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

Abstract

The gravitational instability of a rotating Walters B′ 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 𝑘<5 and 𝑘<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 Jeans’ criterion, an infinite homogenous self-gravitating atmosphere is unstable for all wavenumbers 𝑘 less than Jeans' wavenumber 𝑘_𝑗=√𝐺𝜌/𝑆, where 𝜌 is the density, 𝑆 is the velocity of sound in the gas, and 𝐺 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 B^ liquid. As a result, the usual viscous term in the equation of motion in the case of viscoelastic Walters B^ fluid is replaced by the resistive term (𝜈−𝜈^𝜕/𝜕𝑡)𝐮 where 𝜈 and 𝜈^ are the viscosity and viscoelasticity of the Walters B^ fluid. This type of viscoelastic fluids have been investigated by Sharma and Sunil [18] in the stability of two superposed Walters B^ viscoelastic liquids. Kumar [19] studied stability of two superposed Walters B^ viscoelastic fluid particle mixtures in porous medium. He [20] also studied the thermal convection in Walters B^ 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 B^ 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 B^ fluid of density 𝜌 permeated with neutrals of density 𝜌_𝑑(≪𝜌), and rotating about the 𝑧-axis with angular velocity Ω, 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 are𝜌𝜕𝐮𝜕𝑡=+−∇1𝛿𝑝+(∇×𝐡)×𝐇+𝜌∇𝛿𝑃3𝜇−𝜇^𝜕+𝜌_𝑑𝜕𝑡∇(∇.𝐮)+𝜇𝑣−_𝑐𝜇𝐔_𝑑−𝐮𝜕𝜕𝑡∇²𝜕𝐔_𝑑𝐮−2𝜌(𝛀×𝐮),(1)𝜕𝑡=−𝑣^𝑐𝐔_𝑑𝜕𝐡−𝐮+∇𝛿𝑃,(2)𝜕𝑡=∇×(𝐮×𝐇)+𝜂∇²𝐡−1𝜕𝛿×𝑝[(∇×𝐡)×𝐇],𝑁𝑒∇(3)∇𝜕𝑡=−(∇⋅𝐮)𝜌,(4)₂𝛿𝛿𝑝𝑃==𝑆−₂𝐺𝛿𝜌,(5)𝛿𝜌.(6) In (1), 𝜌_𝑑𝑣_𝑐(𝐔_𝑑−𝐮) represents the frictional effects of the neutral gas on the motion of the ionized fluid while −𝑣_𝑐(𝐔_𝑑−𝐮) in (2) represents the frictional effects of an ionized fluid on the neutral gas. In the above equations, 𝐮(𝑢, 𝑣, 𝑤), 𝛿𝑝, 𝛿𝜌, 𝐡(ℎ_𝑥, ℎ_𝑦, ℎ_𝑧), and 𝛿𝑃 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 𝐔_𝑑 and 𝜌_𝑑. The collision frequency between the two components is represented by 𝑣_𝑐. In these equations, 𝑒 is the electron charge, 𝑁 is the number density, 𝜂 is the electrical resistivity, 𝜇 is the coefficient of viscosity, 𝜇^ is the coefficient of viscoelasticity, and 𝑆 is the velocity of sound in the gas. We assume the oblique magnetic field to be uniform along the 𝑥 and 𝑧-directions 𝐇(𝐻_𝑥, 0, 𝐻_𝑧) We seek the solution of equations (1)–(6) by assuming that all the perturbed quantities depend on 𝑥,𝑧, and 𝑡 as exp[𝑖(𝑘sin𝜃𝑥+𝑘cos𝜃𝑧+𝑛𝑡)],(7) where 𝑘=(𝑘sin𝜃,0,𝑘cos𝜃) is the wavenumber of perturbation making an angle 𝜃 with the 𝑧-axis and 𝑛 is the frequency of perturbation. On using (7) into (1)–(6) and eliminating 𝐔_𝑑, we obtain six equations which can be written as in what follows:𝑖𝑛+𝐷+𝑘3²𝜈−𝑖𝑛𝜈^sin²𝜃+𝑖𝑛𝛽𝑣^𝑐𝑖𝑛+𝑣_𝑐−2Ω𝑣+𝐷+𝑘₂+𝑘²𝜈−𝑖𝑛𝜈^𝑢−𝑖𝑘cos𝜃3𝜈−𝑖𝑛𝜈^cos𝜃sin𝜃𝑤𝜌𝐻^𝑧ℎ_𝑥+𝑖𝑘sin𝜃𝜌𝐻^𝑧2Ω𝑢+𝑖𝑛+𝑘₂ℎ_𝑧=0,−𝑖𝑘𝜈−𝑖𝑛𝜈^+𝑖𝑖𝑛𝑛+𝛽𝑣𝑣^𝑐_𝑐𝑣𝜌𝐻^𝑥𝐷+𝑘₂sin𝜃+𝐻_𝑧cos𝜃ℎ_𝑦=0,+3𝜈𝑖𝑛−+𝑖𝑛𝜈𝐷^+c𝑘o₂s𝜃sin𝜃𝑢3𝜈−𝑖𝑛𝜈^cos²+𝑖𝑛𝛽𝑣_𝑐𝜃𝑖𝑛+𝑣_𝑐+𝑖𝑘+𝑘²𝜈−𝑖𝑛𝜈^𝑤𝜌𝐻^𝑥cos𝜃ℎ_𝑥−𝑖𝑘𝜌𝐻^𝑥−𝑖𝑘cos𝜃𝐻_𝑧sin𝜃ℎ_𝑧=0,𝑢+𝑖𝑘cos𝜃𝐻_𝑥𝑤+𝑖𝑛+𝜂𝑘²ℎ_𝑥+𝑘²𝑁𝑒cos𝜃𝐻^𝑧−𝑖𝑘𝐻_𝑥cos𝜃+𝐻_𝑥sin𝜃ℎ_𝑦=0,−𝑘sin𝜃+𝐻_𝑧cos𝜃₂𝑣𝑁𝑒cos𝜃𝐻^𝑥sin𝜃+𝐻_𝑧cos𝜃ℎ_𝑥+𝑖𝑛+𝜂𝑘²ℎ_𝑦+𝑘²×𝐻𝑁𝑒sin𝜃_𝑥𝑖𝑘𝐻_𝑧sin𝜃+𝐻_𝑧cos𝜃ℎ_𝑧=0,s−in𝜃𝑘𝑢₂−𝑖𝑘sin𝜃𝐻_𝑥𝑤𝑁𝑒sin𝜃𝐻^𝑥sin𝜃+𝐻_𝑧cos𝜃ℎ_𝑦+𝑖𝑛+𝜂𝑘²ℎ_𝑧=0.(8)

