gravitationalinstabilityofrotatingviscoelastic...

International Scholarly Research NetworkISRN Mechanical EngineeringVolume 2011, Article ID 597172, 8 pagesdoi:10.5402/2011/597172

Research Article

Gravitational Instability of Rotating ViscoelasticPartially Ionized Plasma in the Presence ofan Oblique Magnetic Field and Hall Current

M. F. El-Sayed1 and R. A. Mohamed2

1 Department of Mathematics, Faculty of Education, Ain Shams University, Heliopolis, Roxy, Cairo 11757, Egypt2 Department of Physics, Faculty of Education, Ain Shams University, Heliopolis, Roxy, Cairo 11757, Egypt

Correspondence should be addressed to M. F. El-Sayed, [email protected]

Received 29 January 2011; Accepted 17 March 2011

Academic Editor: P. M. Mariano

Copyright © 2011 M. F. El-Sayed and R. A. Mohamed. This is an open access article distributed under the Creative CommonsAttribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work isproperly cited.

The gravitational instability of a rotating Walters B′ viscoelastic partially ionized plasma permeated by an oblique magneticfield has been investigated in the presence of the effects of Hall currents, electrical resistivity, and ion viscosity. The dispersionrelation and numerical calculations have been performed to obtain the dependence of the growth rate of the gravitational unstablemode on the various physical effects. It is found that viscosity and collision frequency of plasma have stabilizing effects, whileviscoelasticity and angular frequency of rotation have destabilizing effect; the electrical resistivity has a destabilizing effect only forsmall wavenumbers; the density of neutral particles and the magnetic field component in z-direction have stabilizing effects forwavenumbers ranges k < 5 and k < 10, respectively; the Hall current has a slightly destabilizing effect. Finally, the inclination angleto z-direction has a destabilizing effect to all physical parameters.

1. Introduction

The gravitational instability problem of an infinite homoge-nous medium was first considered by Jeans [1]. Accordingto Jeans’ criterion, an infinite homogenous self-gravitatingatmosphere is unstable for all wavenumbers k less than

Jeans’ wavenumber kj =√Gρ/S, where ρ is the density,

S is the velocity of sound in the gas, and G is thegravitational constant. This problem has been studied byseveral authors under varying assumptions of hydrody-namics and hydromagnetics, and a comprehensive accountof these investigations has been given by Chandrasekhar[2] in his monograph on problems of hydrodynamic andhydromagnetic stabilities. He showed that Jeans’ criterionremains unaffected by the separate or simultaneous presenceof uniform rotation and uniform magnetic field. Thecombined effects of uniform rotation, Hall currents, finiteconductivity, and finite Larmor radius on gravitational

instability have been studied by Bhatia [3]. Yu and Sanborn[4] studied the internal gravitational instability in a stratifiedanisotropic plasma. Bhatia [5] and also Barbian and Ras-mussen [6] studied the gravitational instability of a rotatinganisotropic plasmas. Ariel [7] studied the gravitationalinstability of a rotating anisotropic plasma with Hall currenteffect.

In cosmic physics, there are several situations such aschromosphere, solar photosphere, and in cool interstellarcloud where the plasma are frequently not fully ionizedbut may instead be partially ionized so that the interactionbetween the ionized fluid and the neutral gas becomesimportant. The importance of such collisions betweenionized fluid and neutral gas on the ionization rate in theseregions have been pointed by Mamun and Shukla [8]. Theystudied a new magnetic Jeans instability in a non-uniformpartially ionized plasma. Pandey et al. [9] studied Jeansinstability of an inhomogeneous streaming dusty plasma.

