Table of Contents
ISRN Mechanical Engineering
Volumeย 2011, Article IDย 597172, 8 pages
http://dx.doi.org/10.5402/2011/597172
Research Article

Gravitational Instability of Rotating Viscoelastic Partially Ionized Plasma in the Presence of an Oblique Magnetic Field and Hall Current

1Department of Mathematics, Faculty of Education, Ain Shams University, Heliopolis, Roxy, Cairo 11757, Egypt
2Department of Physics, Faculty of Education, Ain Shams University, Heliopolis, Roxy, Cairo 11757, Egypt

Received 29 January 2011; Accepted 17 March 2011

Academic Editor: P. M.ย Mariano

Copyright ยฉ 2011 M. F. El-Sayed and R. A. Mohamed. 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.

Abstract

The gravitational instability of a rotating Walters Bโ€ฒ viscoelastic partially ionized plasma permeated by an oblique magnetic field has been investigated in the presence of the effects of Hall currents, electrical resistivity, and ion viscosity. The dispersion relation and numerical calculations have been performed to obtain the dependence of the growth rate of the gravitational unstable mode on the various physical effects. It is found that viscosity and collision frequency of plasma have stabilizing effects, while viscoelasticity and angular frequency of rotation have destabilizing effect; the electrical resistivity has a destabilizing effect only for small wavenumbers; the density of neutral particles and the magnetic field component in z-direction have stabilizing effects for wavenumbers ranges ๐‘˜<5 and ๐‘˜<10, respectively; the Hall current has a slightly destabilizing effect. Finally, the inclination angle to z-direction has a destabilizing effect to all physical parameters.

1. Introduction

The gravitational instability problem of an infinite homogenous medium was first considered by Jeans [1]. According to Jeansโ€™ criterion, an infinite homogenous self-gravitating atmosphere is unstable for all wavenumbers ๐‘˜ less than Jeans' wavenumber ๐‘˜๐‘—=โˆš๐บ๐œŒ/๐‘†, where ๐œŒ is the density, ๐‘† is the velocity of sound in the gas, and ๐บ is the gravitational constant. This problem has been studied by several authors under varying assumptions of hydrodynamics and hydromagnetics, and a comprehensive account of these investigations has been given by Chandrasekhar [2] in his monograph on problems of hydrodynamic and hydromagnetic stabilities. He showed that Jeans' criterion remains unaffected by the separate or simultaneous presence of uniform rotation and uniform magnetic field. The combined effects of uniform rotation, Hall currents, finite conductivity, and finite Larmor radius on gravitational instability have been studied by Bhatia [3]. Yu and Sanborn [4] studied the internal gravitational instability in a stratified anisotropic plasma. Bhatia [5] and also Barbian and Rasmussen [6] studied the gravitational instability of a rotating anisotropic plasmas. Ariel [7] studied the gravitational instability of a rotating anisotropic plasma with Hall current effect.

In cosmic physics, there are several situations such as chromosphere, solar photosphere, and in cool interstellar cloud where the plasma are frequently not fully ionized but may instead be partially ionized so that the interaction between the ionized fluid and the neutral gas becomes important. The importance of such collisions between ionized fluid and neutral gas on the ionization rate in these regions have been pointed by Mamun and Shukla [8]. They studied a new magnetic Jeans instability in a non-uniform partially ionized plasma. Pandey et al. [9] studied Jeans instability of an inhomogeneous streaming dusty plasma. Daughten et al. [10] studied interchange instabilities in a partially ionized plasma. Mamun [11] investigated the effect of temperature and fast ions on gravitational instability in a self-gravitating magnetized dusty plasma. Mamun and Shukla [12] studied instabilities of self-gravitating dusty clouds in magnetized plasmas. The magnetoplasma stability problems have been recently studied by several authors, for example Mamun and Shukla [13], De Juli et al. [14], Cramer and Verheest [15], Azeem and Mirza [16], and El-Sayed and Mohamed [17]. There are many viscoelastic fluids that cannot be characterized by Maxwell's or Oldroyd's constitutive equations. One of such classes of viscoelastic fluids is Walters B๎…ž liquid. As a result, the usual viscous term in the equation of motion in the case of viscoelastic Walters B๎…ž fluid is replaced by the resistive term (๐œˆโˆ’๐œˆ๎…ž๐œ•/๐œ•๐‘ก)๐ฎ where ๐œˆ and ๐œˆ๎…ž are the viscosity and viscoelasticity of the Walters B๎…ž fluid. This type of viscoelastic fluids have been investigated by Sharma and Sunil [18] in the stability of two superposed Walters B๎…ž viscoelastic liquids. Kumar [19] studied stability of two superposed Walters B๎…ž viscoelastic fluid particle mixtures in porous medium. He [20] also studied the thermal convection in Walters B๎…ž viscoelastic fluid permeated by suspended particles in porous medium.

