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ISRN Mathematical Physics
Volume 2012 (2012), Article ID 531250, 18 pages
Research Article

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

1Department of Mathematics, Women's University College, Ain Shams University, Asmaa Fahmy Street, Heliopolis, Cairo 11566, Egypt
2Department of Engineering Physics and Mathematics, Faculty of Engineering, Ain Shams University, Cairo 11566, Egypt
3Department 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
4Department of Engineering Physics and Mathematics, Faculty of Engineering (Mataria), Helwan University, Cairo 11321, Egypt

Received 13 October 2011; Accepted 16 November 2011

Academic Editor: G. Cleaver

Copyright © 2012 Samia S. Elazab et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


The magnetohydrodynamic stability of an ordinary compressible hollow cylinder pervaded by a transverse varying magnetic field, under the influence of capillary, inertia, and Lorentz force, has been developed. The problem is modelized. The basic equations formulated, solved, and, upon applying appropriate boundary conditions, the singular solutions are excluded. The eigenvalue relation has been derived and discussed. The capillary force has destabilizing influence only for long wavelengths in the axisymmetric perturbation but it is stabilizing in the rest and also so in the nonaxisymmetric perturbations. The compressibility increases the stable domains and simultaneously decreases those of instability. The electromagnetic force has different effects due to the axial uniform field and varying transverse one. The axial field is stabilizing for all wavelengths in all kinds of perturbations. The transverse field is stabilizing or not according to restrictions. Here, the high compressibility increases rapidly the magnetodynamic stable domains and leads to shrinking those of instability.

1. 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 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 (1𝑠2𝑘2𝑅20) 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 Radwan [69]. 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.

2. Formulation of the Problem

Consider a gas cylinder of radius 𝑅0 surrounded by a non-viscous, compressible and perfectly conducting liquid. The liquid is pervaded by the uniform magnetic field 𝐻0=(0,0,𝐻0) while the gas cylinder is penetrated by the transverse varying magnetic field 𝐻𝑔0=(0,𝛽𝐻0𝑟/𝑅0,0), where 𝛽 is a parameter satisfying certain restrictions and 𝐻0 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 𝐻0 and 𝐻𝑔0 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.

Figure 1: Sketch for MHD hollow jet.

Equations of motion:𝜕𝑢𝑟+𝑢𝜕𝑡𝑢𝑟𝜇𝜌𝐻𝐻𝑟=𝜕Π𝜕𝑟,(2.1)𝜕𝑢𝜑+𝑢𝜕𝑡𝑢𝜑𝜇𝜌𝐻𝐻𝜑1=𝑟𝜕Π𝜕𝜑,(2.2)𝜕𝑢𝑧+𝑢𝜕𝑡𝑢𝑧𝜇𝜌𝐻𝐻𝑧=𝜕Π,𝜕𝑧(2.3) where 𝜌Π is the total magnetohydrodynamic pressure which is the sum of kinetic and magnetic pressures, given by𝜇𝜌Π=𝑃+2𝐻𝐻.(2.4) Equation of conservation of flux:𝐻=0.(2.5) Equation of conservation of mass of a compressible fluid:𝜕𝜌+𝑢𝜕𝑡𝜌+𝜌𝑢=0.(2.6) Equation of state gives the relation between the pressure 𝑃 and density 𝜌:𝑃=𝜌𝑅𝑐𝑇,(2.7)𝑅𝑐 is a general constant of gas.

Equation of conservation of energy:𝜌𝐶𝑣𝜕𝜕𝑡+𝑢𝑇+𝑃𝑢=0.(2.8) Evaluation equations of a magnetic field (derived from the electromagnetic Maxwell equation) in a perfectly conducting fluid:𝜕𝐻𝑟+𝑢𝜕𝑡𝐻𝑟+𝑢𝐻𝑟=𝐻𝑢𝑟,(2.9)𝜕𝐻𝜑+𝑢𝜕𝑡𝐻𝜑+𝑢𝐻𝜑=𝐻𝑢𝜑,𝜕𝐻𝑧+𝑢𝜕𝑡𝐻𝑧+𝑢𝐻𝑧=𝐻𝑢𝑧.(2.10) In the gas cylinder region, surrounded by a liquid, there is no current flow:𝐻𝑔=0,(2.11)Λ𝐻𝑔=0.(2.12) Along the gas liquid interface, the curvature pressure 𝑃𝑠 due to the capillary force is given by𝑃𝑠1=𝑆𝑟1+1𝑟2,(2.13) with𝑟11+𝑟21=𝑁,(2.14) where 𝑆 is the surface tension coeffient, while 𝑟1 and 𝑟2 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 by𝑁=𝐹(𝑟,𝜑,𝑧,𝑡)||||𝐹(𝑟,𝜑𝑧,𝑡).(2.15) Here, (𝑢𝑟,𝑢𝜑,𝑢𝑧) and (𝐻𝑟,𝐻𝜑,𝐻𝑧) are, respectively, the components of the velocity vector 𝑢 of the fluid and the magnetic field intensity 𝐻, while 𝜌,𝑃,𝐶𝑣, and 𝑇 are the fluid mass density, kinetic pressure, specific heat at a constant volume, and temperature of the fluid, and 𝐻𝑔 is the intensity of the magnetic field in the gas cylinder.