2 ISRN Mechanical Engineering

Daughten et al. [10] studied interchange instabilities ina partially ionized plasma. Mamun [11] investigated theeffect of temperature and fast ions on gravitational instabilityin a self-gravitating magnetized dusty plasma. Mamun andShukla [12] studied instabilities of self-gravitating dustyclouds in magnetized plasmas. The magnetoplasma stabilityproblems 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], andEl-Sayed and Mohamed [17]. There are many viscoelasticfluids that cannot be characterized by Maxwell’s or Oldroyd’sconstitutive equations. One of such classes of viscoelasticfluids is Walters B′ liquid. As a result, the usual viscousterm in the equation of motion in the case of viscoelasticWalters B′ fluid is replaced by the resistive term (ν −ν′∂/∂t)u where ν and ν′ are the viscosity and viscoelasticityof the Walters B′ fluid. This type of viscoelastic fluidshave been investigated by Sharma and Sunil [18] in thestability of two superposed Walters B′ viscoelastic liquids.Kumar [19] studied stability of two superposed WaltersB′ viscoelastic fluid particle mixtures in porous medium.He [20] also studied the thermal convection in Walters B′

viscoelastic fluid permeated by suspended particles in porousmedium.

In all the investigations, the prevalent magnetic field isassumed to act either along the horizontal or the verticaldirection. Chhajlani and Vyas [21] have studied the Kelvin-Helmholtz instability problem in an oblique magnetic field.Khan and Bhatia [22] considered the effects of finiteresistivity and collisions with neutrals on Rayleigh-Taylorinstability of a stratified plasma. Ali and Bhatia [23] havestudied the gravitational instability of a partially ionizedplasma in an oblique magnetic field with the effects of Hallcurrents, magnetic resistivity, and ion viscosity. Therefore,it would be of importance here to examine the effects ofviscosity, viscoelasticity, Hall current, rotation, and finiteconductivity for two components Walters B′ viscoelasticpartially ionized plasma in a uniform oblique magnetic field.This problem, to the best of our knowledge, has not beeninvestigated yet.

2. Problem Formulation andPerturbation Equations

Let us consider the motion of an infinite homogeneousfinitely conducting viscoelastic Walters B′ fluid of density ρpermeated with neutrals of density ρd(� ρ), and rotatingabout the z-axis with angular velocity Ω, in the presence ofHall currents and oblique magnetic field. We assume thatthe two components of the partially ionized plasma (theionized fluid and the neutral gas) behave as a continuumand that their steady state velocities are equal. We assumealso that the magnetic field interacts only with the ionizedcomponents of the plasma and that the frictional force ofthe neutral gas on the ionized fluid is of the same orderas the pressure gradient of the ionized fluid. The force,due to pressure gradient of the neutral gas, is much lessthan the frictional effects between the neutral gas and theionized 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 ofthe instability problem using the above-stated simple modelof the partially ionized plasma. The linearized perturbationequations of partially ionized plasma under the above statedassumptions are

ρ∂u∂t= −∇δp + (∇× h)×H + ρ∇δP

+13

(μ− μ′ ∂

∂t

)∇(∇.u)

+ ρdvc(Ud − u)

+(μ− μ′ ∂

∂t

)∇2u− 2ρ(Ω× u),

(1)

∂Ud

∂t= −vc(Ud − u) +∇δP, (2)

∂h∂t= ∇× (u×H) + η∇2h− 1

Ne∇

× [(∇× h)×H],

(3)

∂δp

∂t= −(∇ · u)ρ, (4)

∇2δP = −Gδρ, (5)

δp = S2δρ. (6)

In (1), ρdvc(Ud − u) represents the frictional effects ofthe neutral gas on the motion of the ionized fluid while−vc(Ud − u) in (2) represents the frictional effects of anionized fluid on the neutral gas. In the above equations, u(u,v, w), δp, δρ, h(hx , hy , hz), and δP are the perturbations,respectively, in the velocity, pressure, density, magneticfield, and gravitational potential of the ionized plasma. Thecorresponding quantities for the neutral gas are denotedby Ud and ρd. The collision frequency between the twocomponents is represented by vc. In these equations, e is theelectron charge, N is the number density, η is the electricalresistivity, μ is the coefficient of viscosity, μ′ is the coefficientof viscoelasticity, and S is the velocity of sound in the gas.We assume the oblique magnetic field to be uniform alongthe x and z-directions H(Hx, 0, Hz). We seek the solutionof equations (1)–(6) by assuming that all the perturbedquantities depend on x, z, and t as

exp[i(k sin θx + k cosθz + nt)], (7)

where k = (k sin θ, 0, k cosθ) is the wavenumber ofperturbation making an angle θ with the z-axis and n is thefrequency of perturbation. On using (7) into (1)–(6) and