3. Dispersion Relation

Equation (8) can be written in the matrix form [𝐴][𝐵]=0,(9) where [𝐵] is a single column matrix in which the elements are (𝑢, 𝑣, 𝑤, ℎ_𝑥, ℎ_𝑦, ℎ_𝑧) and [𝐴] is a sixth order matrix. The elements are𝐴₁₁=𝑖𝑛+𝐷+𝑘²3𝜈−𝑖𝑛𝜈^sin²+𝑖𝑛𝛽𝑣_𝑐𝜃𝑖𝑛+𝑣_𝑐𝐴₁₂+𝑘²𝜈−𝑖𝑛𝜈^,=−2Ω,𝐴₁₃=𝐷+𝑘²𝐴3𝜈−𝑖𝑛𝜈^cos𝜃sin𝜃,₁₄=−𝑖𝑘cos𝜃𝜌𝐻^𝑧,𝐴₁₅=0,𝐴₁₆=𝑖𝑘sin𝜃𝜌𝐻^𝑧𝐴₂₁,=2Ω,𝐴₂₂=𝑖𝑛+𝑘²𝜈−𝑖𝑛𝜈^+𝑖𝑛𝛽𝑣^𝑐𝑖𝑛+𝑣_𝑐𝐴₂₃,=𝐴₂₄=𝐴₂₆=0,𝐴₂₅=−𝑖𝑘𝐴𝜌𝐿𝑁𝑒,₃₁=𝐷+𝑘²𝐴3𝜈−𝑖𝑛𝜈^cos𝜃sin𝜃,₃₂𝐴=𝐴₃₅=0,₃₃=𝑖𝑛+𝐷+𝑘²3𝜈−𝑖𝑛𝜈^cos²+𝑖𝑛𝛽𝑣_𝑐𝜃𝑖𝑛+𝑣_𝑐𝐴₃₄+𝑘²𝜈−𝑖𝑛𝜈^,=𝑖𝑘cos𝜃𝜌𝐻^𝑥,𝐴₃₆=−𝑖𝑘sin𝜃𝜌𝐻^𝑥𝐴₄₁,𝐴=−𝑖𝑘cos𝜃𝐻_𝑧,𝐴₄₂=𝐴₄₆=0,₄₃𝐴₄₅=𝑖𝑘cos𝜃𝐻_𝑥,𝐴₄₄=𝐴₆₆=𝑖𝑛+𝜂𝑘²,𝐴=−𝐴₅₄=𝑘²𝐿cos𝜃,𝐴₅₁=𝐴₅₃=0,₅₂𝐴₅₆=−𝑖𝑘𝐿𝑁𝑒,𝐴₅₅=𝑖𝑛+𝜂𝑘²,𝐴₆₂=−𝐴₆₅=𝑘²𝐿sin𝜃,𝐴₆₁=𝑖𝑘sin𝜃𝐻_𝑧,=𝐴₆₄=0,𝐴₆₃=−𝑖𝑘sin𝜃𝐻_𝑥,(10) where we have written 𝐷=𝑠²𝑘²𝑖𝑛−𝐺𝜌(𝑖1𝑛+𝛽)+𝑖𝑛𝐺+𝜌𝛽𝑣_𝑐,𝛽=𝜌^𝑑𝐿=1𝜌,𝑁𝑒𝐻^𝑥sin𝜃+𝐻_𝑧cos𝜃.(11) The vanishing of |𝐴| gives the following relation as the product𝑖𝑛+𝜂𝑘ײ𝑖𝑛+𝑖𝑛𝑖𝑛+𝛽𝜂𝑣𝑘²_𝑐²+𝑘⁴𝐿²𝑖𝑛+𝑣_𝑐+𝑘²𝜈−𝑖𝑛𝜈^³+𝑖𝑛+𝑖𝑛𝛽𝑣^𝑐𝑖𝑛+𝑣_𝑐+𝑘²𝜈−𝑖𝑛𝜈^²×𝐷+𝑘²3𝜈−𝑖𝑛𝜈^𝑖𝑛+𝜂𝑘²²+𝑘₂+𝑘⁴𝐿²𝑖𝑛+𝜂𝑘²×𝐻_𝑥𝜌+𝑖𝑛+𝑖𝑛c𝛽o𝑣s_𝑐𝜃−𝐻_𝑧sin𝜃²−2𝐿²(𝑁𝑒)²𝑖𝑛+𝑣_𝑐×2𝑘₂+𝑘²𝜈−𝑖𝑛𝜈^𝑖𝑛+𝜂𝑘²+4Ω₂𝜌𝐷+𝑘