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

Abstract

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 (1−𝑠²−𝑘²𝑅²⁰) 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 [6–9]. 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 𝐻^𝑔0=(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 𝐻^𝑔0 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:𝜕𝑢_𝑟𝜕𝑡+𝑢⋅∇𝑢_𝑟−𝜇𝜌𝐻⋅∇𝐻_𝑟=−𝜕Π𝜕𝑢_𝜑𝜕𝑟,(2.1)𝜕𝑡+𝑢⋅∇𝑢_𝜑−𝜇𝜌𝐻⋅∇𝐻_𝜑=−1𝜕𝑢_𝑧𝑟𝜕𝜕Π𝜑,(2.2)𝜕𝑡+𝑢⋅∇𝑢_𝑧−𝜇𝜌𝐻⋅∇𝐻_𝑧=−𝜕Π𝜕𝑧,(2.3) where 𝜌Π is the total magnetohydrodynamic pressure which is the sum of kinetic and magnetic pressures, given by𝜌Π=𝑃+𝜇2𝐻⋅𝐻.(2.4) Equation of conservation of flux:∇⋅𝐻=0.(2.5) Equation of conservation of mass of a compressible fluid:𝜕𝜌𝜕𝑡+𝑢⋅∇𝜌+𝜌∇⋅𝑢=0.(2.6) Equation of state gives the relation between the pressure 𝑃 and density 𝜌:𝑃=𝜌𝑅^𝑐𝑇,(2.7)𝑅^𝑐 is a general constant of gas.

Equation of conservation of energy:𝜌𝐶_𝑣𝜕𝜕𝑡+𝑢⋅∇𝑇+𝑃∇⋅𝑢=0.(2.8) Evaluation equations of a magnetic field (derived from the electromagnetic Maxwell equation) in a perfectly conducting fluid:𝜕𝐻_𝑟𝜕𝑡+𝑢𝜕𝐻_𝜑⋅∇𝐻_𝑟+∇⋅𝑢𝐻_𝑟=𝐻⋅∇𝑢_𝑟,(2.9)𝜕𝑡+𝑢𝜕𝐻_𝑧⋅∇𝐻_𝜑+∇⋅𝑢𝐻_𝜑=𝐻⋅∇𝑢_𝜑,𝜕𝑡+𝑢⋅∇𝐻_𝑧+∇⋅𝑢𝐻_𝑧=𝐻⋅∇𝑢_𝑧.(2.10) In the gas cylinder region, surrounded by a liquid, there is no current flow:∇∇Λ⋅𝐻𝐻^𝑔_𝑔=0,(2.11)=0.(2.12) Along the gas liquid interface, the curvature pressure 𝑃_𝑠 due to the capillary force is given by𝑃_𝑠=𝑆1𝑟₁+1𝑟₂,(2.13) with𝑟¹⁻¹+𝑟²⁻¹=∇⋅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, (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 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=𝑃^𝑔0+𝜇2𝛽²−1𝐻²⁰−𝑆𝑅₀.(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)𝑃^𝑔0 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 𝐻₀≠0, 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 𝑃^𝑔0, otherwise, the model will collapse and the gas may spread through the fluid region.

In the general case, in order that 𝑃₀≥0, the gas kinetic pressure 𝑃^𝑔0 in the initial state must satisfy the restriction:𝑃^𝑔0≥𝑆𝑅₀+𝜇𝐻²⁰21−𝛽².(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=𝛿₀exp[𝜎𝑡+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,𝜎𝑢₁−𝜇𝜌₀𝐻⁰𝜌Π₁⋅∇𝐻₁=−∇Π₁,(3.5)=𝑃₁+𝜇22𝐻⁰∇⋅𝐻₁⋅𝐻₁,(3.6)𝜎𝜌₁=0,(3.7)𝜌+𝜌₀∇⋅𝑢₁=0,(3.8)₀𝑎²𝑃∇⋅𝑢₁=−𝜎𝑃₁,(3.9)₁𝑃₀=𝜌¹𝜌₀+𝑇¹𝑇₀𝜎𝐻₁,(3.10)=∇Λ𝑢₁Λ𝐻₀,(3.11) where 𝑎 is the sound speed in the gas defined by 𝑎=(𝛾𝑃₀/𝜌₀)^1/2, and 𝛾=𝐶_𝑝/𝐶_𝑣 is the ratio of specific heats of the fluids.