ISRN Mechanical Engineering 3

eliminating Ud, we obtain six equations which can be writtenas in what follows:{in +

[D +

k2

3(ν− inν′)

]sin2θ +

inβvcin + vc

+ k2(ν− inν′)

}u

− 2Ωv +

[D +

k2

3(ν− inν′)

]cos θ sin θw

− ik cosθρ

Hzhx +ik sin θρ

Hzhz = 0,

2Ωu +

[in + k2(ν− inν′) +

inβvcin + vc

]v

− ik

ρ(Hx sin θ +Hz cos θ)hy = 0,

[D +

k2

3(ν− inν′)

]cos θ sin θu

+

{in +

[D +

k2

3(ν− inν′)

]cos2θ

+inβvcin + vc

+ k2(ν− inν′)

}w

+ik

ρHx cos θhx − ik

ρHx sin θhz = 0,

− ik cosθHzu + ik cos θHxw +(in + ηk2)hx

+k2

Necosθ(Hz cosθ +Hx sin θ)hy = 0,

− ik(Hx sin θ +Hz cos θ)v

− k2

Necosθ(Hx sin θ +Hz cosθ)hx

+(in + ηk2)hy +

k2

Nesin θ

× (Hx sin θ +Hz cosθ)hz = 0,

ikHz sin θu− ik sin θHxw

− k2

Nesin θ(Hx sin θ +Hz cosθ)hy

+(in + ηk2)hz = 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 elementsare (u, v, w, hx, hy , hz) and [A] is a sixth order matrix. Theelements are

A11 = in +

(D +

k2

3(ν− inν′)

)sin2θ

+inβvcin + vc

+ k2(ν− inν′),

A12 = −2Ω, A13 =(D +

k2

3(ν− inν′)

)cosθ sin θ,

A14 = −ik cosθ

ρHz, A15 = 0, A16 =

ik sin θρ

Hz,

A21 = 2Ω, A22 = in + k2(ν− inν′) +inβvcin + vc

,

A23 = A24 = A26 = 0, A25 = −ik

ρLNe,

A31 =[D +

k2

3(ν− inν′)

]cos θ sin θ,

A32 = A35 = 0,

A33 = in +

[D +

k2

3(ν− inν′)

]cos2θ

+inβvcin + vc

+ k2(ν− inν′),

A34 = ik cosθρ

Hx, A36 = − ik sin θρ

Hx,

A41 = −ik cosθHz, A42 = A46 = 0,

A43 = ik cosθHx, A44 = A66 =(in + ηk2),

A45 = −A54 = k2L cosθ, A51 = A53 = 0,

A52 = −ikLNe, A55 =(in + ηk2),

A56 = −A65 = k2L sin θ, A61 = ik sin θHz,

A62 = A64 = 0, A63 = −ik sin θHx,(10)

where we have written

D = s2k2

in− Gρ

(1 + β

)

in+

Gρβ

in + vc, β = ρd

ρ,

L = 1Ne

(Hx sin θ +Hz cosθ).

(11)


The vanishing of |A| gives the following relation as theproduct

(in + ηk2)

{[(in + ηk2)2

+ k4L2]

×[in +

inβvcin + vc

+ k2(ν− inν′)

]3

+

[in +

inβvcin + vc

+ k2(ν− inν′)

]2

×[(

D +k2

3(ν− inν′)

)[(in + ηk2)2

+ k4L2]

+k2(in + ηk2

)

ρ

×[

(Hx cosθ −Hz sin θ)2 − 2L2(Ne)2]]

+

[in +

inβvcin + vc

+ k2(ν− inν′)

]

×[

2k2(in + ηk2

)

ρ

(D +

k2

3(ν− inν′)

)

+ 4Ω2[(in + ηk2)2

+ k4L2]

