Jeans Instability of Rotating Viscoelastic Fluid in the Presence of Magnetic Field

1 Introduction

The Jeans instability causes the fragmentation of interstellar gas clouds, and subsequent star formation occurs when the internal gas pressure is not sufficient to balance the force of gravity or to prevent gravitational collapse of a region filled with matter. It has importance in astrophysical plasma, in particular in the formation of planetesimals and stars in the interstellar medium (ISM). The study of gravitational instability of an infinite homogeneous medium has been first carried by Jeans [1, 2]. Later, the Jeans criterion of gravitational instability is derived, which confers that an infinite extended homogeneous self-gravitating medium is unstable for all wavenumbers k <  kJ = 4πGρ0/cs (where k_J, G, ρ₀, and c_s are the Jeans wavenumber, universal gravitational constant, fluid density, and sound speed respectively). In addition to this, the effect of uniform rotation and uniform magnetic field on the Jeans instability is studied by Chandrasekhar [3] and found that the Jeans criterion of gravitational instability in an infinite homogeneous medium remains unaffected by the presence of uniform rotation and uniform magnetic field (separately or simultaneously in both cases). In this connection, many authors such as Sharma and Singh [4], Chhajlani and Vyas [5], Prajapati et al. [6], and Tsiklauri [7] have analysed the gravitational instability of a homogeneous fluid, considering the influence of uniform or variable magnetic field and rotation with many other parameters. Prajapati et al. [8] studied the self-gravitational instability of rotating anisotropic heat-conducting plasma and discussed the general dispersion relation for propagation parallel and perpendicular to the direction of magnetic field and rotation. The gravitational instability of rotating viscoelastic partially ionized plasma in the presence of oblique magnetic field is investigated by El Sayed and Mohamed [9]. Most of these works have been carried out using the magnetohydrodynamic (MHD) fluid model.

Moreover, many astrophysical systems are considered to be in a strongly coupled viscoelastic state where the Coulomb coupling parameter approaches 1 < Γ < 170. In recent years, much attention has been devoted on the study of gravitational instability in strongly coupled viscoelastic fluid using the generalised hydrodynamic (GH) model. It is proposed by Frenkel [10] and is now frequently used to investigate the stability of a viscoelastic fluid. The viscoelasticity property of a material generally exhibits both viscous and elastic characteristics that are quite different from the solid properties [11]. The general perception is that viscous properties are associated with liquids and elastic properties are associated with solids, but most of the materials are viscoelastic and show both the properties. In general, liquids exhibit elastic effects especially at short length or time scales [12], but they exhibit viscous effects at long length or time scales.

In recent past years, many authors have introduced new dimensions to the study of waves and instabilities in strongly coupled viscoelastic plasmas. The study of strongly coupled plasmas has importance in the interior of large planets and white dwarfs, highly compressed solids, semiconductor devices, electrical discharge, nuclear matter, dusty plasma, and laser-plasma. The strongly coupled plasma is a multicomponent charged medium at high density and low temperature. Such type of system is characterized by dominant Coulomb forces over their thermal agitation. The coupling parameter Γ (ratio of coulomb potential energy to thermal energy) for strongly coupled plasma is much greater than 1. The GH model becomes relevant in the study of instability in viscoelastic strongly coupled plasmas.

Kaw and Sen [13] investigated the effect of strong correlations on low-frequency collective modes in dusty plasma using the generalised hydrodynamic description. Furthermore, Banerjee et al. [14] have used GH model to analyse the effect of the viscosity gradient on low-frequency collective modes in the strongly coupled dusty plasma. Banerjee et al. [15] discussed the low-frequency modes in a strongly coupled, cold, magnetised dusty plasma. Banerjee et al. [16] have also carried out linear stability analysis of strongly coupled incompressible dusty plasma in the presence of shear flow by employing GH model. Janaki et al. [17] have studied the Jeans instability of viscoelastic fluid with uniform density and found that the presence of viscoelastic effects in the medium contributes to its stability against fluctuations in gravitational potential. Prajapati et al. [18] studied the Jeans instability of self-gravitating magnetised strongly coupled plasma and analysed the dispersion relation for transverse and longitudinal mode of propagations. Prajapati and Chhajlani [19] also studied the combined effects of finite electrical resistivity, magnetic field, and shear viscosity on the Jeans instability of strongly coupled viscoelastic fluid.