In all the investigations, the prevalent magnetic field is assumed to act either along the horizontal or the vertical direction. Chhajlani and Vyas [21] have studied the Kelvin-Helmholtz instability problem in an oblique magnetic field. Khan and Bhatia [22] considered the effects of finite resistivity and collisions with neutrals on Rayleigh-Taylor instability of a stratified plasma. Ali and Bhatia [23] have studied the gravitational instability of a partially ionized plasma in an oblique magnetic field with the effects of Hall currents, magnetic resistivity, and ion viscosity. Therefore, it would be of importance here to examine the effects of viscosity, viscoelasticity, Hall current, rotation, and finite conductivity for two components Walters B๎…ž viscoelastic partially ionized plasma in a uniform oblique magnetic field. This problem, to the best of our knowledge, has not been investigated yet.

2. Problem Formulation and Perturbation Equations

Let us consider the motion of an infinite homogeneous finitely conducting viscoelastic Walters B๎…ž fluid of density ๐œŒ permeated with neutrals of density ๐œŒ๐‘‘(โ‰ช๐œŒ), and rotating about the ๐‘ง-axis with angular velocity ฮฉ, in the presence of Hall currents and oblique magnetic field. We assume that the two components of the partially ionized plasma (the ionized fluid and the neutral gas) behave as a continuum and that their steady state velocities are equal. We assume also that the magnetic field interacts only with the ionized components of the plasma and that the frictional force of the neutral gas on the ionized fluid is of the same order as the pressure gradient of the ionized fluid. The force, due to pressure gradient of the neutral gas, is much less than the frictional effects between the neutral gas and the ionized plasma in the pressure of an oblique magnetic field. The investigation of instability problem is quite complex; consequently, we have carried out here an investigation of the instability problem using the above-stated simple model of the partially ionized plasma. The linearized perturbation equations of partially ionized plasma under the above stated assumptions are๐œŒ๐œ•๐ฎ+1๐œ•๐‘ก=โˆ’โˆ‡๐›ฟ๐‘+(โˆ‡ร—๐ก)ร—๐‡+๐œŒโˆ‡๐›ฟ๐‘ƒ3๎‚€๐œ‡โˆ’๐œ‡๎…ž๐œ•๎‚๐œ•๐‘กโˆ‡(โˆ‡.๐ฎ)+๐œŒ๐‘‘๐‘ฃ๐‘๎€ท๐”๐‘‘๎€ธ+๎‚€โˆ’๐ฎ๐œ‡โˆ’๐œ‡๎…ž๐œ•๎‚โˆ‡๐œ•๐‘ก2๐ฎโˆ’2๐œŒ(๐›€ร—๐ฎ),(1)๐œ•๐”๐‘‘๐œ•๐‘ก=โˆ’๐‘ฃ๐‘๎€ท๐”๐‘‘๎€ธโˆ’๐ฎ+โˆ‡๐›ฟ๐‘ƒ,(2)๐œ•๐ก๐œ•๐‘ก=โˆ‡ร—(๐ฎร—๐‡)+๐œ‚โˆ‡21๐กโˆ’โˆ‡ร—[],๐‘๐‘’(โˆ‡ร—๐ก)ร—๐‡(3)๐œ•๐›ฟ๐‘โˆ‡๐œ•๐‘ก=โˆ’(โˆ‡โ‹…๐ฎ)๐œŒ,(4)2๐›ฟ๐‘ƒ=โˆ’๐บ๐›ฟ๐œŒ,(5)๐›ฟ๐‘=๐‘†2๐›ฟ๐œŒ.