3²𝜈−𝑖𝑛𝜈^+4𝑘₄𝑖𝑛+𝜂𝑘²²+𝑘⁴𝐿²𝐿Ω𝐻_𝑧𝜌𝐻^𝑥+𝑘sin𝜃+𝐻_𝑧cos𝜃₄𝐻^2𝑥+𝐻^2𝑧𝜌²𝐻_𝑥sin𝜃+𝐻_𝑧cos𝜃²+𝐷+𝑘₂×4Ω₂3𝜈−𝑖𝑛𝜈^+4𝑘𝐿𝜌𝑖₋₁𝑛+𝜂𝑘²²cos²𝜃+4𝑘⁴𝐿²Ω²cos²𝜃Ωcos𝜃𝐿²(𝑁𝑒)²×𝑘⁴𝜌⁻²𝐻^2𝑥sin²𝜃−𝐻^2𝑧cos²𝜃²+4Ω+₂4𝐻_𝑥𝐻_𝑧cos𝜃sin𝜃𝐿²(𝑁𝑒)²𝐻^2𝑥𝑘²𝜌𝑖𝑛+𝜂𝑘²=0.(12) The first factor of equation (12) gives𝑛=𝑖𝜂𝑘²,(13) which corresponds to viscous type of damped mode modified by finite conductivity. By writing 𝑛=𝑖𝑊 and using the value of 𝐷 in the second factor of equation (12), we obtain the resulting dispersion relation which is an equation of the tenth degree in 𝑊 of the form𝑑₀𝑊¹⁰−𝑑₁𝑊⁹+𝑑₂𝑊⁸−𝑑₃𝑊⁷+𝑑₄𝑊⁶−𝑑₅𝑊⁵+𝑑₆𝑊⁴−𝑑₇𝑊³+𝑑₈𝑊²−𝑑₉𝑊+𝑑₁₀=0,(14) where the coefficients 𝑑₀-𝑑₁₀ in (14) can obviously be obtained from (12). We state here explicitly only the coefficients 𝑑₀ and 𝑑₁₀, since we discuss the nature of the roots of 𝑊 with their help, that is, 𝑑₀=13𝑘²𝑑₁₀𝜈^−1²3−4𝑘²𝜈^,=𝑘⁶𝜂𝑣^3𝑐𝜌²×𝜌𝑘²𝑆²−𝐺𝜌(1+𝛽)₂+2𝜌𝐿²𝐻+_𝑥𝜂²𝑘⁴𝜈²+4Ω²cos²𝜃×𝑘sin𝜃+𝐻_𝑧cos𝜃₂+𝐻𝜂𝜈+2𝐿Ωcos𝜃_2𝑥sin²𝜃−𝐻^2𝑧cos²𝜃²(15) and the coefficients 𝑑₂-𝑑₉ are not given here because they are very lengthy.