Therefore, looking at the above studies, the Jeans instability problem for a viscoelastic medium in the presence of magnetic field and rotation may have applications in many astrophysical situations. Thus, in the present paper, we investigated the influence of magnetic field and rotation on the Jeans instability of self-gravitating strongly coupled viscoelastic fluid.

2 Linearised Perturbation Equations

Consider an infinite extending strongly coupled viscoelastic, homogeneous self-gravitating fluid of uniform mass density ρ and angular velocity Ω(Ω_x, 0, Ω_z) is embedded in uniform magnetic field H (0, 0, H). We assume the perturbation in physical quantities as p=p₀+p₁, ρ=ρ₀+ρ₁, H=H₀+h, V=V₀+V₁ , ϕ=ϕ₀+ϕ₁, where p₀, ρ₀, H₀ , V₀ , ϕ₀ denote the equilibrium state and p₁, ρ₁, h (h_x, h_y, h_z), V₁ (v_x, v_y, v_z), ϕ₁ are the perturbations in pressure, density, magnetic field, velocity, and gravitational potential, respectively. In the linearization process, we neglect the equilibrium flow velocity (V₀ = 0) to discuss the stability of the system. Using the GH model, we write the modified momentum transfer equation as

(1)(1 + τ∂∂t)[ρ0∂V1∂t  + ρ0∇φ1 + ∇p1 − 14π(∇ · h) · H  − 2ρ0(V1 · Ω)]= η∇2·.V1 + (ξ + η3)∇(∇·V1). (1)

The continuity equation is written as

(2)∂ρ1∂t + (∇·V1)ρ0 = 0. (2)

The Poisson’s equation for self-gravitational potential is

(3)∇2φ1 = 4πGρ1. (3)

The equation for magnetic field is given by

(4)∂h∂t = ∇ · (V1 · H0), (4)

where η (shear viscosity), ξ (bulk viscosity coefficient), and τ (relaxation time) are used to describe the properties of the strongly coupled viscoelastic medium. In (1) the term (1 + τ∂/∂t) is referred to as Frenkel or memory term, which is used as viscoelastic operator. The linear form of this term is invariant under the Galilean transformation.

We seek the solution of (1)–(4) for an infinite homogeneous medium by assuming that all the perturbed quantities depend on z and t. Using p₁ = c²ρ₁ (where c = γp/ρ₀ is the sound speed) and the perturbation of the form of exp(ikz+ σt) in (1) to (4), where k is the wavenumber of perturbation and σ is the frequency of perturbation, then we obtain

(5)[(1 + τσ)ρ0σ +k2η]vx − 2ρ0(1 + τσ)Ωzvy  − (1 + τσ)ikHhx4π = 0, (5)

(6)2ρ0(1 + τσ)Ωzvx + [(1 + τσ)ρ0σ +k2η]vy  − (1 + τσ)ikHhy4π = 0, (6)

(7)− 2ρ0(1 + τσ)Ωxvy + [(1 + τσ)ρ0σ +k2(ξ + 4η3)]vz  + (1 + τσ)(ikρ0φ1 +ikc2ρ1)=0, (7)

(8)ρ1 = − ikρ0vzσ, (8)

(9)φ1 = i4πGρ0vzkσ, (9)

(10)hx = ikHvxσ,hy = ikHvyσ,hz = 0. (10)

From (5) to (10), we obtain three linear homogeneous equations in terms of velocity components and get a matrix equation in the form of [M]⋅[N] = 0. Here [M] is a third-order square matrix of coefficients of velocity components and [N] is a column matrix of velocity components v_x, v_y, and v_z. For nontrivial solution of the system, we set |M| = 0, and after expanding and reducing the determinant, we get the following equation