(6) In (1), ๐œŒ๐‘‘๐‘ฃ๐‘(๐”๐‘‘โˆ’๐ฎ) represents the frictional effects of the neutral gas on the motion of the ionized fluid while โˆ’๐‘ฃ๐‘(๐”๐‘‘โˆ’๐ฎ) in (2) represents the frictional effects of an ionized fluid on the neutral gas. In the above equations, ๐ฎ(๐‘ข, ๐‘ฃ, ๐‘ค), ๐›ฟ๐‘, ๐›ฟ๐œŒ, ๐ก(โ„Ž๐‘ฅ, โ„Ž๐‘ฆ, โ„Ž๐‘ง), and ๐›ฟ๐‘ƒ are the perturbations, respectively, in the velocity, pressure, density, magnetic field, and gravitational potential of the ionized plasma. The corresponding quantities for the neutral gas are denoted by ๐”๐‘‘ and ๐œŒ๐‘‘. The collision frequency between the two components is represented by ๐‘ฃ๐‘. In these equations, ๐‘’ is the electron charge, ๐‘ is the number density, ๐œ‚ is the electrical resistivity, ๐œ‡ is the coefficient of viscosity, ๐œ‡๎…ž is the coefficient of viscoelasticity, and ๐‘† is the velocity of sound in the gas. We assume the oblique magnetic field to be uniform along the ๐‘ฅ and ๐‘ง-directions ๐‡(๐ป๐‘ฅ, 0, ๐ป๐‘ง) We seek the solution of equations (1)โ€“(6) by assuming that all the perturbed quantities depend on ๐‘ฅ,๐‘ง, and ๐‘ก as [],exp๐‘–(๐‘˜sin๐œƒ๐‘ฅ+๐‘˜cos๐œƒ๐‘ง+๐‘›๐‘ก)(7) where ๐‘˜=(๐‘˜sin๐œƒ,0,๐‘˜cos๐œƒ) is the wavenumber of perturbation making an angle ๐œƒ with the ๐‘ง-axis and ๐‘› is the frequency of perturbation. On using (7) into (1)โ€“(6) and eliminating ๐”๐‘‘, we obtain six equations which can be written as in what follows:๎‚ป๎‚ธ๐‘˜๐‘–๐‘›+๐ท+23๎€ท๐œˆโˆ’๐‘–๐‘›๐œˆ๎…ž๎€ธ๎‚นsin2๐œƒ+๐‘–๐‘›๐›ฝ๐‘ฃ๐‘๐‘–๐‘›+๐‘ฃ๐‘+๐‘˜2๎€ท๐œˆโˆ’๐‘–๐‘›๐œˆ๎…ž๎€ธ๎‚ผ๐‘ข๎‚ธ๐‘˜โˆ’2ฮฉ๐‘ฃ+๐ท+23๎€ท๐œˆโˆ’๐‘–๐‘›๐œˆ๎…ž๎€ธ๎‚นโˆ’cos๐œƒsin๐œƒ๐‘ค๐‘–๐‘˜cos๐œƒ๐œŒ๐ป๐‘งโ„Ž๐‘ฅ+๐‘–๐‘˜sin๐œƒ๐œŒ๐ป๐‘งโ„Ž๐‘ง๎‚ธ=0,2ฮฉ๐‘ข+๐‘–๐‘›+๐‘˜2๎€ท๐œˆโˆ’๐‘–๐‘›๐œˆ๎…ž๎€ธ+๐‘–๐‘›๐›ฝ๐‘ฃ๐‘๐‘–๐‘›+๐‘ฃ๐‘๎‚น๐‘ฃโˆ’๐‘–๐‘˜๐œŒ๎€ท๐ป๐‘ฅsin๐œƒ+๐ป๐‘ง๎€ธโ„Žcos๐œƒ๐‘ฆ๎‚ธ๐‘˜=0,๐ท+23๎€ท๐œˆโˆ’๐‘–๐‘›๐œˆ๎…ž๎€ธ๎‚น+๎‚ป๎‚ธ๐‘˜cos๐œƒsin๐œƒ๐‘ข๐‘–๐‘›+๐ท+23๎€ท๐œˆโˆ’๐‘–๐‘›๐œˆ๎…ž๎€ธ๎‚นcos2๐œƒ+๐‘–๐‘›๐›ฝ๐‘ฃ๐‘๐‘–๐‘›+๐‘ฃ๐‘+๐‘˜2๎€ท๐œˆโˆ’๐‘–๐‘›๐œˆ๎…ž๎€ธ๎‚ผ๐‘ค+๐‘–๐‘˜๐œŒ๐ป๐‘ฅcos๐œƒโ„Ž๐‘ฅโˆ’๐‘–๐‘˜๐œŒ๐ป๐‘ฅsin๐œƒโ„Ž๐‘ง=0,โˆ’๐‘–๐‘˜cos๐œƒ๐ป๐‘ง๐‘ข+๐‘–๐‘˜cos๐œƒ๐ป๐‘ฅ๎€ท๐‘ค+๐‘–๐‘›+๐œ‚๐‘˜2๎€ธโ„Ž๐‘ฅ+๐‘˜2๎€ท๐ป๐‘๐‘’cos๐œƒ๐‘งcos๐œƒ+๐ป๐‘ฅ๎€ธโ„Žsin๐œƒ๐‘ฆ๎€ท๐ป=0,โˆ’๐‘–๐‘˜๐‘ฅsin๐œƒ+๐ป๐‘ง๎€ธ๐‘ฃโˆ’๐‘˜cos๐œƒ2๎€ท๐ป๐‘๐‘’cos๐œƒ๐‘ฅsin๐œƒ+๐ป๐‘ง๎€ธโ„Žcos๐œƒ๐‘ฅ+๎€ท๐‘–๐‘›+๐œ‚๐‘˜2๎€ธโ„Ž๐‘ฆ+๐‘˜2ร—๎€ท๐ป๐‘๐‘’sin๐œƒ๐‘ฅsin๐œƒ+๐ป๐‘ง๎€ธโ„Žcos๐œƒ๐‘ง=0,๐‘–๐‘˜๐ป๐‘งsin๐œƒ๐‘ขโˆ’๐‘–๐‘˜sin๐œƒ๐ป๐‘ฅ๐‘คโˆ’๐‘˜2๎€ท๐ป๐‘๐‘’sin๐œƒ๐‘ฅsin๐œƒ+๐ป๐‘ง๎€ธโ„Žcos๐œƒ๐‘ฆ+๎€ท๐‘–๐‘›+๐œ‚๐‘˜2๎€ธโ„Ž๐‘ง=0.(8)