For the gas region surrounded by the fluid,∇⋅𝐻^𝑔1=0,(3.12)∇Λ𝐻^𝑔1=0(thereisnocurrent).(3.13) Along the gas-liquid perturbed interface,𝑃_1𝑠=𝑆𝑅²⁰𝑅¹+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)=𝑄^∗1(𝑟)exp[𝜎𝑡+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) yield𝜎+Ω𝜎^2𝐴𝑢¹=−∇Π₁+𝑖Ω^2𝐴𝑃₁𝑘𝑎²𝜌₀𝜎𝐻₁𝑒_𝑧,(3.16)=𝑖𝑘𝐻₀𝑢₁+𝜎𝐻⁰𝜌₀𝑎²𝑃₁𝑒_𝑧,(3.17) where 𝑒_𝑧 is a unit vector in the 𝑧-direction and Ω_𝐴 is the Alfven wave frequency defined in terms of 𝐻₀ asΩ_𝐴=𝜇𝐻²⁰𝑘²𝜌₀^1/2.(3.18) Combining the 𝑧-components of (3.16) and (3.17) which are𝜎1+Ω^𝐴𝜎²𝑢^1𝑧=−𝑖𝑘Π₁+𝑖Ω^2𝐴𝑘𝑎²𝜌₀𝜎𝐻_1𝑧𝑃₁,=𝐻₀𝑖𝑘𝑢^1𝑧+𝜎𝑃¹𝑎²𝜌₀,(3.19) we obtain𝐻_1𝑧=(𝑖𝑘)²𝐻₀𝜎²+Ω^2𝐴−Π¹+𝜇𝐻²⁰𝑎²𝜌₀𝑃₁+𝑎𝐻²𝜌⁰₀𝑃₁.(3.20) By substituting (3.6) into (3.20), we finally obtain𝐻_1𝑧=𝜎²+𝑎²𝑘²𝐻₀𝜎²𝑎²𝜌₀𝑃₁.(3.21) Again, inserting (3.21) into (3.6) yieldsΠ₁=𝜉𝜌₀𝑃¹.(3.22) With𝜉=1+𝜇𝐻²⁰𝜎²+𝑎²𝑘²𝜎²𝑎²𝜌₀.(3.23) Consequently, the components 𝑢_1𝑟, 𝑢_1𝜑, and 𝑢_1𝑧 of 𝑢₁ are given, from the vector equation (3.16), by𝜎𝜎²+Ω^2𝐴𝑢_1𝑟=−𝜎𝜕𝜕Π𝑟¹₂,+Ω^2𝐴𝑢_1𝜑=−𝑖𝑚𝜎𝑟Π¹𝜎₂,+Ω^2𝐴𝑢_1𝑧=−𝑖𝑘𝜎−𝑎²𝜉+𝜇𝐻²⁰𝑎²𝜉Π¹.(3.24) By inserting (3.24) into the nonsolenoid equation (3.9) which is1𝑟𝜕𝑟𝑢^1𝑟𝜕𝑟+1𝑟𝜕𝜕𝑢𝜑^1𝜑+𝜕𝜕𝑢𝑧^1𝑧=−𝑎𝜎²𝜌𝑃¹₀,(3.25) and by taking into account the space (𝜑,𝑧)-dependence (3.15), we get1𝑟𝑑𝑑𝑟𝑟𝑑𝑑Π𝑟¹−𝑚𝑟²²+𝜂²Π¹=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)=𝐴𝐾_𝑚(𝜂𝑟).(3.28) Therefore, Π₁(𝑟,𝜑,𝑧,𝑡) is beingΠ₁(𝑟,𝜑,𝑧,𝑡)=𝐴𝐾_𝑚(𝜂𝑟)exp[𝜎𝑡+𝑖(𝑘𝑧+𝑚𝜑)],(3.29) 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 𝑢₁(=(𝑢_1𝑟,𝑢_1𝜑,𝑢_1𝑧)) and 𝐻₁(=(𝐻_1𝑟,𝐻_1𝜑,𝐻_1𝑧)) and also 𝑃₁ could be identified from (3.16), (3.17), and (3.22) explicitly.

