a liquid, there is no current flow: ∇ · H g 0, 2.11 Along the gas liquid interface, the curvature pressure P s due to the capillary force is given by where S is the surface tension coeffient, while r 1 and r 2 are the principle radii of curvature of the gas fluid and N the unit outward drawn normal vector to the perturbed interface, F r, ϕ, z, t 0, given by Here, u r , u ϕ , u z and H r , H ϕ , H z are, respectively, the components of the velocity vector u of the fluid and the magnetic field intensity H, while ρ, P, C v , and T are the fluid mass density, kinetic pressure, specific heat at a constant volume, and temperature of the fluid, and H g is the intensity of the magnetic field in the gas cylinder. The unperturbed state is studied by simplifying the fundamental equation in view of u 0 0, ∂/∂ϕ 0 and ∂/∂z 0 and integrating the resulting differential equations. Finally, by means of the continuity of the normal component of the stress tensor across the gas-fluid interface at r R 0 , the kinetic pressure of the fluid in the unperturbed state is obtained, namely, Equation 2.16 gives the kinetic pressure P 0 of the fluid and it is a simple linear combination of the contribution of the different forces effects.
a −S/R 0 is the contribution due to the capillary force.
b μH 2 0 /2 β 2 is the contribution due to the magnetodynamic force acting in the gas region.
c μH 2 0 /2 −1 is the contribution due to the magnetodynamic force acting in the fluid region. d P g 0 is the gas kinetic pressure and it must be suitably strong otherwise the model collapses and the gas spreads into fluid region surrounding the gas cylinder.
One has to refer here that the net magnetodynamic force acting on the model has no contribution in the balance of the total pressure cf. 2.16 in the unperturbed state in the following cases: To maintain and keep the model without collapsing in the unperturbed state, it must be P 0 > 0, this means that the contribution of the capillary force S/R 0 must be less than the gas pressure P g 0 , otherwise, the model will collapse and the gas may spread through the fluid region. In the general case, in order that P 0 ≥ 0, the gas kinetic pressure P g 0 in the initial state must satisfy the restriction: Otherwise, the model collapses and will be a homogeneous fluid medium.

Perturbation Analysis
For small departures from the unperturbed state, due to the perturbation along the gas-fluid interface, every variable quantity Q r, ϕ, z, t may be expressed as its unperturbed part plus a fluctuation part see 11, 12 , viz.
Here, Q stands for each of u, H, ρ, P, T, and the perturbed cross-section radius distance of the gas-fluid interface. The amplitude δ of the perturbation is given by where δ 0 δ at t 0 is the initial amplitude and σ is the temporal amplification at any instant of time t. Consider a sinusoidal wave along the gas-fluid interface, for a single Fourier term, the perturbed cylindrical radial distance is described by For the fluid surrounding the gas jet, where a is the sound speed in the gas defined by a γP 0 /ρ 0 1/2 , and γ C p /C v is the ratio of specific heats of the fluids.

ISRN Mathematical Physics 7
For the gas region surrounded by the fluid, Along the gas-liquid perturbed interface, 3.14 By the aid of the series expansion 3.1 and the time-space dependence 3.4 , based on the linear perturbation technique, every physical quantity Q r, ϕ, z, t could be expressed as This means that any perturbed quantity could be expressed as an amplitude function of r times the space-time dependence exp σt i kz mϕ . Upon utilizing the expansion, 3.15 , 3.5 , and 3.11 yield where e z is a unit vector in the z-direction and Ω A is the Alfven wave frequency defined in terms of H 0 as

3.18
Combining the z-components of 3.16 and 3.17 which are we obtain

ISRN Mathematical Physics
By substituting 3.6 into 3.20 , we finally obtain Again, inserting 3.21 into 3.6 yields Consequently, the components u 1r , u 1ϕ , and u 1z of u 1 are given, from the vector equation 3.16 , by

3.24
By inserting 3.24 into the nonsolenoid equation 3.9 which is 1 r and by taking into account the space ϕ, z -dependence 3.15 , we get .26 is an ordinary second-order differential equation; its solution is given in terms of the ordinary Bessel functions of order m with imaginary argument. For the problem under consideration, apart from the singular solution, the finite solution of 3.26 is given by Therefore, Π 1 r, ϕ, z, t is being where A is a constant of integration to be determined, while K m ηr is modified Bessel function of second kind of order m.
It is worthwhile to mention here that by means of 3.29 , the components of u 1 u 1r , u 1ϕ , u 1z and H 1 H 1r , H 1ϕ , H 1z and also P 1 could be identified from 3.16 , 3.17 , and 3.22 explicitly. Equation 3.13 means that the magnetic field H g 1 in the perturbation state can be derived by means of a scalar function that, by using 3.12 , satisfies Laplace's equation. The latter, by means of the expansion 3.15 , transforms to an ordinary second-order differential equation whose solution is given in terms of cylindrical functions. For the problem under consideration, the nonsingular solution gives H g 1 in the form: where B is an arbitrary constant of integration to be identified, while I m kr is the modified Bessel function of the first kind of order m. Based on the space dependence 3.15 , the curvature pressure due to the capillary force along the gas-fluid interface is given, from 3.14 , in the form:

Boundary Conditions
The solution of the relevant perturbation equations 3.5 -3.14 represented by 3.15 -3.31 and the solution of the unperturbed system of equations represented by 2.16 must satisfy appropriate boundary conditions. Under the present circumstances for the problem at hand, these boundary conditions are given as follows.
i The normal component of the magnetic field must be continuous across the gasfluid interface 3.3 at r R 0 . This condition may be formulated as follows: where N N 0 εN 1 is a unit outward vector normal to the gas-fluid interface given by 2.15 , with being the equation of the perturbed interface. Then, where x kR 0 is the dimensionless longitudinal wave number.
ii The normal component u r of the velocity u must be compatible with the velocity of the deformed gas-fluid interface 3.3 at r R 0 . This condition reads By substituting u 0 , u 1 , N 0 , N 1 ,and r into 4.5 , this yields As we seen, complete nonsingular solutions for the variables of the problem have been obtained. Here, for the aim of stability theory, one has to make one more step to identify the stability criterion upon applying some compatibility condition.
iii This compatibility condition states that the normal component of the total stress tensor concerning the kinetic and magnetic pressures must be discontinuous by the curvature pressure due to the capillary force, across the gas-fluid interface 3.3 at r R 0 .
Mathematically, this reads

4.7
Upon substituting into the condition 4.7 about the different variables, the following eigenvalue relation is obtained: where y ηR 0 is the dimensionless longitudinal wave number due to compressibility.

Limiting Cases
where a → ∞ , we find y → x . Upon using the recurrence relation, The relation 5.3 is indicated by Chandrasekhar 2, page 540 , for the first time as a dispersion relation for the mirror case of a full fluid cylinder surrounded by vacuum. For the discussion of this relation, we may refer to Radwan and Elazab 6 as we neglect the contribution of the viscosity there. If we suppose that a → ∞ so y → x , H 0 0 for m / 0, the relation 4.8 gives This relation coincides with the relation given by Drazin and Reid 3 . The capillary eigenvalue relation of a compressible hollow cylinder is given from 4.8 as H 0 0, in the form: The magnetodynamic eigenvalue relation of a compressible hollow cylinder is given from the general relation 4.8 as 0, in the form

Stability Discussions
The general eigenvalue relation 4.8 of the model under consideration is a quadratic relation in σ 2 . Therefore, we have to distinguish between the following different cases.
i The model will be ordinary stable, as σ 2 is negative.
ii The model will be ordinary unstable, as σ 2 is positive.
iii The model will be marginally stable, as σ 2 is zero.
In order to judge such cases and investigate the stability of the present model, we have to write down about the character and behavior of the modified Bessel functions I m and K m and their derivatives.
Consider the recurrence relations cf. 16 : For each nonzero real value of x. Also I m x is positive definite and monotonic increasing, while K m x is monotonic decreasing but never negative: Therefore, we may see, for a nonzero real value of x, that In view of 3.27 , we see that y → x as a → ∞, where the fluid in this case is an incompressible one. Consequently, the stability of the hollow cylinder under the action of the combined effects of the capillary and electromagnetic forces could be discussed.
As the model of a hollow cylinder is acted by the capillary force while the effect of the electromagnetic force is neglected, the eigenvalue relation 4.8 reduces to 5.5 . By means of the relation, the capillary stable and unstable regions of the hollow cylinder could be identified. By an appeal to the relations 6.1 and the inequalities 6.2 and 6.3 , the sign of σ 2 depends on the sign of the quantity 1 − m 2 − x 2 . Henceforth, we have the different cases: for m 0, is the dimensionless growth rate at the instant of time t.

ISRN Mathematical Physics 13
This means that the hollow cylinder is stable in the case 1 ≤ x < ∞ for m 0 and 0 ≤ x < ∞ for m ≥ 1, while it is unstable only as 0 ≤ x ≤ 1 for m 0. This results, of course, as a −1 → 0. However, for ordinary values of a: in such case, we have the modified Bessel functions with argument y which in turn is a function of the compressibility factor a γP 0 /ρ 0 1/2 . Now, for x / 0, y / 0 and y > x, we have I m y > I m x , K m x > K m y , I m y > I m x , K m x > K m y .