3. Dispersion Relation

Equation (8) can be written in the matrix form [๐ด๐ต]][=0,(9) where [๐ต] is a single column matrix in which the elements are (๐‘ข, ๐‘ฃ, ๐‘ค, โ„Ž๐‘ฅ, โ„Ž๐‘ฆ, โ„Ž๐‘ง) and [๐ด] is a sixth order matrix. The elements are๐ด11๎‚ต๐‘˜=๐‘–๐‘›+๐ท+23๎€ท๐œˆโˆ’๐‘–๐‘›๐œˆ๎…ž๎€ธ๎‚ถsin2๐œƒ+๐‘–๐‘›๐›ฝ๐‘ฃ๐‘๐‘–๐‘›+๐‘ฃ๐‘+๐‘˜2๎€ท๐œˆโˆ’๐‘–๐‘›๐œˆ๎…ž๎€ธ,๐ด12=โˆ’2ฮฉ,๐ด13=๎‚ต๐‘˜๐ท+23๎€ท๐œˆโˆ’๐‘–๐‘›๐œˆ๎…ž๎€ธ๎‚ถ๐ดcos๐œƒsin๐œƒ,14=โˆ’๐‘–๐‘˜cos๐œƒ๐œŒ๐ป๐‘ง,๐ด15=0,๐ด16=๐‘–๐‘˜sin๐œƒ๐œŒ๐ป๐‘ง,๐ด21=2ฮฉ,๐ด22=๐‘–๐‘›+๐‘˜2๎€ท๐œˆโˆ’๐‘–๐‘›๐œˆ๎…ž๎€ธ+๐‘–๐‘›๐›ฝ๐‘ฃ๐‘๐‘–๐‘›+๐‘ฃ๐‘,๐ด23=๐ด24=๐ด26=0,๐ด25=โˆ’๐‘–๐‘˜๐œŒ๐ด๐ฟ๐‘๐‘’,31=๎‚ธ๐‘˜๐ท+23๎€ท๐œˆโˆ’๐‘–๐‘›๐œˆ๎…ž๎€ธ๎‚น๐ดcos๐œƒsin๐œƒ,32=๐ด35๐ด=0,33๎‚ธ๐‘˜=๐‘–๐‘›+๐ท+23๎€ท๐œˆโˆ’๐‘–๐‘›๐œˆ๎…ž๎€ธ๎‚นcos2๐œƒ+๐‘–๐‘›๐›ฝ๐‘ฃ๐‘๐‘–๐‘›+๐‘ฃ๐‘+๐‘˜2๎€ท๐œˆโˆ’๐‘–๐‘›๐œˆ๎…ž๎€ธ,๐ด34=๐‘–๐‘˜cos๐œƒ๐œŒ๐ป๐‘ฅ,๐ด36=โˆ’๐‘–๐‘˜sin๐œƒ๐œŒ๐ป๐‘ฅ,๐ด41=โˆ’๐‘–๐‘˜cos๐œƒ๐ป๐‘ง,๐ด42=๐ด46๐ด=0,43=๐‘–๐‘˜cos๐œƒ๐ป๐‘ฅ,๐ด44=๐ด66=๎€ท๐‘–๐‘›+๐œ‚๐‘˜2๎€ธ,๐ด45=โˆ’๐ด54=๐‘˜2๐ฟcos๐œƒ,๐ด51=๐ด53๐ด=0,52=โˆ’๐‘–๐‘˜๐ฟ๐‘๐‘’,๐ด55=๎€ท๐‘–๐‘›+๐œ‚๐‘˜2๎€ธ,๐ด56=โˆ’๐ด65=๐‘˜2๐ฟsin๐œƒ,๐ด61=๐‘–๐‘˜sin๐œƒ๐ป๐‘ง,๐ด62=๐ด64=0,๐ด63=โˆ’๐‘–๐‘˜sin๐œƒ๐ป๐‘ฅ,(10) where we have written ๐‘ ๐ท=2๐‘˜2โˆ’๐‘–๐‘›๐บ๐œŒ(1+๐›ฝ)+๐‘–๐‘›๐บ๐œŒ๐›ฝ๐‘–๐‘›+๐‘ฃ๐‘๐œŒ,๐›ฝ=๐‘‘๐œŒ,1๐ฟ=๎€ท๐ป๐‘๐‘’๐‘ฅsin๐œƒ+๐ป๐‘ง๎€ธ.cos๐œƒ(11) The vanishing of |๐ด| gives the following relation as the product๎€ท๐‘–๐‘›+๐œ‚๐‘˜2๎€ธ๎‚ป๎‚ƒ๎€ท๐‘–๐‘›+๐œ‚๐‘˜2๎€ธ2+๐‘˜4๐ฟ2๎‚„ร—๎‚ธ๐‘–๐‘›+๐‘–๐‘›๐›ฝ๐‘ฃ๐‘๐‘–๐‘›+๐‘ฃ๐‘+๐‘˜2๎€ท๐œˆโˆ’๐‘–๐‘›๐œˆ๎…ž๎€ธ๎‚น3+๎‚ธ๐‘–๐‘›+๐‘–๐‘›๐›ฝ๐‘ฃ๐‘๐‘–๐‘›+๐‘ฃ๐‘+๐‘˜2๎€ท๐œˆโˆ’๐‘–๐‘›๐œˆ๎…ž๎€ธ๎‚น2ร—๐‘˜๎‚ธ๎‚ต๐ท+23๎€ท๐œˆโˆ’๐‘–๐‘›๐œˆ๎…ž๎€ธ๎‚ถ๎‚ƒ๎€ท๐‘–๐‘›+๐œ‚๐‘˜2๎€ธ2+๐‘˜4๐ฟ2๎‚„+๐‘˜2๎€ท๐‘–๐‘›+๐œ‚๐‘˜2๎€ธ๐œŒร—๎