(11)[(1 + τσ)(σ2 +VA2k2) + k2ησρ0]2[(1 + τσ)(σ2 +k2c2 − 4πGρ0) + k2σρ0(ξ + 4η3)]+ 4σ2(1 + τσ)2[{(1 + τσ)(σ2 + VA2k2) + k2ησρ0}Ω2+ {k2σρ0(ξ + η3) + (1 + τσ)(k2c2 − 4πGρ0 − VA2k2)}Ωz2] = 0, (11)

where VA2 = H2/4πρ0 is the square of Alfvén speed. We have removed ‘0’ from subscript of H just for simplicity. Also, Ω2 = Ωx2 + Ωz2.

Equation (11) is the general dispersion relation for homogeneous self-gravitating, rotating, and magnetised viscoelastic fluid. This dispersion relation shows significant changes in the Jeans instability of viscoelastic magnetised plasma due to the inclusion of rotation. If we neglect the effect of rotation, then this dispersion relation is reduced to the dispersion relation of Prajapati et al. [18] in longitudinal mode of propagation. In the absence of shear viscosity and bulk viscosity coefficients, (11) reduces to Chandrasekhar [3] in some limiting cases. The dispersion relation (11) is a modified form of dispersion relation of Janaki et al. [17] with combined effects of magnetic field and rotation.

The general dispersion relation (11) is discussed for the axis of rotation parallel and perpendicular to the magnetic field.

3 Discussion of General Dispersion Relation

3.1 Axis of Rotation Parallel to the Magnetic Field (Ω_x=0, Ω=Ω_z)

Putting Ω_x = 0, Ω = Ω_z in (11), the general dispersion relation is reduced to the equation,

(12)[(1 + τσ)(σ2 + k2c2 − 4πGρ0) + k2σρ0(ξ + 4η3)] · [{(1 + τσ)(σ2 + VA2k2) + k2ησρ0}2 + 4σ2(1 + τσ)2Ω2] = 0. (12)

The dispersion relation (12) shows the combined effect of uniform rotation and uniform magnetic field on the self-gravitating strongly coupled viscoelastic fluid. If we ignore the effect of rotation, then this result is similar to (9) of Prajapati et al. [18]. In (12), there are two independent factors that represent different modes of propagation. These two factors are discussed below separately.

On equating the first factor of (12) to zero, we get,

(13)(1 + τσ)(σ2 + k2c2 − 4πGρ0) + k2σρ0(ξ + 4η3) =​ 0. (13)

This equation shows a gravitating shear viscous mode independent of rotation and magnetic field. This equation is discussed below for weakly coupled (hydrodynamic τσ << 1) and strongly coupled (kinetic τσ >> 1) limits.

3.1.1 Weakly Coupled (Hydrodynamic) Limit (τσ << 1)

In this limit, using (13), we get,

(14)σ2 + k2σρ0(ξ + 4η3) + k2c2 − 4πGρ0 = 0. (14)

In the hydrodynamic limit (τσ << 1), with the axis of rotation parallel to the magnetic field, the dispersion relation does not depend upon magnetic field and uniform rotation. Thus, the dispersion characteristics and growth rates of Jeans instability both are independent of magnetic field and rotation parameters. From the constant term of (14), we derive the condition of instability. If the constant term is negative, then the system will be unstable provided the condition of Jeans instability

k2c2 − 4πGρ0 < 0.

It is clear that there is instability for all wavenumbers k<k_J, where kJ = 4πGρ0/c is the critical Jeans wavenumber. This condition of Jeans instability is identical to the condition of instability given by Chandrasekhar [3]. Thus, the Jeans criterion of instability is unaffected due to the presence of viscoelastic parameters, uniform rotation, and magnetic field in weakly coupled limit.