4. Stability Analysis and Discussion

Equation (15) shows that when 3−4𝑘²𝜈^≷0 and 𝑘²𝑆²−𝐺𝜌(1+𝛽)<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 𝑘<√𝐺𝜌(1+𝛽)/𝑆, which is precisely the Jeans' criterion for a partially ionized plasma, irrespective of the kinematic viscoelasticity value. However, (15) shows also that when 3−4𝑘²𝜈^>0 and 𝑘²𝑆²−𝐺𝜌(1+𝛽)>0, that is, for the wavenumber range (1/2)√3/𝜈^>𝑘>√𝐺𝜌(1+𝛽)/𝑆, 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(𝑊) is always positive as the equation satisfied by Re(𝑊) turns out to be one which has its coefficients alternatively positive and negative. Hence, Jeans' criterion for gravitational stability 𝑘>√𝐺𝜌(1+𝛽)/𝑆 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 𝑘<(1/2)√3/𝜈^ depends on the kinematic viscoelasticity appears due to the viscoelastic Walters B^ 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 𝑊 against the wavenumber 𝑘 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]): 𝜌=1.7×10⁻²¹kg m⁻³, 𝐺=6.658×10⁻¹¹(kg)⁻¹ m³ s⁻², and 𝑆²=2.5×10⁸ m² s⁻². These calculations are presented in Figures 1–5 and Tables 1–4, to show the variation of the growth rate with wavenumber of the considered system for different values of viscosity 𝜇, viscoelasticity 𝜇^, electrical resistivity 𝜂, density of neutral particles 𝜌_𝑑, the magnetic field component 𝐻_𝑧 in the 𝑧-direction, inclination angle 𝜃 (between the wavenumber vector and the 𝑧-axis), Hall currents 𝐿, angular frequency of rotation Ω, and collisions with neutrals 𝑣_𝑐, respectively.

