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

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.


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.

ISRN Mechanical Engineering
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.

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 are In (1), ρ d v c (U d − u) represents the frictional effects of the neutral gas on the motion of the ionized fluid while (2) represents the frictional effects of an ionized fluid on the neutral gas. In the above equations, u(u, v, w), δ p, 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 where k = (k sin θ, 0, k cos θ) 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:

Dispersion Relation
Equation (8) can be written in the matrix form where we have written

ISRN Mechanical Engineering
The vanishing of |A| gives the following relation as the product The first factor of equation (12) gives 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 form where the coefficients d 0 -d 10 in (14) can obviously be obtained from (12). We state here explicitly only the coefficients d 0 and d 10 , since we discuss the nature of the roots of W with their help, that is, and the coefficients d 2 -d 9 are not given here because they are very lengthy.

Stability Analysis and Discussion
Equation (15) shows that when 3 − 4k 2 ν 0 and k 2 S 2 − 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 − 4k 2 ν > 0 and k 2 S 2 − 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   vector and the z-axis), Hall currents L, angular frequency of rotation Ω, and collisions with neutrals v c , respectively. 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 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 cm 2 sec −1 , and H z = 0.1 Amp cm −1 .  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 Table 2: Variation of W, given by (14), with the wavenumber for different values of L. 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.

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.