3.1.2 Strongly Coupled (Kinetic) Limit (τσ >> 1)

Equation (13) is reduced in the limit τσ >> 1,

(15)σ2 + k2c2 − 4πGρ0 + k2τρ0(ξ + 4η3) = 0. (15)

The dispersion relation (15) shows a gravitating mode for strongly coupled viscoelastic plasma modified due to the viscoelastic effects. If the effects of relaxation time, shear, and bulk viscosity terms are ignored in (15), then this becomes identical to the dispersion relation that is obtained by Chandrasekhar [3]. If the value of the constant term is negative, then the considered system described by (15) will be unstable. Thus, the condition of Jeans instability obtained from the constant term of (15) is

(16)k2c2 − 4πGρ0 + k2τρ0(ξ + 4η3) < 0. (16)

The critical Jeans wavenumber in the strongly coupled (kinetic) limit is

(17)kJ1 =  (4πGρ0c2 + vc2)1/2, (17)

where vc = (ξ + 4η/3)/τρ0 is the velocity of compressional viscoelastic modes. In the strongly coupled viscoelastic medium, the compressibility leads to an additional wave mode along with the sound speed. In general, the kinematic viscosity in a viscous plasma yields the modifications in the growth rate of the system, but the shear and viscous effects in the compressible strongly coupled viscoelastic medium give this additional wave mode, which is often observed in many studies. This velocity increases as the bulk and shear viscosity coefficients (ξ, η) increases, but it decreases with increase in the relaxation time (τ). In [13–18], the compressional viscoelastic modes are obtained in the presence of various physical parameters.

It is clear that the condition of Jeans instability and critical Jeans wavenumber depend on bulk viscosity, shear viscosity, and relaxation time, but it does not depend upon rotation and magnetic field. The critical Jeans wavenumbers also dependent upon the velocity of compressional viscoelastic modes. The condition of instability or the criterion of Jeans instability in the kinetic regime is k < k_J1. The critical Jeans wavenumber given by (17) is not identical to the critical Jeans wavenumber that is obtained in hydrodynamic limit because of the presence of velocity of compressional viscoelastic mode. If we neglect shear viscosity, bulk viscosity, and relaxation time or neglect the influence of compressional velocity, then the Jeans condition of instability and expression of critical Jeans wavenumber in kinetic limit will be similar to those that are obtained in the hydrodynamic limit.

Now we perform the numerical calculation to see the effect of Mach number (M) on the growth rate of Jeans instability. We write dispersion relation (15) in the dimensionless form as

(18)σ∗2 + k∗2+ k∗2M2 − 1 = 0, (18)

where the following dimensionless parameters are used:

(19)σ∗ = σ4πGρ0,    k∗ = kc4πGρ0,    M = 1c[1τρ0(ξ + 4η3)]12 = vcc. (19)

It is clear from (19) that growth rate for the kinetic regime in the case of rotation axis parallel to magnetic field is governed by the Mach number (ratio of velocity of compressional viscoelastic mode to the sound velocity). The effect of Mach number on the growth rate of Jeans instability of viscoelastic fluid is analysed in Figure 1. This graph is plotted taking M=0.0, 0.3, and 0.6 in dimensionless equation (18), where M=0 means nonviscoelastic medium. The graph is plotted between growth rate (σ*) and wavenumber (k^*) with variation in Mach number for the kinetic limit. We find that the growth rate is decreasing on increasing the value of Mach number; hence, Mach number has stabilising effect on the growth rate of Jeans instability.

Figure 1

The growth rate (σ*) versus wavenumber (k*) for different values of Mach number (M) when axis of rotation is parallel to the magnetic field.

The second factor of (12) represents a nongravitating, damped viscous mode that includes relaxation time, rotation, and magnetic field. This provides the known results that are discussed in earlier studies.