+4k4LΩHz

ρ[Hx sin θ +Hz cosθ]

+k4(H2x +H2

z

)

ρ2(Hx sin θ +Hz cos θ)2

]

+

[D +

k2

3(ν− inν′)

]

×[

4Ω2(in + ηk2)2cos2θ + 4k4L2Ω2cos2θ

]

+ 4kLρ−1Ω cos θL2(Ne)2

×{k4ρ−2

[(H2x sin2θ −H2

z cos2θ)2

+4HxHz cosθ sin θL2(Ne)2]}

+4Ω2H2xk2

ρ

(in + ηk2)

}= 0.

(12)

The first factor of equation (12) gives

n = iηk2, (13)

which corresponds to viscous type of damped mode modi-fied by finite conductivity. By writing n = iW and using thevalue of D in the second factor of equation (12), we obtain

the resulting dispersion relation which is an equation of thetenth degree in W of the form

d0W10 − d1W

9 + d2W8 − d3W

7 + d4W6 − d5W

5

+ d6W4 − d7W3 + d8W2 − d9W + d10 = 0,(14)

where the coefficients d0–d10 in (14) can obviously beobtained from (12). We state here explicitly only thecoefficients d0 and d10, since we discuss the nature of theroots of W with their help, that is,

d0 = 13

(k2ν′ − 1

)2(3− 4k2ν′

),

d10 =k6ηv3

c

ρ2

[k2S2 −Gρ(1 + β

)]

×{ρ2(L2 + η2)(k4ν2 + 4Ω2cos2θ

)

+ 2ρ(Hx sin θ +Hz cosθ)

× (k2ην + 2LΩ cos θ)

+(H2x sin2θ −H2

z cos2θ)2}

(15)

and the coefficients d2–d9 are not given here because they arevery 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 kinematicviscoelasticity value. However, (15) shows also that when3 − 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 conjugateroots. The real roots correspond to stable modes. Thecomplex roots also correspond to stable modes, since Re(W)is always positive as the equation satisfied by Re(W) turns outto be one which has its coefficients alternatively positive andnegative. 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 ionviscosity when the plasma is permeated by an obliquemagnetic field. In addition to this usual Jeans’ criterionfor stability there exists a new stability condition whichoccurs for wavenumber values k < (1/2)

√3/ν′ depends on

the kinematic viscoelasticity appears due to the viscoelasticWalters B′ plasma model.

In order to study the influences of various physicalparameters on the growth rate of an unstable mode, wehave performed numerical calculations of the dispersionrelation (14), using Mathematica 6, to locate the roots of


1

1.1

1.2

1.3

1.4

1.5

W

0 2 4 6 8 10

k

μ = 0.1μ = 0.11μ = 0.12

Figure 1: Variation of W , given by (14), with the wavenumber fordifferent values of viscosity μ 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,ρd = 0.001 gm cm−3, L = 0.1 cm2 sec−1, and θ = 0◦.

0.8

0.9

1

1.1

1.2

1.3

W

0 2 4 6 8 10

k

μ′ = 0.1μ′ = 0.11μ′ = 0.13

Figure 2: Variation of W , given by (14), with the wavenumberfor 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◦.

the growth rate W against the wavenumber k for variousvalues of the parameters included in the analysis. For thesecalculations, we take the numerical values for the followingphysical quantities which correspond to the conditions ingalaxies Bhatia and Hazarika [24]): ρ = 1.7 × 10−21 kg 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 wavenumberof the considered system for different values of viscosity μ,viscoelasticity μ′, electrical resistivity η, density of neutralparticles ρd, the magnetic field component Hz in thez-direction, inclination angle θ (between the wavenumber

1

1.025

1.05

1.075

1.1

1.125

1.15

1.175

W

0 2 4 6 8 10

k

η = 1η = 2η = 3

Figure 3: Variation of W , given by (14), with the wavenum-ber 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◦.