Table 1 

Variation of 𝑊, given by (14), with the wavenumber for different values of 𝜃. At 𝜇^=0.1 gm cm⁻¹ sec⁻², 𝜇=0.1 gm cm⁻¹ sec⁻¹, 𝜂=1 mH cm⁻¹, 𝑣_𝑐=0.1 sec⁻¹, Ω=0.05 rad sec⁻¹, 𝜌_𝑑=0.001 gm cm⁻³, 𝐿=0.1 cm² sec⁻¹, and 𝐻_𝑧=0.1 Amp cm⁻¹.

Table 2 

Variation of 𝑊, given by (14), with the wavenumber for different values of 𝐿. At 𝜇^=0.1 gm cm⁻¹ sec⁻², 𝜇=0.1 gm cm⁻¹ sec⁻¹, 𝜂=1 mH cm⁻¹, 𝑣_𝑐=0.1 sec⁻¹, Ω=0.05 rad sec⁻¹, 𝜌_𝑑=0.001 gm cm⁻³, 𝜃=0^∘, and 𝐻_𝑧=0.1 Amp cm⁻¹.

Table 3 

Variation of 𝑊, given by (14), with the wavenumber for different values of Ω. At 𝜇^=0.1 gm cm⁻¹ sec⁻², 𝜇=0.1 gm cm⁻¹ sec⁻¹, 𝜂=1 mH cm⁻¹, 𝑣_𝑐=0.1 sec⁻¹, 𝐿=0.1 cm² sec⁻¹, 𝜌_𝑑=0.001 gm cm⁻³, 𝜃=0^∘, and 𝐻_𝑧=0.1 Amp cm⁻¹.

Table 4 

Variation of 𝑊, given by (14), with the wavenumber for different values of 𝑣_𝑐. At 𝜇^=0.1 gm cm⁻¹ sec⁻², 𝜇=0.1 gm cm⁻¹ sec⁻¹, 𝜂=1 mH cm⁻¹, Ω=0.05 rad sec⁻¹, 𝐿=0.1 cm² sec⁻¹, 𝜌_𝑑=0.001 gm cm⁻³, 𝜃=0^∘, and 𝐻_𝑧=0.1 Amp cm⁻¹.

Figure 1 

Variation of 𝑊, given by (14), with the wavenumber for different values of viscosity 𝜇 when 𝜇=0.1gm cm−1 sec−2, 𝜂=1 mH cm−1, 𝑣𝑐=0.1 sec−1, Ω=0.05 rad sec−1, 𝐻𝑧=0.1 Amp cm−1, 𝜌𝑑=0.001 gm cm−3, 𝐿=0.1 cm2 sec−1, and 𝜃=0∘.

Figure 2 

Variation of 𝑊, given by (14), with the wavenumber for different values of viscoelasticity 𝜇 when 𝜂=1 mH cm−1, 𝜇=0.1 gm cm−1 sec−1, 𝑣𝑐=0.1 sec−1, Ω=0.05 rad sec−1, 𝐻𝑧=0.1 Amp cm−1, 𝜌𝑑=0.001 gm cm−3, 𝐿=0.1 cm2 sec−1, and 𝜃=0∘.

Figure 3 

Variation of 𝑊, given by (14), with the wavenumber for different values of electrical resistivity 𝜂 when 𝜇=0.1 gm cm−1 sec−2, 𝜇=0.1 gm cm−1 sec−1, 𝑣𝑐=0.1 sec−1, Ω=0.05 rad sec−1, 𝐻𝑧=0.1 Amp cm−1, 𝜌𝑑=0.001gm cm−3, 𝐿=0.1 cm2 sec−1, and 𝜃=0∘.

Figure 4 

Variation of 𝑊, given by (14), with the wavenumber for different values of 𝜌𝑑 when 𝜇=0.1 gm cm−1 sec−2, 𝜂=1 mH cm−1, 𝑣𝑐=0.1 sec−1, Ω=0.05 rad sec−1, 𝐻𝑧=0.1 Amp cm−1, 𝜇=0.1 gm cm−1 sec−1, 𝐿=0.1 cm2 sec−1 and 𝜃=0∘.