6.6
In view of these inequalities, the discussion of the relation 5.5 reveals that the compressibility has a stabilizing effect in all m 0 and m / 0 modes of perturbation for all wavelengths. The stabilizing effect becomes stronger for high compressibility and, therefore, it acts to overcome the capillary destabilizing influence.
As the hollow cylinder model is acted upon the electromagnetic force due to the pervading magnetic fields in the initial states and pressure gradient forces as S 0, the eigenvalue relation of such case is given, in its general form, by the relation 5.6 . The effect of the axial magnetic field in the liquid region is represented by the term −x 2 following μH 2 0 /ρ 0 R 2 0 . It contributes as a negative part in σ 2 , that is, σ is imaginary in this case. This means that it has a stabilizing effect on the model. The transverse varying magnetic field pervaded into the gas region is presented by the terms −β 2 yK m y /K m y and m 2 β 2 yI m x K m y /xI m x K m y following the natural quantity μH 2 0 /ρ 0 R 2 0 . In the axisymmetric mode, m 0, the transverse magnetic field is purely destabilizing. In the nonaxisymmetric modes, m ≥ 1, the transverse magnetic field is purely destabilizing in the term −β 2 yK m y /K m y while it is stabilizing due to the other terms. Therefore, we conclude that the transverse magnetic field is stabilizing or destabilizing according to restrictions.
Consequently, the electromagnetic forces in the gas and liquid regions are strongly stabilizing the model in the axisymmetric mode m 0, while they have a destabilizing influence in m ≥ 1 modes in the gas region only. Also, in such magnetodynamic case the compressibility has a stabilizing tendency not only in m 0 mode but also in the non-axisymmetric modes m ≥ 1. Its influence is decreasing the magnetodynamic unstable domain in m ≥ 1 and simultaneously increasing those of stability in m 0 mode.
One has to mention here it is argued that when the effects of materials compressibility are considered, the growth rate value may be reduced in comparison with the incompressible case. This is due to the fact that compression absorbs some of the energy which would, otherwise, go into fluid motion and causes more instability. For this reason, it is stated that the compressibility has a stabilizing tendency.
It is worth to mention here that the stabilizing effect of the electromagnetic force μ ∇ΛH ΛH may be interpreted as follows.
This force is interpreted as arising from the action on the fluid of Maxwell's stresses: a magnetic tension μ H · H /2 per unit area along the magnetic lines of force and equal magnetic pressure acting in all the directions in the conducting fluid. Taking into account that the latter is not perpendicular to the magnetic lines of force and acting in all directions because the diffusion term is neglected in the evolution equations of the magnetic field 2.9 . Due to these stresses, the lines of force are able to endow the fluid with a sort of rigidity.

Numerical Analysis
The main purpose of the numerical analysis is that one could determine exactly where are the MHD stable and unstable domains of the model under consideration influenced by the combined effects of the electromagnetic and capillary forces. In addition, we may identify the critical points which separate the stable and unstable domains for different values of the problem parameters.
To perform that, we have to write down the eigenvalue relation 4.8 in dimensionless form in the most important mode m 0 of perturbation. Now, since the natural quantity S/ρ 0 R 3 0 −1/2 has a unit of time −1 , we have σ * σ S/ρ 0 R 3 0 −1/2 which is the dimensionless growth rate. For m 0, the relation 4.8 leads to where σ * is the nondimensional growth rate.
If σ * 2 is negative, we write σ * iω * with i √ −1 being the imaginary factor where ω/2π is the oscillation frequency. The notation H 2 s S/μR 0 is with H s as a unit of magnetic field, and we have used the notation I 0 I 1 and K 0 −K 1 . The dispersion relation 7.1 has been computed for all short and long wavelengths in the new form in relation 7.2 which is The values of σ * corresponding the unstable domains and those of corresponding the stable domains are collected, tabulated, and presented graphically. It is found that there are many features of interest in this numerical analysis as we see in the following.

Conclusion
From the numerical discussion, we deduce that the compressibility has a strong stabilizing tendency for all wavelengths. The capillary force is destabilizing for a small domain of long wavelengths in the axisymmetric mode while it is stabilizing in all the rest. The electromagnetic force interior the gas is stabilizing. The electromagnetic force in the liquid region is stabilizing also. These results are in good agreement with the analytical discussions of relation 4.8 .