1

1.1

1.2

1.3

1.4

W

0 2 4 6 8 10

k

ρd = 0.001ρd = 0.1ρd = 0.2

Figure 4: Variation of W , given by (14), with the wavenumber fordifferent 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◦.

vector and the z-axis), Hall currents L, angular frequency ofrotation Ω, and collisions with neutrals vc, respectively.

Figure 1 shows the variation of the growth rate W withthe wavenumber k for various values of viscosity μ andindicates that the growth rate W increases by increasing theviscosity μ for the same θ and k, showing thereby the stabi-lizing effect of the viscosity μ. Figure 2 shows the variationof the growth rate W with wavenumber k for various valuesof viscoelasticity μ′ and indicates that the growth rate Wdecreases by increasing the viscoelasticity μ′ for the sameθ and k, showing thereby the destabilizing influence of theviscoelasticity 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, vc = 0.1 sec−1, Ω = 0.05 rad sec−1, ρd = 0.001 gm cm−3, L = 0.1 cm2 sec−1, and Hz = 0.1 Amp cm−1.

k θ = 0◦ θ = 30◦ θ = 45◦ θ = 60◦ θ = 90◦

0.1 1.36664 1.41271 1.43026 1.43829 1.42172

0.2 1.31426 1.35344 1.36885 1.37685 1.36133

0.3 1.2968 1.3341 1.3488 1.3568 1.34164

0.4 1.28205 1.31794 1.33196 1.34996 1.32517

0.5 1.26666 1.30106 1.31422 1.32222 1.30795

0.6 1.25032 1.28308 1.29511 1.30311 1.28955

0.7 1.23339 1.26431 1.27489 1.28289 1.2703

0.8 1.21631 1.24528 1.25393 1.26193 1.25073

0.9 1.19955 1.22648 1.23151 1.23951 1.23135

1 1.18344 1.20832 1.21332 1.22132 1.2126

1

1.2

1.4

1.6

1.8

W

0 2 4 6 8 10

k

Hz = 0.1Hz = 0.2Hz = 0.3

Figure 5: Variation of W , given by (14), with the wavenumberfor different values of magnetic field in z direction Hz whenμ′ = 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◦.

rate W with the wavenumber k for various values of theelectrical resistivity η and indicates, for the same θ and k,that the growth rate W decreases by increasing the electricalresistivity η only for small wavenumber values, showingthereby its destabilizing effect for small wavenumbers, whileelectrical resistivity η is found to has a slightly destabilizinginfluence for higher wavenumber values. Figure 4 illustratesthe behavior of the growth rate W with the wavenumberk for various values of the density of neutral particles ρd..It is clear from Figure 4 that the growth rate W increasesby increasing the density of neutral particles ρd only forsmall wavenumber values less than 5, showing thereby itsstabilizing effect for the wavenumbers range 0 < k < 5,while it shows also that the density of neutral particles ρdhas no effect on the stability of the considered system forwavenumber values k > 5. Also, Figure 5 illustrates thebehavior of the growth rate W with the wavenumber k for

Table 2: Variation of W , given by (14), with the wavenumberfor 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, vc = 0.1 sec−1, Ω =0.05 rad sec−1, ρd = 0.001 gm cm−3, θ = 0◦, and Hz =0.1 Amp cm−1.