‚ƒ๎€ท๐ป๐‘ฅcos๐œƒโˆ’๐ป๐‘ง๎€ธsin๐œƒ2โˆ’2๐ฟ2(๐‘๐‘’)2๎‚„๎‚น+๎‚ธ๐‘–๐‘›+๐‘–๐‘›๐›ฝ๐‘ฃ๐‘๐‘–๐‘›+๐‘ฃ๐‘+๐‘˜2๎€ท๐œˆโˆ’๐‘–๐‘›๐œˆ๎…ž๎€ธ๎‚นร—๎ƒฌ2๐‘˜2๎€ท๐‘–๐‘›+๐œ‚๐‘˜2๎€ธ๐œŒ๎‚ต๐‘˜๐ท+23๎€ท๐œˆโˆ’๐‘–๐‘›๐œˆ๎…ž๎€ธ๎‚ถ+4ฮฉ2๎‚ƒ๎€ท๐‘–๐‘›+๐œ‚๐‘˜2๎€ธ2+๐‘˜4๐ฟ2๎‚„+4๐‘˜4๐ฟฮฉ๐ป๐‘ง๐œŒ๎€บ๐ป๐‘ฅsin๐œƒ+๐ป๐‘ง๎€ป+๐‘˜cos๐œƒ4๎€ท๐ป2๐‘ฅ+๐ป2๐‘ง๎€ธ๐œŒ2๎€ท๐ป๐‘ฅsin๐œƒ+๐ป๐‘ง๎€ธcos๐œƒ2๎ƒญ+๎‚ธ๐‘˜๐ท+23๎€ท๐œˆโˆ’๐‘–๐‘›๐œˆ๎…ž๎€ธ๎‚นร—๎‚ƒ4ฮฉ2๎€ท๐‘–๐‘›+๐œ‚๐‘˜2๎€ธ2cos2๐œƒ+4๐‘˜4๐ฟ2ฮฉ2cos2๐œƒ๎‚„+4๐‘˜๐ฟ๐œŒโˆ’1ฮฉcos๐œƒ๐ฟ2(๐‘๐‘’)2ร—๎‚†๐‘˜4๐œŒโˆ’2๎‚ƒ๎€ท๐ป2๐‘ฅsin2๐œƒโˆ’๐ป2๐‘งcos2๐œƒ๎€ธ2+4๐ป๐‘ฅ๐ป๐‘งcos๐œƒsin๐œƒ๐ฟ2(๐‘๐‘’)2๎€ป๎‚‡+4ฮฉ2๐ป2๐‘ฅ๐‘˜2๐œŒ๎€ท๐‘–๐‘›+๐œ‚๐‘˜2๎€ธ๎‚ผ=0.(12) The first factor of equation (12) gives๐‘›=๐‘–๐œ‚๐‘˜2,(13) which corresponds to viscous type of damped mode modified by finite conductivity. By writing ๐‘›=๐‘–๐‘Š and using the value of ๐ท in the second factor of equation (12), we obtain the resulting dispersion relation which is an equation of the tenth degree in ๐‘Š of the form๐‘‘0๐‘Š10โˆ’๐‘‘1๐‘Š9+๐‘‘2๐‘Š8โˆ’๐‘‘3๐‘Š7+๐‘‘4๐‘Š6โˆ’๐‘‘5๐‘Š5+๐‘‘6๐‘Š4โˆ’๐‘‘7๐‘Š3+๐‘‘8๐‘Š2โˆ’๐‘‘9๐‘Š+๐‘‘10=0,(14) where the coefficients ๐‘‘0-๐‘‘10 in (14) can obviously be obtained from (12). We state here explicitly only the coefficients ๐‘‘0 and ๐‘‘10, since we discuss the nature of the roots of ๐‘Š with their help, that is, ๐‘‘0=13๎€ท๐‘˜2๐œˆ๎…ž๎€ธโˆ’12๎€ท3โˆ’4๐‘˜2๐œˆ๎…ž๎€ธ,๐‘‘10=๐‘˜6๐œ‚๐‘ฃ3๐‘๐œŒ2๎€บ๐‘˜2๐‘†2๎€ปร—๎‚†๐œŒโˆ’๐บ๐œŒ(1+๐›ฝ)2๎€ท๐ฟ2+๐œ‚2๐‘˜๎€ธ๎€ท4๐œˆ2+4ฮฉ2cos2๐œƒ๎€ธ๎€ท๐ป+2๐œŒ๐‘ฅsin๐œƒ+๐ป๐‘ง๎€ธร—๎€ท๐‘˜cos๐œƒ2๎€ธ+๎€ท๐ป๐œ‚๐œˆ+2๐ฟฮฉcos๐œƒ2๐‘ฅsin2๐œƒโˆ’๐ป2๐‘งcos2๐œƒ๎€ธ2๎‚‡(15) and the coefficients ๐‘‘2-๐‘‘9 are not given here because they are very lengthy.