The unperturbed state is studied by simplifying the fundamental equation in view of 𝑢0=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=𝑅0, the kinetic pressure of the fluid in the unperturbed state is obtained, namely,𝑃0=𝑃𝑔0+𝜇2𝛽2𝐻120𝑆𝑅0.(2.16) Equation (2.16) gives the kinetic pressure 𝑃0 of the fluid and it is a simple linear combination of the contribution of the different forces effects.(a)𝑆/𝑅0 is the contribution due to the capillary force. (b)(𝜇𝐻20/2)𝛽2 is the contribution due to the magnetodynamic force acting in the gas region.(c)(𝜇𝐻20/2)(1) is the contribution due to the magnetodynamic force acting in the fluid region.(d)𝑃𝑔0 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:(1)as 𝐻0=0,(2)as 𝐻00, 𝛽=1.

To maintain and keep the model without collapsing in the unperturbed state, it must be 𝑃0>0, this means that the contribution of the capillary force 𝑆/𝑅0 must be less than the gas pressure 𝑃𝑔0, otherwise, the model will collapse and the gas may spread through the fluid region.

In the general case, in order that 𝑃00, the gas kinetic pressure 𝑃𝑔0 in the initial state must satisfy the restriction:𝑃𝑔0𝑆𝑅0+𝜇𝐻2021𝛽2.(2.17) Otherwise, the model collapses and will be a homogeneous fluid medium.

3. 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.)𝑄(𝑟,𝜑,𝑧,𝑡)=𝑄0(𝑟)+𝛿(𝑡)𝑄1(𝑟,𝜑,𝑧).(3.1)

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 by𝛿=𝛿0exp(𝜎𝑡),(3.2) where 𝛿0(=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 by𝑟=𝑅0+𝑅1,(3.3) with𝑅1=𝛿0[]exp𝜎𝑡+𝑖(𝑘𝑧+𝑚𝜑)(3.4) being the elevation of the surface wave measured from the unperturbed position, where 𝑘 and 𝑚 are the axial and transverse wave numbers. From the view point of the foregoing expansions (3.1)–(3.4), the relevant perturbation equations of the present case could be deduced from the fundamental equations (2.1)–(2.15) as follows.

For the fluid surrounding the gas jet,𝜎𝑢1𝜇𝜌0𝐻0𝐻1=Π1,(3.5)𝜌Π1=𝑃1+𝜇22𝐻0𝐻1,(3.6)𝐻1=0,(3.7)𝜎𝜌1+𝜌0𝑢1𝜌=0,(3.8)0𝑎2𝑢1=𝜎𝑃1𝑃,(3.9)1𝑃0=𝜌1𝜌0+𝑇1𝑇0,(3.10)𝜎𝐻1𝑢=Λ1Λ𝐻0,(3.11) where 𝑎 is the sound speed in the gas defined by 𝑎=(𝛾𝑃0/𝜌0)1/2, and 𝛾=𝐶𝑝/𝐶𝑣 is the ratio of specific heats of the fluids.