k L = 0.1 L = 0.5 L = 1

0.1 1.36664 1.36663 1.36662

0.2 1.31426 1.31419 1.31398

0.3 1.2968 1.29648 1.29547

0.4 1.28205 1.28113 1.27828

0.5 1.26666 1.2647 1.25866

0.6 1.25032 1.2469 1.23638

0.7 1.23339 1.22814 1.21228

0.8 1.21631 1.20908 1.18765

0.9 1.19955 1.19034 1.16375

1 1.18344 1.17244 1.14156

various values of the magnetic field component Hz, andit shows that the growth rate W increases by increasingthe magnetic field component Hz in the z-direction onlyfor small wavenumber values less than 10, showing therebyits stabilizing effect for such wavenumbers, while it showsalso that Hz has a slightly stabilizing effect afterwards forwavenumber values k > 10. Table 1 shows the changes of thegrowth rate W with the wavenumber k for various valuesof the inclination angle θ. It is clear from this table thatthe growth rate W increases by increasing the inclinationangle θ (between the wavenumber vector and the z-axis),showing thereby the stabilizing effect of the inclination angleθ. It is clear also that the growth rate W for longitudinaldisturbances (i.e., when θ = π/2) is larger than its value fortransverse disturbances (i.e., when θ = 0), which means thatthe system is stable in the first case faster than in the latercase. Also, it can be seen from Tables 2 and 3 that the growthrates W slightly decrease by increasing the Hall currentparameter L as well as the angular frequency of rotation Ω,showing thereby the slightly destabilizing effects of the Hallcurrent L and the rotation Ω. Finally, Table 4 indicates thatthe growth rate W increases by increasing the collisions withneutrals parameter vc, which means that the collisions with


Table 3: Variation of W , given by (14), with the wavenumberfor 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, vc = 0.1 sec−1, L =0.1 cm2 sec−1, ρd = 0.001 gm cm−3, θ = 0◦, and Hz =0.1 Amp cm−1.

k Ω = 0.05 Ω = 1018 Ω = 1019

0.1 1.36664 1.31438 1.29901

0.2 1.31426 1.30745 1.25293

0.3 1.2968 1.29528 1.25156

0.4 1.28205 1.28146 1.25063

0.5 1.26666 1.26633 1.24872

0.6 1.25032 1.2501 1.23963

0.7 1.23339 1.23321 1.22644

0.8 1.21631 1.21616 1.21143

0.9 1.19955 1.19942 1.19589

1 1.18344 1.18332 1.18056

Table 4: Variation of W , given by (14), with the wavenumberfor different values of vc. 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, ρd = 0.001 gm cm−3, θ = 0◦, and Hz =0.1 Amp cm−1.

k vc = 0.1 vc = 0.1 vc = 0.2

0.1 1.36664 1.423 1.47494

0.2 1.31426 1.32782 1.33983

0.3 1.2968 1.30284 1.30814

0.4 1.28205 1.28547 1.28847

0.5 1.26666 1.26887 1.27081

0.6 1.25032 1.25188 1.25325

0.7 1.23339 1.23455 1.23556

0.8 1.21631 1.21722 1.218

0.9 1.19955 1.20027 1.20089

1 1.18344 1.18403 1.18454

neutrals vc has a stabilizing effect. Finally, it should be notedthat these results are in agreement with earlier observationsof Chhonkar and Bhatia [25], Ali and Bhatia [23], and Bhatiaand Hazarika [24] in absence of kinematic viscoelasticity,and the additional behaviors of other physical parametersarose from the presence of viscoelasticity due to the usedviscoelastic Walters B′ model.

5. Conclusion

The gravitational instability of a rotating Walters B′ partiallyionized plasma permeated by an oblique magnetic fieldhas been investigated in the presence of the effects ofHall currents, electrical resistivity, and ion viscosity. Thedispersion relation is obtained and numerical calculationshave been performed to obtain the dependence of the growthrate of the gravitational unstable mode on the variousphysical effects. We found that

(1) viscosity and collision frequency of the two compo-nent partially ionized plasma have stabilizing effects,

(2) viscoelasticity and angular frequency of rotation havedestabilizing effects,

(3) the electrical resistivity has a destabilizing effect onlyfor small wavenumbers, while it has a slightly desta-bilizing influence for higher wavenumber values,

(4) the density of neutral particles ρd only for smallwavenumber values less than 5, showing thereby itsstabilizing effect for the wavenumbers range 0 <k < 5, while it has no effect on the stability of theconsidered system for wavenumber values k > 5,

(5) the magnetic field component Hz in the z-directionhas a stabilizing effect only for small wavenumbervalues less than 10, while it has a slightly stabilizingeffect afterwards for k > 10,

(6) the Hall current parameter has a slightly destabilizingeffect,

(7) the increase in the angle θ (between the wavenumberof perturbation and he z-direction) has a destabiliz-ing effect on the stability of the considered system forall physical parameters.

Acknowledgment

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

References

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