4. Stability Analysis and Discussion

Equation (15) shows that when 3โˆ’4๐‘˜2๐œˆ๎…žโ‰ท0 and ๐‘˜2๐‘†2โˆ’๐บ๐œŒ(1+๐›ฝ)<0, then the product of the roots is negative. Therefore, equation (14) has always one negative real root. The considered plasma is, therefore, unstable when โˆš๐‘˜<๐บ๐œŒ(1+๐›ฝ)/๐‘†, which is precisely the Jeans' criterion for a partially ionized plasma, irrespective of the kinematic viscoelasticity value. However, (15) shows also that when 3โˆ’4๐‘˜2๐œˆ๎…ž>0 and ๐‘˜2๐‘†2โˆ’๐บ๐œŒ(1+๐›ฝ)>0, that is, for the wavenumber range โˆš(1/2)3/๐œˆ๎…žโˆš>๐‘˜>๐บ๐œŒ(1+๐›ฝ)/๐‘†, then the product of roots is positive. Equation (14) has, therefore, either all positive real roots or pairs of complex conjugate roots. The real roots correspond to stable modes. The complex roots also correspond to stable modes, since Re(๐‘Š) is always positive as the equation satisfied by Re(๐‘Š) turns out to be one which has its coefficients alternatively positive and negative. Hence, Jeans' criterion for gravitational stability โˆš๐‘˜>๐บ๐œŒ(1+๐›ฝ)/๐‘† remains unchanged in the presence of the effects of magnetic resistivity, Hall currents, and ion viscosity when the plasma is permeated by an oblique magnetic field. In addition to this usual Jeans' criterion for stability there exists a new stability condition which occurs for wavenumber values โˆš๐‘˜<(1/2)3/๐œˆ๎…ž depends on the kinematic viscoelasticity appears due to the viscoelastic Walters B๎…ž plasma model.

In order to study the influences of various physical parameters on the growth rate of an unstable mode, we have performed numerical calculations of the dispersion relation (14), using Mathematica 6, to locate the roots of the growth rate ๐‘Š against the wavenumber ๐‘˜ for various values of the parameters included in the analysis. For these calculations, we take the numerical values for the following physical quantities which correspond to the conditions in galaxies Bhatia and Hazarika [24]): ๐œŒ=1.7ร—10โˆ’21kgโ€‰mโˆ’3, ๐บ=6.658ร—10โˆ’11(kg)โˆ’1โ€‰m3โ€‰sโˆ’2, and ๐‘†2=2.5ร—108โ€‰m2โ€‰sโˆ’2. These calculations are presented in Figures 1โ€“5 and Tables 1โ€“4, to show the variation of the growth rate with wavenumber of the considered system for different values of viscosity ๐œ‡, viscoelasticity ๐œ‡๎…ž, electrical resistivity ๐œ‚, density of neutral particles ๐œŒ๐‘‘, the magnetic field component ๐ป๐‘ง in the ๐‘ง-direction, inclination angle ๐œƒ (between the wavenumber vector and the ๐‘ง-axis), Hall currents ๐ฟ, angular frequency of rotation ฮฉ, and collisions with neutrals ๐‘ฃ๐‘, respectively.