3.2 Axis of Rotation Perpendicular to the Magnetic Field (Ω_z = 0, Ω_x = Ω)

Substituting Ω_z= 0, Ω_x= Ω in (11), the general dispersion relation is reduced to

(20)[(1 + τσ)(σ2 + VA2k2) + k2ησρ0][{(1 + τσ)(σ2 + k2c2 − 4πGρ0) + k2σρ0(ξ + 4η3)}{(1 + τσ)(σ2 + VA2k2) + k2ησρ0}+ 4σ2(1 + τσ)2Ω2] = 0. (20)

This shows the dispersion relation for self-gravitating, magnetised, rotating, infinitely conducting viscoelastic fluid in longitudinal mode of wave propagation in the case of rotation axis perpendicular to the magnetic field. It is clear from (20) that the dispersion relation is affected by the presence of magnetic field and rotation. There are two factors in which one is gravitating and the other one is non-gravitating mode. Setting these factors equal to zero, we get

(21)(1 + τσ)(σ2 + VA2k2) + k2ησρ0  = 0, (21)

(22){(1 + τσ)(σ2 + k2c2 − 4πGρ0) + k2σρ0(ξ + 4η3)} {(1 + τσ)(σ2 + VA2k2) + k2ησρ0} + 4σ2(1 + τσ)2Ω2 =  0. (22)

Equation (21) represents the nongravitating, damped viscous mode modified due to the presence of magnetic field in the absence of bulk viscosity and rotation. In the hydrodynamic limit, this gives a shear viscous Alfvén non-gravitating mode independent of rotation. It is obvious that rotation and self-gravitation do not have any role in this mode of propagation.

The dispersion relation (22) represents a gravitating shear viscous mode modified due to the presence of rotation. In absence of magnetic field and rotation, (22) reduces to the dispersion relation of Janaki et al. [17]. Now we discuss (22) in weakly coupled (hydrodynamic) and strongly coupled (kinetic) limits.

3.2.1 Weakly Coupled (Hydrodynamic) Limit (τσ << 1)

In the limit when τσ << 1, (22) reduces to

(23)σ4 + σ3{k2ρ0(ξ + 4η3) + k2ηρ0} + σ2{VA2k2 + k2c2 − 4πGρ0 + 4Ω2 + k4ηρ02(ξ + 4η3)}+ σ{k2ρ0(ξ + 4η3)VA2k2 + (k2c2 − 4πGρ0)k2ηρ0} + VA2k2(k2c2 − 4πGρ0) = 0. (23)

The dispersion relation given by (23) represents self-gravitating mode modified due to shear viscous effect, relaxation time, and the rotation in the hydrodynamic limit. The condition of Jeans instability is obtained from the constant term of (23), which is identical to the one obtained in the hydrodynamic limit of axis of rotation parallel to the magnetic field. Thus, in this case, the condition of Jeans instability and expression of critical Jeans wavenumber are independent of rotation and magnetic field. The condition of Jeans instability in this mode of propagation is similar to the condition of instability obtained by Chandrasekhar [3].

3.2.2 Strongly Coupled (Kinetic) Limit (τσ >> 1)

In this case, we get the following dispersion relation:

(24)σ4 + σ2{k2c2 − 4πGρ0 + VA2k2 + 4Ω2 + k2τρ0(ξ + 4η3) + k2ητρ0}+ (VA2k2 + k2ητρ0){k2c2 − 4πGρ0 + k2τρ0(ξ + 4η3)} = 0. (24)

The above dispersion relation represents the influence of rotation, magnetic field, and viscoelastic effects on the Jeans instability of system. The condition of Jeans instability and expression of critical Jeans wavenumber are obtained. In this case, the condition of Jeans instability or critical wavenumber also depends on shear viscosity and relaxation time. The condition of Jeans instability and expression of critical Jeans wavenumber are similar to those that are obtained in the case of axis of rotation perpendicular to the magnetic field. Thus, in the kinetic limit, the critical wavenumber does not depend on uniform magnetic field and uniform rotation in rotation axis perpendicular or parallel to the direction of the magnetic field.