Equation (3.13) means that the magnetic field 𝐻^𝑔1 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^𝑔1 in the form:𝐻^𝑔1=𝐵∇𝐼_𝑚(𝑘𝑟)exp[𝜎𝑡+𝑖(𝑘𝑧+𝑚𝜑)],(3.30) 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:𝑃_1𝑠=𝑆𝑅²⁰1−𝑚²−𝑘²𝑅²⁰𝑅₁.(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: 𝑁_𝑜⋅𝐻₁+𝑁₁⋅𝐻_𝑜=𝑁_𝑜⋅𝐻^𝑔1+𝑁₁⋅𝐻^𝑔𝑜,(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)=𝑟−𝑅₀−𝑅₁=0(4.2) being the equation of the perturbed interface. Then, 𝑁_𝑜=(1,0,0),𝑁₁=0,−𝑖𝑚𝑅_𝑜,−𝑖𝑘𝑅₁.(4.3)

By substituting from (4.3) and the required expressions concerning 𝐻₀,𝐻₁,𝐻^𝑔0, and 𝐻^𝑔1 into (4.1), we obtain 𝐵=𝑖𝑚𝛽𝐻⁰𝑥𝐼^′_𝑚(𝑥),(4.4) where 𝑥(=𝑘𝑅₀) 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=𝑅₀. This condition reads 𝑁₀⋅𝑢₁+𝑁₁⋅𝑢₀=𝜕𝑟𝜕𝑡.(4.5)

By substituting 𝑢₀,𝑢₁,𝑁₀,𝑁₁,and 𝑟 into (4.5), this yields 𝐴=−𝜎²+Ω^2𝐴𝜂𝐾^𝑚𝜂𝑅₀.(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𝜕𝑟+𝜇22𝐻⁰⋅𝐻₁+𝑅₁𝜕𝜕𝑟𝐻⁰=𝑃_1𝑠⋅𝐻₀+𝜇22𝐻^𝑔0⋅𝐻^𝑔1+𝑅₁𝜕𝜕𝑟𝐻^𝑔0⋅𝐻^𝑔0.(4.7)

Upon substituting into the condition (4.7) about the different variables, the following eigenvalue relation is obtained: 𝜎²=−𝑆𝜌𝑅³⁰1−𝑚²−𝑥²𝑦𝐾^𝑚(𝑦)𝐾_𝑚(𝑦)+𝜌𝜇_𝑜𝐻𝑅^202𝑜−𝑥²+−𝛽²+(𝑚𝛽)²𝐼_𝑚(𝑥)𝑥𝐼^𝑚(𝑥)𝑦𝐾^𝑚(𝑦)𝐾_𝑚(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 𝐼_𝑚 and 𝐾_𝑚 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) yields𝜎²=−𝑆𝜌𝑅³⁰𝑥𝐾^0(𝑥)𝐾₀(𝑥)1−𝑥²,(5.1) where (𝑎→∞), we find (𝑦→𝑥). Upon using the recurrence relation,𝐾^0(𝑥)=−𝐾₁(𝑥),(5.2) the relation (5.1) becomes𝜎²=𝑆𝜌𝑅³⁰𝑥𝐾¹(𝑥)𝐾₀(𝑥)1−𝑥².(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 (𝑦→𝑥), 𝐻₀=0 for 𝑚≠0, the relation (4.8) gives𝜎²=−𝑆𝜌𝑅³⁰𝑥𝐾^𝑚(𝑥)𝐾_𝑚(𝑥)1−𝑚²−𝑥².(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=−𝑆𝜌𝑅³⁰1−𝑚²−𝑥²𝑦𝐾^𝑚(𝑦)𝐾_𝑚(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=𝜇𝐻²⁰𝜌_𝑜𝑅^2𝑜−𝑥²+−𝛽²+(𝑚𝛽)²𝐼_𝑚(𝑥)𝑥𝐼^𝑚(𝑥)𝑦𝐾^𝑚(𝑦)𝐾_𝑚(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 𝐼_𝑚 and 𝐾_𝑚 and their derivatives.