tab1
Table 1: Variation of ๐‘Š, given by (14), with the wavenumber for different values of ๐œƒ. At ๐œ‡๎…ž=0.1โ€‰gmโ€‰cmโˆ’1โ€‰secโˆ’2, ๐œ‡=0.1โ€‰gmโ€‰cmโˆ’1โ€‰secโˆ’1, ๐œ‚=1โ€‰mHโ€‰cmโˆ’1, ๐‘ฃ๐‘=0.1โ€‰secโˆ’1, ฮฉ=0.05โ€‰radโ€‰secโˆ’1, ๐œŒ๐‘‘=0.001โ€‰gmโ€‰cmโˆ’3, ๐ฟ=0.1โ€‰cm2โ€‰secโˆ’1, and ๐ป๐‘ง=0.1โ€‰Ampโ€‰cmโˆ’1.
tab2
Table 2: Variation of ๐‘Š, given by (14), with the wavenumber for different values of ๐ฟ. At ๐œ‡๎…ž=0.1โ€‰gmโ€‰cmโˆ’1โ€‰secโˆ’2, ๐œ‡=0.1โ€‰gmโ€‰cmโˆ’1โ€‰secโˆ’1, ๐œ‚=1โ€‰mHโ€‰cmโˆ’1, ๐‘ฃ๐‘=0.1โ€‰secโˆ’1, ฮฉ=0.05โ€‰radโ€‰secโˆ’1, ๐œŒ๐‘‘=0.001โ€‰gmโ€‰cmโˆ’3, ๐œƒ=0โˆ˜, and ๐ป๐‘ง=0.1โ€‰Ampโ€‰cmโˆ’1.
tab3
Table 3: Variation of ๐‘Š, given by (14), with the wavenumber for different values of ฮฉ. At ๐œ‡๎…ž=0.1โ€‰gmโ€‰cmโˆ’1โ€‰secโˆ’2, ๐œ‡=0.1โ€‰gmโ€‰cmโˆ’1โ€‰secโˆ’1, ๐œ‚=1โ€‰mHโ€‰cmโˆ’1, ๐‘ฃ๐‘=0.1โ€‰secโˆ’1, ๐ฟ=0.1โ€‰cm2โ€‰secโˆ’1, ๐œŒ๐‘‘=0.001โ€‰gmโ€‰cmโˆ’3, ๐œƒ=0โˆ˜, and ๐ป๐‘ง=0.1โ€‰Ampโ€‰cmโˆ’1.
tab4
Table 4: Variation of ๐‘Š, given by (14), with the wavenumber for different values of ๐‘ฃ๐‘. At ๐œ‡๎…ž=0.1โ€‰gmโ€‰cmโˆ’1โ€‰secโˆ’2, ๐œ‡=0.1โ€‰gmโ€‰cmโˆ’1โ€‰secโˆ’1, ๐œ‚=1โ€‰mHโ€‰cmโˆ’1, ฮฉ=0.05โ€‰radโ€‰secโˆ’1, ๐ฟ=0.1โ€‰cm2โ€‰secโˆ’1, ๐œŒ๐‘‘=0.001โ€‰gmโ€‰cmโˆ’3, ๐œƒ=0โˆ˜, and ๐ป๐‘ง=0.1โ€‰Ampโ€‰cmโˆ’1.
597172.fig.001
Figure 1: Variation of ๐‘Š, given by (14), with the wavenumber for different values of viscosity ๐œ‡ when ๐œ‡๎…ž=0.1gmโ€‰cmโˆ’1โ€‰secโˆ’2, ๐œ‚=1โ€‰mHโ€‰cmโˆ’1, ๐‘ฃ๐‘=0.1โ€‰secโˆ’1, ฮฉ=0.05โ€‰radโ€‰secโˆ’1, ๐ป๐‘ง=0.1โ€‰Ampโ€‰cmโˆ’1, ๐œŒ๐‘‘=0.001 gmโ€‰cmโˆ’3, ๐ฟ=0.1 cm2โ€‰secโˆ’1, and ๐œƒ=0โˆ˜.
597172.fig.002
Figure 2: Variation of ๐‘Š, given by (14), with the wavenumber for different values of viscoelasticity ๐œ‡๎…ž when ๐œ‚=1 mHโ€‰cmโˆ’1, ๐œ‡=0.1โ€‰gmโ€‰cmโˆ’1โ€‰secโˆ’1, ๐‘ฃ๐‘=0.1โ€‰secโˆ’1, ฮฉ=0.05โ€‰radโ€‰secโˆ’1, ๐ป๐‘ง=0.1โ€‰Ampโ€‰cmโˆ’1, ๐œŒ๐‘‘=0.001โ€‰gmโ€‰cmโˆ’3, ๐ฟ=0.1 cm2โ€‰secโˆ’1, and ๐œƒ=0โˆ˜.
597172.fig.003
Figure 3: Variation of ๐‘Š, given by (14), with the wavenumber for different values of electrical resistivity ๐œ‚ when ๐œ‡๎…ž=0.1โ€‰gmโ€‰cmโˆ’1โ€‰secโˆ’2, ๐œ‡=0.1โ€‰gmโ€‰cmโˆ’1โ€‰secโˆ’1, ๐‘ฃ๐‘=0.1โ€‰secโˆ’1, ฮฉ=0.05โ€‰radโ€‰secโˆ’1, ๐ป๐‘ง=0.1โ€‰Ampโ€‰cmโˆ’1, ๐œŒ๐‘‘=0.001gmโ€‰cmโˆ’3, ๐ฟ=0.1โ€‰cm2โ€‰secโˆ’1, and ๐œƒ=0โˆ˜.
597172.fig.004
Figure 4: Variation of ๐‘Š, given by (14), with the wavenumber for different values of ๐œŒ๐‘‘ when ๐œ‡๎…ž=0.1โ€‰gmโ€‰cmโˆ’1โ€‰secโˆ’2, ๐œ‚=1โ€‰mHโ€‰cmโˆ’1, ๐‘ฃ๐‘=0.1โ€‰secโˆ’1, ฮฉ=0.05โ€‰radโ€‰secโˆ’1, ๐ป๐‘ง=0.1โ€‰Ampโ€‰cmโˆ’1, ๐œ‡=0.1โ€‰gmโ€‰cmโˆ’1โ€‰secโˆ’1, ๐ฟ=0.1โ€‰cm2โ€‰secโˆ’1 and ๐œƒ=0โˆ˜.
597172.fig.005
Figure 5: Variation of ๐‘Š, given by (14), with the wavenumber for different values of magnetic field in ๐‘ง direction ๐ป๐‘ง when ๐œ‡๎…ž=0.1โ€‰gmโ€‰cmโˆ’1โ€‰secโˆ’2, ๐œ‚=1โ€‰mHโ€‰cmโˆ’1, ๐‘ฃ๐‘=0.1โ€‰secโˆ’1, ฮฉ=0.05โ€‰radโ€‰secโˆ’1, ๐œ‡=0.1โ€‰gmโ€‰cmโˆ’1โ€‰secโˆ’1, ๐œŒ๐‘‘=0.001โ€‰gmโ€‰cmโˆ’3, ๐ฟ=0.1โ€‰cm2โ€‰secโˆ’1 and ๐œƒ=0โˆ˜.