The dimensionless form of dispersion relation (24) is

(25)σ∗4 + σ∗2(k∗2 + k∗2ξ∗ + k∗2c∗2 + k∗2η∗ + 4Ω∗2 − 1) + (k∗2 + k∗2η∗)(k∗2ξ∗ + k∗2c∗2 − 1) = 0, (25)

where

σ ∗= σ4πGρ0, k∗ = kVA4πGρ0, c∗ = cVA, ξ∗ = 1τρ0VA2(ξ + 4η3), Ω∗ = Ω4πGρ0, η∗ = ητρ0VA2.

The effect of different parameters on the growth rate of Jeans instability of viscoelastic fluid is analysed in Figures 2–4. In Figure 2, the effect of rotation on the growth rate of Jeans instability is illustrated. We chose Ω* = 0.0, 0.1, and 0.2, with constant parameters ξ* = 0.1, c* = 0.2, and η* = 0.1 in (25) and plot growth rate against wavenumber. We find that growth rate is decreasing on increasing the value of angular velocity in perpendicular direction to the magnetic field. Hence, rotation has stabilising effect on the growth rate of Jeans instability. It is obvious that the peak value of the growth rate remains constant for each value of the rotational frequency parameter.

Figure 2

The growth rate (σ*) versus wavenumber (k*) for different values of angular velocity (Ω*) when axis of rotation is perpendicular to the magnetic field.

Figure 3

The growth rate (σ*) versus wavenumber (k*) for different values of ξ* when axis of rotation is perpendicular to the magnetic field.

Figure 4

The growth rate (σ*) versus wavenumber (k*) for different values of sound velocity (c*) when axis of rotation is perpendicular to the magnetic field.

In Figure 3, the graph is plotted between dimensionless growth rate and wavenumber for Ω* = 0.25, ξ* = 0.1, 0.2, 0.3, c* = 0.2, η* = 0.1. From the curves, we find that growth rate decreases on increasing the value of shear viscosity; hence, it also has a stabilising influence on the growth rate of Jeans instability. In order to see the effect of sound speed, in Figure 4, the growth rates are plotted for parameters c* = 0.2, 0.4, 0.6, and Ω* = 0.25, ξ* = 0.1, η* = 0.1. The similar stabilising influence of sound speed is observed on the growth rate of the Jeans instability.

4 Conclusion

In this paper, we have studied the linear theory for self-gravitational Jeans instability of magnetised viscoelastic medium in the presence of rotation. The set of linearised generalized hydrodynamic equations is used for analysing the problem, and a dispersion relation for such a medium is obtained using normal mode analysis. The general dispersion relation is reduced for the cases when the direction of rotation axis is parallel and perpendicular to the magnetic field. The reduced dispersion relations are further discussed for weakly coupled (hydrodynamic) and strongly coupled (kinetic) limits.

In the case of axis of rotation parallel to the magnetic field, we find that nongravitating, damped viscous mode is affected by relaxation time, rotation, and magnetic field, and gravitating shear viscous mode is unaffected due to the presence of rotation and magnetic field. The condition of Jeans instability is influenced by bulk viscosity, shear viscosity coefficients, and relaxation time, but it is unaffected by the presence of rotation and magnetic field. From the curves in kinetic limit, we observe that the Mach number has stabilising effect on the growth rate of the Jeans instability for a magnetised, rotating viscoelastic fluid.

In the case of axis of rotation perpendicular to the magnetic field, we find that the nongravitating, damped viscous mode is modified due to magnetic field and the gravitating shear viscous mode is modified due to the presence of rotation and bulk viscosity. The Jeans criterion of instability is obtained in hydrodynamic and kinetic limits. In the hydrodynamic limit, this criterion remains unchanged, and in the kinetic limit, it depends upon shear viscosity, but it is independent of magnetic field and rotation. From the curves, it is observed that shear viscosity, sound velocity, and rotation have stabilising influence on the growth rate of the Jeans instability of a magnetised, rotating viscoelastic fluid.