Consider the recurrence relations (cf. [16]):2𝐾2_𝑚𝐼^𝑚(𝑥)=𝐼_𝑚−1(𝑥)+𝐼_𝑚+1(𝑥),(𝑥)=−𝐾_𝑚−1(𝑥)−𝐾_𝑚+1(𝑥).(6.1) 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:𝐼_𝑚(𝑥)>0,𝐾_𝑚(𝑥)>0.(6.2) Therefore, we may see, for a nonzero real value of 𝑥, that𝐼^𝑚(𝑥)>0,𝐾^𝑚(𝑥)<0.(6.3)

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 𝜎² depends on the sign of the quantity (1−𝑚²−𝑥²). Henceforth, we have the different cases:⎧⎪⎪⎨⎪⎪⎩𝜎𝜎^∗2=0,as𝑥=1,_∗2𝜎<0as1<𝑥<∞_∗2𝜎>0as0<𝑥<1,for𝑚=0,_∗2<0,as0≤𝑥<∞,for𝑚≠0,(6.4) where𝜎^∗=𝜎𝑆𝜌₀𝑅³⁰^−1/2(6.5) is the dimensionless growth rate at the instant of time 𝑡.

This means that the hollow cylinder is stable in the case 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(=(𝛾𝑃₀/𝜌₀)^1/2). Now, for 𝑥≠0,𝑦≠0 and 𝑦>𝑥, we have𝐼𝐼_𝑚(𝑦)>𝐼_𝑚(𝑥),𝐾_𝑚(𝑥)>𝐾_𝑚(𝑦),_′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 −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 −𝛽²𝑦𝐾^𝑚(𝑦)/𝐾_𝑚(𝑦) and (𝑚²𝛽²)(𝑦𝐼_𝑚(𝑥)𝐾^𝑚(𝑦)/𝑥𝐼^𝑚(𝑥)𝐾_𝑚(𝑦)) following the natural quantity 𝜇𝐻²⁰/𝜌₀𝑅²⁰. 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 −𝛽²𝑦𝐾^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.

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 to𝜎^∗2=1−𝑥²𝑦𝐾¹(𝑦)𝐾₀(𝑦)+𝐻𝐻⁰_𝑠²−𝑥²+𝛽²𝑦𝐾₁(𝑦)𝐾₀(𝑦),(7.1) 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^2𝑠=𝑆/𝜇𝑅₀ is with 𝐻_𝑠 as a unit of magnetic field, and we have used the notation 𝐼^0=𝐼₁ and 𝐾^0=−𝐾₁. The dispersion relation (7.1) has been computed for all short and long wavelengths in the new form in relation (7.2) which is𝜎^∗2=1−𝑥²𝑦𝐾¹(𝑦)𝐾₀(𝑦)+𝑀−𝑥²+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 1.6209≤𝑦<∞, 1.6209≤𝑦<∞, 1.6209≤𝑦<∞, 1.6209≤𝑦<∞, 1.6209≤𝑦<∞, and 1.6209≤𝑥<∞ 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 1.1383≤𝑦<∞, 1.1383≤𝑦<∞, 1.1383≤𝑦<∞, 1.1383≤𝑦<∞, 1.1383≤𝑦<∞, and 1.1383≤𝑥<∞ 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 0.829≤𝑦<∞, 0.829≤𝑦<∞, 0.829≤𝑦<∞, 0.829≤𝑦<∞, 0.829≤𝑦<∞, and 0.829≤𝑥<∞ 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 0.6179≤𝑦<∞, 0.6179≤𝑦<∞, 0.6179≤𝑦<∞, 0.6179≤𝑦<∞, 0.6179≤𝑦<∞, and 0.6179≤𝑥<∞ 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 0.4439≤𝑦<∞, 0.4439≤𝑦<∞, 0.4439≤𝑦<∞, 0.4439≤𝑦<∞, 0.4439≤𝑦<∞, and 0.4439≤𝑥<∞ 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).