Figure 1 shows the variation of the growth rate ๐‘Š with the wavenumber ๐‘˜ for various values of viscosity ๐œ‡ and indicates that the growth rate ๐‘Š increases by increasing the viscosity ๐œ‡ for the same ๐œƒ and ๐‘˜, showing thereby the stabilizing effect of the viscosity ๐œ‡. Figure 2 shows the variation of the growth rate ๐‘Š with wavenumber ๐‘˜ for various values of viscoelasticity ๐œ‡๎…ž and indicates that the growth rate ๐‘Š decreases by increasing the viscoelasticity ๐œ‡๎…ž for the same ๐œƒ and ๐‘˜, showing thereby the destabilizing influence of the viscoelasticity ๐œ‡๎…ž. Figure 3 shows the variation of the growth rate ๐‘Š with the wavenumber ๐‘˜ for various values of the electrical resistivity ๐œ‚ and indicates, for the same ๐œƒ and ๐‘˜, that the growth rate ๐‘Š decreases by increasing the electrical resistivity ๐œ‚ only for small wavenumber values, showing thereby its destabilizing effect for small wavenumbers, while electrical resistivity ๐œ‚ is found to has a slightly destabilizing influence for higher wavenumber values. Figure 4 illustrates the behavior of the growth rate ๐‘Š with the wavenumber ๐‘˜ for various values of the density of neutral particles ๐œŒ๐‘‘.. It is clear from Figure 4 that the growth rate ๐‘Š increases by increasing the density of neutral particles ๐œŒ๐‘‘ only for small wavenumber values less than 5, showing thereby its stabilizing effect for the wavenumbers range 0<๐‘˜<5, while it shows also that the density of neutral particles ๐œŒ๐‘‘ has no effect on the stability of the considered system for wavenumber values ๐‘˜>5. Also, Figure 5 illustrates the behavior of the growth rate ๐‘Š with the wavenumber ๐‘˜ for various values of the magnetic field component ๐ป๐‘ง, and it shows that the growth rate ๐‘Š increases by increasing the magnetic field component ๐ป๐‘ง in the ๐‘ง-direction only for small wavenumber values less than 10, showing thereby its stabilizing effect for such wavenumbers, while it shows also that ๐ป๐‘ง has a slightly stabilizing effect afterwards for wavenumber values ๐‘˜>10. Table 1 shows the changes of the growth rate ๐‘Š with the wavenumber ๐‘˜ for various values of the inclination angle ๐œƒ. It is clear from this table that the growth rate ๐‘Š increases by increasing the inclination angle ๐œƒ (between the wavenumber vector and the ๐‘ง-axis), showing thereby the stabilizing effect of the inclination angle ๐œƒ. It is clear also that the growth rate ๐‘Š for longitudinal disturbances (i.e., when ๐œƒ=๐œ‹/2) is larger than its value for transverse disturbances (i.e., when ๐œƒ=0), which means that the system is stable in the first case faster than in the later case. Also, it can be seen from Tables 2 and 3 that the growth rates ๐‘Š slightly decrease by increasing the Hall current parameter ๐ฟ as well as the angular frequency of rotation ฮฉ, showing thereby the slightly destabilizing effects of the Hall current ๐ฟ and the rotation ฮฉ. Finally, Table 4 indicates that the growth rate ๐‘Š increases by increasing the collisions with neutrals parameter ๐‘ฃ๐‘, which means that the collisions with neutrals ๐‘ฃ๐‘ has a stabilizing effect. Finally, it should be noted that these results are in agreement with earlier observations of Chhonkar and Bhatia [25], Ali and Bhatia [23], and Bhatia and Hazarika [24] in absence of kinematic viscoelasticity, and the additional behaviors of other physical parameters arose from the presence of viscoelasticity due to the used viscoelastic Walters ๐ต๎…ž model.

5. Conclusion

The gravitational instability of a rotating Walters B๎…ž partially ionized plasma permeated by an oblique magnetic field has been investigated in the presence of the effects of Hall currents, electrical resistivity, and ion viscosity. The dispersion relation is obtained and numerical calculations have been performed to obtain the dependence of the growth rate of the gravitational unstable mode on the various physical effects. We found that(1) viscosity and collision frequency of the two component partially ionized plasma have stabilizing effects,(2) viscoelasticity and angular frequency of rotation have destabilizing effects,(3) the electrical resistivity has a destabilizing effect only for small wavenumbers, while it has a slightly destabilizing influence for higher wavenumber values,(4) the density of neutral particles ๐œŒ๐‘‘ only for small wavenumber values less than 5, showing thereby its stabilizing effect for the wavenumbers range 0<๐‘˜<5, while it has no effect on the stability of the considered system for wavenumber values ๐‘˜>5,(5) the magnetic field component ๐ป๐‘ง in the ๐‘ง-direction has a stabilizing effect only for small wavenumber values less than 10, while it has a slightly stabilizing effect afterwards for ๐‘˜>10,(6) the Hall current parameter has a slightly destabilizing effect,(7) the increase in the angle ๐œƒ (between the wavenumber of perturbation and he ๐‘ง-direction) has a destabilizing effect on the stability of the considered system for all physical parameters.

Acknowledgment

The authors would like to thank Professor M. A. Kamel (Ain Shams University, Egypt) for his critical reading of the manuscript and for his valuable comments and discussion.

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