For the gas region surrounded by the fluid,𝐻𝑔1=0,(3.12)Λ𝐻𝑔1=0(thereisnocurrent).(3.13) Along the gas-liquid perturbed interface,𝑃1𝑠=𝑆𝑅20𝑅1+𝑅20𝜕2𝑅1𝜕𝑧2+𝜕2𝑅1𝜕𝜑2.(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 as𝑄1(𝑟,𝜑,𝑧,𝑡)=𝑄1[](𝑟)exp𝜎𝑡+𝑖(𝑘𝑧+𝑚𝜑).(3.15) 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) yieldΩ𝜎+2𝐴𝜎𝑢1=Π1+𝑖Ω2𝐴𝑃1𝑘𝑎2𝜌0𝑒𝑧,(3.16)𝜎𝐻1=𝑖𝑘𝐻0𝑢1+𝜎𝐻0𝜌0𝑎2𝑃1𝑒𝑧,(3.17) where 𝑒𝑧 is a unit vector in the 𝑧-direction and Ω𝐴 is the Alfven wave frequency defined in terms of 𝐻0 asΩ𝐴=𝜇𝐻20𝑘2𝜌01/2.(3.18) Combining the 𝑧-components of (3.16) and (3.17) which are𝜎Ω1+𝐴𝜎2𝑢1𝑧=𝑖𝑘Π1+𝑖Ω2𝐴𝑘𝑎2𝜌0𝑃1,𝜎𝐻1𝑧=𝐻0𝑖𝑘𝑢1𝑧+𝜎𝑃1𝑎2𝜌0,(3.19) we obtain𝐻1𝑧=(𝑖𝑘)2𝐻0𝜎2+Ω2𝐴Π1+𝜇𝐻20𝑎2𝜌0𝑃1+𝐻0𝑎2𝜌0𝑃1.(3.20) By substituting (3.6) into (3.20), we finally obtain𝐻1𝑧=𝜎2+𝑎2𝑘2𝐻0𝜎2𝑎2𝜌0𝑃1.(3.21) Again, inserting (3.21) into (3.6) yieldsΠ1=𝜉𝜌0𝑃1.(3.22) With𝜉=1+𝜇𝐻20𝜎2+𝑎2𝑘2𝜎2𝑎2𝜌0.(3.23) Consequently, the components 𝑢1𝑟, 𝑢1𝜑, and 𝑢1𝑧 of 𝑢1 are given, from the vector equation (3.16), by𝜎2+Ω2𝐴𝑢1𝑟=𝜎𝜕Π1,𝜎𝜕𝑟2+Ω2𝐴𝑢1𝜑=𝑖𝑚𝜎𝑟Π1,𝜎2+Ω2𝐴𝑢1𝑧=𝑖𝑘𝜎𝑎2𝜉+𝜇𝐻20𝑎2𝜉Π1.(3.24) By inserting (3.24) into the nonsolenoid equation (3.9) which is1𝑟𝜕𝑟𝑢1𝑟+1𝜕𝑟𝑟𝜕𝑢1𝜑+𝜕𝜑𝜕𝑢1𝑧=𝜕𝑧𝜎𝑃1𝑎2𝜌0,(3.25) and by taking into account the space (𝜑,𝑧)-dependence (3.15), we get1𝑟𝑑𝑟𝑑𝑟𝑑Π1𝑚𝑑𝑟2𝑟2+𝜂2Π1=0,(3.26) with𝜂2=𝑘2+𝜎2𝑎2𝜉.(3.27) Equation (3.26) 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 byΠ1(𝑟)=𝐴𝐾𝑚(𝜂𝑟).(3.28) Therefore, Π1(𝑟,𝜑,𝑧,𝑡) is beingΠ1(𝑟,𝜑,𝑧,𝑡)=𝐴𝐾𝑚[](𝜂𝑟)exp𝜎𝑡+𝑖(𝑘𝑧+𝑚𝜑),(3.29) where 𝐴 is a constant of integration to be determined, while 𝐾𝑚(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 𝑢1(=(𝑢1𝑟,𝑢1𝜑,𝑢1𝑧)) and 𝐻1(=(𝐻1𝑟,𝐻1𝜑,𝐻1𝑧)) and also 𝑃1 could be identified from (3.16), (3.17), and (3.22) explicitly.

Equation (3.13) means that the magnetic field 𝐻𝑔1 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 𝐻𝑔1 in the form:𝐻𝑔1𝐼=𝐵𝑚[](𝑘𝑟)exp𝜎𝑡+𝑖(𝑘𝑧+𝑚𝜑),(3.30) where 𝐵 is an arbitrary constant of integration to be identified, while 𝐼𝑚(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:𝑃1𝑠=𝑆𝑅201𝑚2𝑘2𝑅20𝑅1.(3.31)

