Fluid Dynamical Instabilities in Magnetized Partially Ionized Dense Dusty Plasma

Received: 05/Dec/2018, Accepted: 13/Dec/2018, Online: 31/Dec/2018 Abstract— Fluid dynamical instabilities in magnetized partially ionized dense dusty plasma are studied by taking into account relative flow between dust and neutral gas. Following Hurwitz criterion, the onset criteria for instabilities are derived for different densities of the neutral gas and dust components across the interface. It is found that in case of no significant magnetic field stabilization occurs not only due to dust neutral gas collisions but due to relative flow also. Our result might be useful in many situations of astrophysical magnetized dusty plasma namely comets and circumsteller dusty disk e.g. T-Tauri stars.


I. INTRODUCTION
Instabilities are ubiquitous in partially ionized dusty plasmas. The study of instabilities in partially ionized dusty plasmas has drawn considerable interest in recent past in many different astrophysical contexts such as accretion disks [1] and relativistic jets [2]. The Rayleigh-Taylor instability may be responsible for the formation of waves and bubbles in the Earth's equatorial region [3,4]. D'Angelo [5] studied the Rayleigh-Taylor instability in such a dusty plasma where the dust grains have been assumed to be massive. He investigated the effects of negatively and positively charged dust grains on the gravitational Rayleigh-Taylor instability and found that negatively charged dust have stabilizing effects. A flute like instability which is different from the usual Rayleigh-Taylor instability is investigated by Varma and Shukla [6]. Another important instability, which occurs when adjacent layers of fluid are in relative motion, is called Kelvin-Helmholtz instability and has been analyzed for a conductive magnetized incompressible fluids streaming along the direction of the magnetic field [7]. Goertz [8] described various phenomena and instabilities occurring in dusty plasma of solar system in his review paper. Shear flows play an important role in the dynamics of partially ionized dusty plasma because they induce the unstable Kelvin-Helmholtz modes in various physical situations namely, superwinds of primeval galaxies in the intergalactic medium [9] and the amplification of self induced magnetic fields in the early Universe [10]. The existence of fluid dynamical instabilities for the partially ionized flow have been discussed by Kamaya and Nishi [11]. They found that the instability of the Alfve'n wave for any n and the two fluid instability for any '' k if 1 n  . The Alfven instability appears when its wave number is smaller than a critical value.
Birk [12] derived criteria for unstable Rayleigh-Taylor modes in partially ionized dusty plasma for different density characteristics of the neutral gas and dust components across the interface and found that dust-neutral gas collisions limit the range of unstable wavelengths. Shear flow instabilities in magnetized partially dense dusty plasma have been studied by Birk and Wiechen [13]. They derived onset criteria for instabilities with and without electrical resistivity and found that momentum exchange between the dust and neutral gas stabilized long wavelength perturbations. Excited unstable modes lead to the formation of current sheets and vortices.
In the present paper, we study the fluid dynamical instabilities in magnetized partially ionized dense dusty plasma by taking relative flow between dust and neutral gas. The plan of the paper is as follows. In section 2, the problem is formulated in terms of basic equations governing the motion. Instabilities criteria are obtained and compared with previous studies in section 3.
where , , , d i e n p are the dust, ion and neutral gas pressures. The symbols , Bg and dn  are the magnetic field, the gravitational acceleration and the effective elastic collision frequency between the dust and neutral gas particles respectively. The electron partial pressure is usually negligible in the total pressure of charged components The resistivity, Hall effect as well as other small effect of magnetic field generation are not considered. We consider equilibrium state where the homogeneous magnetic field 0 B is taken along the z -axis and the homogeneous gravitational field ˆy ge  g We consider interface along the z -axis i.e along the equilibrium flow and magnetic field. The other equilibrium quantities on either side of the interface are of the form

III. DERIVATION OF ONSET CRITERIA
A constant relative velocity of dust to the neutral gas is assumed 0 v with motion d0 v in dust and n0 v in neutral gas and  d0 n0 vv is taken along the z -axis and is such that Let ,,   vB and p denote the perturbed quantities for velocity, magnetic field, density and pressure due to a small disturbance to the system. Linearizing equations (1)-(6) about the equilibrium, we obtain Fourier analyzing the perturbations by taking the following form where  represents any of the perturbed field quantities of the considered dusty plasma,  is the frequency, and x k and z k are the wavenumbers along x  and z  axes, we obtain The determinant 0 C  gives the following characteristic complex polynomial To discuss the instability of the system, we follow the Hurwitz criterion described by Hurwitz [14] and Giaretta [15], and construct the test sequence 0 Hence the instability condition derived from equation (24)  This condition for instability is same to that condition (15) obtained by Birk [12]. It can be observed from condition (25) that stabilization occurs for modes with 0 z k  due to dust -neutral gas collisions only. The relative flow between dust and neutral gas tries to quench the instability, as one can see in case of no significant magnetic field the modes with wavenumber