Quantum Effects on the Magnetogravitational Instability of Viscoelastic fluid through a Porous Medium

Joginder Singh Dhiman, Rajni Sharma

Department of Mathematics, Himachal Pradesh University, Summerhill, Shimla -171005, INDIA

*Corresponding Author Email: jsdhiman66@gmail.com, rajni22_sharma@yahoo.com

ABSTRACT:

In the present analysis, the effect of quantum corrections on the gravitational instability of a viscoelastic fluid through porous medium in the presence of uniform magnetic field has been studied in the transverse and longitudinal mode of wave propagation. For the mathematical formulation of the physical problem Generalized Hydrodynamic and Hass model have been used. A general dispersion relation has been derived by using the normal mode analysis. The general dispersion relation is discussed separately for both the modes of wave propagation under the strongly and weakly coupling limits. It is found that the porosity of the medium and quantum effects modifies the Jeans criterion of instability for both the modes of wave propagation under the strongly and weakly coupling limits. Further, the effects of various parameters on the growth rate of gravitational instability has been numerically studied and depicted graphically.

1. INTRODUCTION:

It is a well established fact that the gravitation plays a key role in understanding the process of fragmentation of interstellar medium, formation of stars, planets, asteroids, comets and other astrophysical objects (Kaothekar and Chhajlani [1]). Chandrasekhar [2] has discussed the effect of uniform rotation and magnetic field simultaneously and/or individually on the self gravitational instability of a plasma medium. After the pioneer work of Chandrasekhar [2], several authors studied the self gravitational instability problem through various kinds of parameters and plasma environments viz. gaseous plasma, fluid plasma, dusty plasma, and Fermi degenerate quantum plasma. An extensive study related to the Jeans instability has been done theoretically and experimentally by various authors in the strongly coupled plasma.

The behavior of strongly coupled plasma have been discussed by the using the Generalized Hydrodynamic model (GH) as proposed by Frenkel 1946 [3]. Many authors including Kaw and Sen [4], Janeki et al [5], Rosenberg and Shukla [6] and Prajapti and Chhajlani [7] have studied the gravitational instability problems using the GH model. Recently, Dhiman and Sharma [8] studied the onset of gravitational instability of a magnetized viscoelastic medium in the longitudinal and transverse mode of wave propagation under the strongly and weakly coupling limits in the presence of rotation. They observed that the instability criteria get modified due to the viscoelastic effects under the strongly coupling limit, whereas rotation has stabilizing effect on the growth rate of instability.

The self gravitational instability problem through porous medium is of great importance in the study of interplanetary dust, meteorites, and comets. McDonnell [9] reported that the physical properties of comets, meteorites, and interplanetary dust strongly suggest the importance of porosity in an astrophysical context. Chhajlani and Vyas [10] have studied the magnetogravitational instability problem of thermally conducting rotating plasma through a porous medium in the presence of suspended particle and observed that the porosity reduces the effects of both rotation and magnetic field. Also, Sharma and Rana [11] have studied the magnetogravitational instability of a thermally conducting rotating Rivlin–Ericksen fluid with Hall current in porous medium and concluded that due to thermal conductivity adiabatic sound velocity is replaced by the isothermal velocity. Recently, El-Sayed and Mohamed [12] have investigated magnetogravitational instability of a thermally conducting rotating viscoelastic fluid with Hall current in Brinkman porous medium. El-Sayed and Hussian [13] studied the magnetogravitational instability of a Walters B′ viscoelastic rotating anisotropic heat-conducting fluid in Brinkman porous medium and obtained the Jeans criterion for different cases of wave propagation. Thus, we can say that the medium porosity plays a significant role on the stability investigation of self-gravitating fluids.

The field of quantum plasmas is a subject of great interest in both the space and astrophysical plasmas. Quantum effects show major role in the structure formation through gravitational collapsing [14] process of astrophysical objects, such as white dwarfs, neutron stars, magneto stars and supernovas, where the density can reach several orders of magnitude that of ordinary solids [15]. The Quantum Hydrodynamic (QHD) model generalizes the fluid model of plasma with inclusion of a quantum correction term known as Bohm potential in momentum transfer equation to describe quantum diffraction effects (Shukla [16]). Similarly, quantum statistical effects appear in the QHD model through an equation of state with zero temperature Fermi gas. The work done by Haas et al. [17] is concerned with two-species quantum plasmas described by the QHD model. A lot of researchers used QHD model to describe quantum plasma to discuss the instabilities in the multi component plasma system (Ali Shan and Mushtaq [18], Shukla and Stenflo [19] and Salimullah et al [20]. Furthermore, Hass [21] has extended the QHD model of charge particles in presence of magnetic field and derived quantum corrections due to both statistical and diffraction effects to MHD set of equations starting from QHD model with two fluid system incorporating magnetic fields.

From the above studies of various authors and the importance of quantum and porous medium in the various astrophysical objects, we are motivated to study the combined effect of quantum and porosity in the viscoelastic medium under the strongly and weakly coupling limits. In the present analysis, the effect of quantum corrections on the gravitational instability of a viscoelastic fluid through porous medium in the presence of uniform magnetic field has been studied in the transverse and longitudinal mode of wave propagation.

5. CONCLUSIONS AND RESULTS:

In the present paper, we have studied the effect of quantum corrections, uniform magnetic field and porosity on the gravitational instability of a viscoelastic fluid in the porous medium. Generalized Hydrodynamic and Hass model is used to describe the physical configuration of the system. A general dispersion relation has been derived by using the normal mode analysis for both the modes of wave propagation. The general dispersion relation is discussed separately for both the modes of wave propagation under the strongly and weakly coupling limits. It is found that the porosity of the medium and quantum effects modifies the Jeans criterion of instability for both the modes of wave propagation under the strongly and weakly coupling limits. Whereas the magnetic field modifies the Jeans criterion in the transverse mode of wave propagation under both the strongly and weakly coupling limits. It is also found that the porosity and quantum corrections have stabilizing effect on the growth rate of Jeans instability, whereas medium permeability has destabilizing effect on the system.

Figure 1, 2 represents the variation in growth rate with wave number () for SCP in transverse and longitudinal mode of wave propagation mode under strongly coupling limit for various values of quantum corrections respectively. It is observed that as the values of quantum corrections increases the growth rate of instability decreases. Hence quantum corrections have stabilizing effect on the growth rate of gravitational instability on both the modes of wave propagation.

The effect of porosity of the medium has been observed in both the transverse and longitudinal mode of wave propagation under strongly coupling limit. Figure 3, 4 represents the variation in the growth rate with wave numbers for different values of porosity in the transverse and longitudinal mode under strongly coupling limits. It is observed that as the values of porosity increases the growth rate decreases. Hence porosity has stabilizing effect on the growth rate of gravitational instability on both the modes of wave propagation.

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Received on 25.08.2016 Accepted on 10.09.2016

Int. J. Tech. 2016; 6(2): 199-205.

DOI: 10.5958/2231-3915.2016.00030.4