4. 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 gas-fluid interface (3.3) at 𝑟=𝑅0. This condition may be formulated as follows: 𝑁𝑜𝐻1+𝑁1𝐻𝑜=𝑁𝑜𝐻𝑔1+𝑁1𝐻𝑔𝑜,(4.1) where 𝑁(=𝑁0+𝜀𝑁1) is a unit outward vector normal to the gas-fluid interface given by (2.15), with 𝐹(𝑟,𝜑,𝑧,𝑡)=𝑟𝑅0𝑅1=0(4.2) being the equation of the perturbed interface. Then, 𝑁𝑜=(1,0,0),𝑁1=0,𝑖𝑚𝑅𝑜𝑅,𝑖𝑘1.(4.3)

By substituting from (4.3) and the required expressions concerning 𝐻0,𝐻1,𝐻𝑔0, and 𝐻𝑔1 into (4.1), we obtain 𝐵=𝑖𝑚𝛽𝐻0𝑥𝐼𝑚(𝑥),(4.4) where 𝑥(=𝑘𝑅0) is the dimensionless longitudinal wave number.(ii)The normal component 𝑢𝑟 of the velocity 𝑢 must be compatible with the velocity of the deformed gas-fluid interface (3.3) at 𝑟=𝑅0. This condition reads 𝑁0𝑢1+𝑁1𝑢0=𝜕𝑟𝜕𝑡.(4.5)

By substituting 𝑢0,𝑢1,𝑁0,𝑁1,and 𝑟 into (4.5), this yields 𝜎𝐴=2+Ω2𝐴𝜂𝐾𝑚𝜂𝑅0.(4.6)

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=𝑅0.

Mathematically, this reads 𝑃1+𝑅1𝜕𝑃0+𝜇𝜕𝑟22𝐻0𝐻1+𝑅1𝜕𝐻𝜕𝑟0𝐻0=𝑃1𝑠+𝜇22𝐻𝑔0𝐻𝑔1+𝑅1𝜕𝐻𝜕𝑟𝑔0𝐻𝑔0.(4.7)

Upon substituting into the condition (4.7) about the different variables, the following eigenvalue relation is obtained: 𝜎2=𝑆𝜌𝑅301𝑚2𝑥2𝑦𝐾𝑚(𝑦)𝐾𝑚+(𝑦)𝜇𝐻20𝜌𝑜𝑅2𝑜𝑥2+𝛽2+(𝑚𝛽)2𝐼𝑚(𝑥)𝑥𝐼𝑚(𝑥)𝑦𝐾𝑚(𝑦)𝐾𝑚(𝑦),(4.8) where 𝑦(=𝜂𝑅0) is the dimensionless longitudinal wave number due to compressibility.

5. Limiting Cases

Equation (4.8) is the required capillary instability eigenvalue relation for a compressible gas cylinder surrounded by a liquid and pervaded by a transverse varying magnetic field. It relates the temporal amplification 𝜎 or alternatively the oscillation frequency 𝜔 (i.e., if 𝜎(=𝑖𝜔) is imaginary) with the dimensionless (ordinary) longitudinal 𝑥 and 𝑦 (compressible) wave numbers, azimuthal wave number 𝑚, the fundamental quantity (𝑆/𝜌0𝑅30)1/2, as well as (𝜇𝐻20/𝜌0𝑅20)1/2 as a unit of time, the modified Bessel functions 𝐼𝑚 and 𝐾𝑚 of the first and second kind of order 𝑚 and their derivatives of different arguments, the parameter 𝛽 of the magnetic field pervading the interior of the gas cylinder, and with the parameters 𝑆,𝜌0,𝑅0,𝜇, and 𝐻0 of the problem.

Several reported works may be obtained as limiting cases from the general result (4.8).

For incompressible (𝑎), nonconducting (𝜇=0,𝐻0=0) fluid and axisymmetric perturbation 𝑚=0, the relation (4.8) yields𝜎2𝑆=𝜌𝑅30𝑥𝐾0(𝑥)𝐾0(𝑥)1𝑥2,(5.1) where (𝑎), we find (𝑦𝑥). Upon using the recurrence relation,𝐾0(𝑥)=𝐾1(𝑥),(5.2) the relation (5.1) becomes𝜎2=𝑆𝜌𝑅30𝑥𝐾1(𝑥)𝐾0(𝑥)1𝑥2.(5.3) 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 (𝑦𝑥), 𝐻0=0 for 𝑚0, the relation (4.8) gives𝜎2𝑆=𝜌𝑅30𝑥𝐾𝑚(𝑥)𝐾𝑚(𝑥)1𝑚2𝑥2.(5.4) 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 𝐻0=0, in the form:𝜎2=𝑆𝜌𝑅301𝑚2𝑥2𝑦𝐾𝑚(𝑦)𝐾𝑚(𝑦).(5.5) The magnetodynamic eigenvalue relation of a compressible hollow cylinder is given from the general relation (4.8) as =0, in the form𝜎2=𝜇𝐻20𝜌𝑜𝑅2𝑜𝑥2+𝛽2+(𝑚𝛽)2𝐼𝑚(𝑥)𝑥𝐼𝑚(𝑥)𝑦𝐾𝑚(𝑦)𝐾𝑚(𝑦).(5.6)

