On the Effect of Non-Uniform Magnetic field on the Jeans Instability of a Viscoelastic Medium
Joginder Singh Dhiman and Rajni Sharma
Department of Mathematics, Himachal Pradesh University, Summerhill, Shimla-171005, India.
*Corresponding Author E-mail: jsdhiman@hpuniv.ac.in; rajni22_sharma@yahoo.com
ABSTRACT:
In the present paper the problem of Jeans instability of a self gravitating viscoelastic medium in the presence of non-uniform magnetic field for both longitudinal and transverse mode of wave propagation under the kinetic and hydrodynamic limits has been investigated, using the generalized hydrodynamic model. It is found that magnetic field has no effect on the Jeans criterion for the onset of gravitational instability in case of longitudinal mode of wave propagation, whereas it modifies the Jeans criterion in the case of transverse mode of wave propagation and has stabilizing effect on the onset of instability. Further, it is observed that the magnetic field has no effect on the growth rate of Jeans instability of a viscoelastic medium. The effects of shear viscosity, bulk viscosity and Mach number on the Jeans criterion and on the growth rate of Jeans instability have also been studied numerically and the obtained results are depicted graphically, for both strongly coupled plasma (SCP) and weakly coupled plasma (WCP).
KEYWORDS: Jeans instability; viscoelastic medium; non-uniform magnetic field; wave propagation; coupled plasma.
1. INTRODUCTION:
During last century, the problem of self gravitational
instability has been a broad area of research in astrophysics, plasma physics
and many other crucial phenomenon of the interstellar medium (Dangarh et al [1]).
Jeans [2] in 1902 studied the gravitational instability of homogeneous gaseous
medium and derived a criterion for the onset of instability, now known as Jeans
criterion. This criterion states that the medium becomes unstable and
breaks up for perturbations of the wave number 
less than Jeans wave number
. Here, 
is the density, 
the velocity of the sound and 
is the gravitational constant.
Since then several researchers have studied the problem of gravitational
instability of different medium such as viscoelastic medium, plasma, anisotropic
plasma, ionized plasma and dusty plasma. Rosenberg and Shukla [3] studied the
instabilities in strongly coupled ultracold neutral plasmas and reported that
in the field of cosmic physics strongly coupled plasmas (SCP) is of
considerable interest because of possible applications to various astronomical objects
like white dwarf matter, interior of heavy planets, atmosphere of neutron star
and ultra cold neutral plasma. These objects are supposed to be composed of
viscoelastic fluid which is strongly coupled and show both viscous and elastic
behaviour. They observed that the strongly coupled state depends upon the
Coulomb coupling parameter(Г), which characterizes the ratio of the electrostatic Coulomb
interaction between neighboring plasma particles to the thermal (kinetic)
energy of the plasma particles.When the coupling parameter Г
then plasma behaves as strongly
coupled plasma (SCP), when Г
then plasma behaves as weakly
coupled plasma (WCP). Kaw and Sen [4], Janeki et al [5] and Prajapati and Chhajlani [6] studied the
gravitational instability in the viscoelastic medium, using the Generalized Hydrodynamic model
(GH).The GH model describes the effects of strong correlations through the
introduction of viscoelastic coefficients. Further, the GH model is valid for
both the SCP as well as weakly coupled plasma (WCP).
Larson [7] observed that the theoretical studies have shown that the magnetic field plays a vital role in self- gravitating star forming regions and in the evolution of interstellar clouds into self-gravitating star forming regions. In astrophysics the problems that are considered generally assume magnetic field/rotation to be uniform. This idealization of the uniform character in theoretical investigations is valid only for laboratory purposes; however in the interstellar interior and atmosphere, the magnetic field/rotation may be variable and may alter the nature of the instability. Chandrasekhar [8] examined the individual/simultaneous effects of uniform magnetic field and rotation on the gravitational instability of the homogeneous medium and found that Jeans criterion remains unaffected. Bhatia [9] also studied the gravitational instability of plasma in the presence of a uniform rotation and uniform magnetic field and showed these have no effect on that Jeans criterion. Bel and Schatzman [10] investigated the effect of non-uniform rotation on the gravitational instability of a gaseous medium and found that the Jeans criterion gets modified due to the presence of non-uniform rotation. Further, Dhiman and Dadwal [11, 12] studied the effect of non-uniform magnetic field on the problem of gravitational instability considered by Bel and Schatzman [10] and discussed the fate of Jeans criterion under these simultaneous effects. Dhiman and Dadwal [13] studied the gravitational instability of a stratified heat conducting medium in the presence of non-uniform rotation and magnetic field and found that magnetic field, whether uniform or non-uniform, have no effects on the Jeans criterion.
The present paper is primarily motivated by the works of Janeki et al [5], Prajapati and Chhajlani [6], Dhiman and Dadwal [13] and the desire to investigate the effect of non-uniform magnetic field on the onset of Jeans instability in viscoelastic medium. Both longitudinal and transverse modes of wave propagation have been considered and Jeans criterion is investigated under the effect of both kinetic and hydrodynamic limits. The effects of various parameters of the problem on the growth rate of Jeans instability have also been investigated numerically and the results obtained are depicted graphically.
2. Mathematical Model and Basic Equations
Consider
an infinite homogeneous, self gravitating viscoelastic fluid with infinite
electrical conductivity. The system is permeated with the action of non-
uniform magnetic field 
. Let 
respectively denote the
components of the velocity and magnetic field intensity in the x, y and z
directions, in a rectangular co- ordinate system. Generalized hydrodynamic (GH)
model has been used to treat the viscoelastic properties of the medium. The
general basic equations of continuity, motions, magnetic induction and Poisson
equation governing the above physical problem are given by (cf. Chapter V and Chapter XIII of Chandrasekhar [8], Janeki et al [5] and Dhiman and Dadwal [13])