Figure 5 

Variation of 𝑊, given by (14), with the wavenumber for different values of magnetic field in 𝑧 direction 𝐻𝑧 when 𝜇=0.1 gm cm−1 sec−2, 𝜂=1 mH cm−1, 𝑣𝑐=0.1 sec−1, Ω=0.05 rad sec−1, 𝜇=0.1 gm cm−1 sec−1, 𝜌𝑑=0.001 gm cm−3, 𝐿=0.1 cm2 sec−1 and 𝜃=0∘.

Figure 1 shows the variation of the growth rate 𝑊 with the wavenumber 𝑘 for various values of viscosity 𝜇 and indicates that the growth rate 𝑊 increases by increasing the viscosity 𝜇 for the same 𝜃 and 𝑘, showing thereby the stabilizing effect of the viscosity 𝜇. Figure 2 shows the variation of the growth rate 𝑊 with wavenumber 𝑘 for various values of viscoelasticity 𝜇^ and indicates that the growth rate 𝑊 decreases by increasing the viscoelasticity 𝜇^ for the same 𝜃 and 𝑘, showing thereby the destabilizing influence of the viscoelasticity 𝜇^. Figure 3 shows the variation of the growth rate 𝑊 with the wavenumber 𝑘 for various values of the electrical resistivity 𝜂 and indicates, for the same 𝜃 and 𝑘, that the growth rate 𝑊 decreases by increasing the electrical resistivity 𝜂 only for small wavenumber values, showing thereby its destabilizing effect for small wavenumbers, while electrical resistivity 𝜂 is found to has a slightly destabilizing influence for higher wavenumber values. Figure 4 illustrates the behavior of the growth rate 𝑊 with the wavenumber 𝑘 for various values of the density of neutral particles 𝜌_𝑑.. It is clear from Figure 4 that the growth rate 𝑊 increases by increasing the density of neutral particles 𝜌_𝑑 only for small wavenumber values less than 5, showing thereby its stabilizing effect for the wavenumbers range 0<𝑘<5, while it shows also that the density of neutral particles 𝜌_𝑑 has no effect on the stability of the considered system for wavenumber values 𝑘>5. Also, Figure 5 illustrates the behavior of the growth rate 𝑊 with the wavenumber 𝑘 for various values of the magnetic field component 𝐻_𝑧, and it shows that the growth rate 𝑊 increases by increasing the magnetic field component 𝐻_𝑧 in the 𝑧-direction only for small wavenumber values less than 10, showing thereby its stabilizing effect for such wavenumbers, while it shows also that 𝐻_𝑧 has a slightly stabilizing effect afterwards for wavenumber values 𝑘>10. Table 1 shows the changes of the growth rate 𝑊 with the wavenumber 𝑘 for various values of the inclination angle 𝜃. It is clear from this table that the growth rate 𝑊 increases by increasing the inclination angle 𝜃 (between the wavenumber vector and the 𝑧-axis), showing thereby the stabilizing effect of the inclination angle 𝜃. It is clear also that the growth rate 𝑊 for longitudinal disturbances (i.e., when 𝜃=𝜋/2) is larger than its value for transverse disturbances (i.e., when 𝜃=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 𝑊 slightly decrease by increasing the Hall current parameter 𝐿 as well as the angular frequency of rotation Ω, showing thereby the slightly destabilizing effects of the Hall current 𝐿 and the rotation Ω. Finally, Table 4 indicates that the growth rate 𝑊 increases by increasing the collisions with neutrals parameter 𝑣_𝑐, which means that the collisions with neutrals 𝑣_𝑐 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 𝐵^ model.

5. Conclusion

The gravitational instability of a rotating Walters B^ 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 𝜌_𝑑 only for small wavenumber values less than 5, showing thereby its stabilizing effect for the wavenumbers range 0<𝑘<5, while it has no effect on the stability of the considered system for wavenumber values 𝑘>5,(5) the magnetic field component 𝐻_𝑧 in the 𝑧-direction has a stabilizing effect only for small wavenumber values less than 10, while it has a slightly stabilizing effect afterwards for 𝑘>10,(6) the Hall current parameter has a slightly destabilizing effect,(7) the increase in the angle 𝜃 (between the wavenumber of perturbation and he 𝑧-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.