6. Stability Discussions

The general eigenvalue relation (4.8) of the model under consideration is a quadratic relation in 𝜎2. Therefore, we have to distinguish between the following different cases.(i)The model will be ordinary stable, as 𝜎2 is negative.(ii)The model will be ordinary unstable, as 𝜎2 is positive.(iii)The model will be marginally stable, as 𝜎2 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 𝐼𝑚 and 𝐾𝑚 and their derivatives.

Consider the recurrence relations (cf. [16]):2𝐼𝑚(𝑥)=𝐼𝑚1(𝑥)+𝐼𝑚+1(𝑥),2𝐾𝑚(𝑥)=𝐾𝑚1(𝑥)𝐾𝑚+1(𝑥).(6.1) For each nonzero real value of 𝑥. Also 𝐼𝑚(𝑥) is positive definite and monotonic increasing, while 𝐾𝑚(𝑥) is monotonic decreasing but never negative:𝐼𝑚(𝑥)>0,𝐾𝑚(𝑥)>0.(6.2) Therefore, we may see, for a nonzero real value of 𝑥, that𝐼𝑚(𝑥)>0,𝐾𝑚(𝑥)<0.(6.3)

In view of (3.27), we see that 𝑦𝑥 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 𝜎2 depends on the sign of the quantity (1𝑚2𝑥2). Henceforth, we have the different cases:𝜎2𝜎=0,as𝑥=1,2𝜎<0as1<𝑥<2𝜎>0as0<𝑥<1,for𝑚=0,2<0,as0𝑥<,for𝑚0,(6.4) where𝜎𝑆=𝜎𝜌0𝑅301/2(6.5) is the dimensionless growth rate at the instant of time 𝑡.

This means that the hollow cylinder is stable in the case 1𝑥< for 𝑚=0 and 0𝑥< for 𝑚1, while it is unstable only as 0𝑥1 for 𝑚=0. This results, of course, as 𝑎10. 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(=(𝛾𝑃0/𝜌0)1/2). Now, for 𝑥0,𝑦0 and 𝑦>𝑥, we have𝐼𝑚(𝑦)>𝐼𝑚(𝑥),𝐾𝑚(𝑥)>𝐾𝑚𝐼(𝑦),𝑚(𝑦)>𝐼𝑚(𝑥),𝐾𝑚(𝑥)>𝐾𝑚(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 𝑚0) 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 𝑥2 following 𝜇𝐻20/𝜌0𝑅20. It contributes as a negative part in 𝜎2, 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 𝛽2𝑦𝐾𝑚(𝑦)/𝐾𝑚(𝑦) and (𝑚2𝛽2)(𝑦𝐼𝑚(𝑥)𝐾𝑚(𝑦)/𝑥𝐼𝑚(𝑥)𝐾𝑚(𝑦)) following the natural quantity 𝜇𝐻20/𝜌0𝑅20. In the axisymmetric mode, 𝑚=0, the transverse magnetic field is purely destabilizing. In the nonaxisymmetric modes, 𝑚1, the transverse magnetic field is purely destabilizing in the term 𝛽2𝑦𝐾𝑚(𝑦)/𝐾𝑚(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 𝑚1 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 𝑚1. Its influence is decreasing the magnetodynamic unstable domain in 𝑚1 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(𝐻𝐻)/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.

7. 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/𝜌0𝑅30)1/2 has a unit of (time)−1, we have 𝜎=𝜎(𝑆/𝜌0𝑅30)1/2 which is the dimensionless growth rate. For 𝑚=0, the relation (4.8) leads to𝜎2=1𝑥2𝑦𝐾1(𝑦)𝐾0(+𝐻𝑦)0𝐻𝑠2𝑥2+𝛽2𝑦𝐾1(𝑦)𝐾0(𝑦),(7.1) where 𝜎 is the nondimensional growth rate.

If 𝜎2 is negative, we write 𝜎=𝑖𝜔 (with 𝑖=1 being the imaginary factor) where 𝜔/2𝜋 is the oscillation frequency. The notation 𝐻2𝑠=𝑆/𝜇𝑅0 is with 𝐻𝑠 as a unit of magnetic field, and we have used the notation 𝐼0=𝐼1 and 𝐾0=𝐾1. The dispersion relation (7.1) has been computed for all short and long wavelengths in the new form in relation (7.2) which is𝜎2=1𝑥2𝑦𝐾1(𝑦)𝐾0(𝑦)+𝑀𝑥2+𝛽2𝑦𝐾1(𝑦)𝐾0(𝑦)(7.2) such calculations have been elaborated for different values of 𝛽 and 𝑀 for regular values of 𝑦. 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.(i)For 𝛽=0.7 and 𝑥=1, see Figure 2. Corresponding to 𝑀=0.5, 1, 1.5, 2, 2.5, and 3, it is found that the unstable domains are 0<𝑦<1.6209,0<𝑦<1.6209, 0<𝑦<1.6209, 0<𝑦<1.6209, 0<𝑦<1.6209, and 0<𝑦<1.6209 while the neighbouring stable domains are 1.6209𝑦<, 1.6209𝑦<, 1.6209𝑦<, 1.6209𝑦<, 1.6209𝑦<, and 1.6209𝑥< where the equalities correspond to the marginal stability states.(ii)For 𝛽=0.8 and 𝑥=1, see Figure 3. Corresponding to 𝑀=0.5, 1, 1.5, 2, 2.5, and 3, it is found that the unstable domains are 0<𝑦<1.1383,0<𝑦<1.1383, 0<𝑦<1.1383, 0<𝑦<1.1383, 0<𝑦<1.1383, and 0<𝑦<1.1383 while the neighbouring stable domains are 1.1383𝑦<, 1.1383𝑦<, 1.1383𝑦<, 1.1383𝑦<, 1.1383𝑦<, and 1.1383𝑥< where the equalities correspond to the marginal stability states.(iii)For 𝛽=0.9 and 𝑥=1, see Figure 4. Corresponding to 𝑀=0.5, 1, 1.5, 2, 2.5, and 3, it is found that the unstable domains are 0<𝑦<0.829,0<𝑦<0.829, 0<𝑦<0.829, 0<𝑦<0.829, 0<𝑦<0.829, and 0<𝑦<0.829 while the neighbouring stable domains are 0.829𝑦<, 0.829𝑦<, 0.829𝑦<, 0.829𝑦<, 0.829𝑦<, and 0.829𝑥< where the equalities correspond to the marginal stability states.(iv)For 𝛽=1 and 𝑥=1, see Figure 5. Corresponding to 𝑀=0.5, 1, 1.5, 2, 2.5, and 3, it is found that the unstable domains are 0<𝑦<0.6179,0<𝑦<0.6179, 0<𝑦<0.6179, 0<𝑦<0.6179, 0<𝑦<0.6179, and 0<𝑦<0.6179 while the neighbouring stable domains are 0.6179𝑦<, 0.6179𝑦<, 0.6179𝑦<, 0.6179𝑦<, 0.6179𝑦<, and 0.6179𝑥< where the equalities correspond to the marginal stability states.(v)For 𝛽=1.1 and 𝑥=1, see Figure 6. Corresponding to 𝑀=0.5, 1, 1.5, 2, 2.5, and 3, it is found that the unstable domains are 0<𝑦<0.4439,0<𝑦<0.4439, 0<𝑦<0.4439, 0<𝑦<0.4439, 0<𝑦<0.4439, and 0<𝑦<0.4439 while the neighbouring stable domains are 0.4439𝑦<, 0.4439𝑦<, 0.4439𝑦<, 0.4439𝑦<, 0.4439𝑦<, and 0.4439𝑥< where the equalities correspond to the marginal stability states.

Figure 2: Stable and unstable domains for 𝛽=0.7 and 𝑥=1.
Figure 3: Stable and unstable domains for 𝛽=0.8 and 𝑥=1.
Figure 4: Stable and unstable domains for 𝛽=0.9 and 𝑥=1.
Figure 5: Stable and unstable domains for 𝛽=1 and 𝑥=1.
Figure 6: Stable and unstable domains for 𝛽=1.1 and 𝑥=1.

8. 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).


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