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Scholarly Research Journal for Interdisciplinary Studies, Online ISSN 2278-8808, SJIF 2021 = 7.380, www.srjis.com PEER REVIEWED & REFEREED JOURNAL, NOV-DEC, 2021, VOL- 9/68

STABILITY OF FLUID/PLASMA IN THE PRESENCE OF QUANTUM PHYSICS SATURATING A POROUS MEDIU Urmil Kumari & Prakash Chand Chopra P.hd Schlor Mathematics, Carrier Point University Kota Head of Department, AS & H Govt Polytechnic College Sundernagar

Paper Received On: 21 DEC 2021 Peer Reviewed On: 31 DEC 2021 Published On: 1 JAN 2022

The present investigation deals with the quantum effects on the Rayleigh –Taylor instability in an infinitely electrically conducting inhomogeneous stratified incompressible viscoelastic fluid/plasma through a porous medium. The linear growth rate is derived for the case where a plasma with exponential density, viscosity, viscoelasticity and quantum parameter distribution is confined between two rigid planes. The solution of the linearized equations of the system together with the appropriate boundary conditions leads to derive the dispersion relation (the relation between the normalized growth rate and square normalized wavenumber) using normal mode technique. The behavior of growth rate with respect to quantum effect and kinematic viscoelasticity are examined in the presence of porous medium, medium permeability and kinematic viscoelasticity. It is observed that the quantum effects bring more stability for a certain wave number band on the growth rate on the unstable configuration.

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1. Introduction Rayleigh-Taylor instability arises from the character of equilibrium of an incompressible heavy fluid of variable density (i.e. of a heterogeneous fluid). The simplest, nevertheless important, example demonstrating the Rayleigh-Taylor instability is when, we consider two fluids of different densities superposed one over the other (or accelerated towards each other); the instability of the plane interface between the two fluids, if it occurs, is known as Rayleigh Taylor instability. Rayleigh (1900) [1] was the first to investigate the character of equilibrium of an inviscid, non- heat conducting as well as incompressible heavy fluid of variable density, which is continuously stratified in the vertical direction. The case of (i) two uniform fluids of different densities superposed one over the other and (ii) an exponentially varying density distribution, was also treated by him. The main result in all cases is that the configuration is stable or unstable with respect to infinitesimal small perturbations according as the higher density fluid underlies or overlies the lower density fluid. Taylor (1950) [2] carried out the theoretical investigation further and studied the instability of liquid surfaces when accelerated in a direction perpendicular to their planes. The experimental Copyright © 2021, Scholarly Research Journal for Interdisciplinary Studies

Urmil Kumari & Prakash Chand Chopra

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demonstration of the development of the Rayleigh –Taylor instability (in case of heavier fluid overlaying a lighter one, is accelerated towards it) is described by Lewis (1950) [3]. This instability has been further studied by many authors e.g. Kruskal and Schwarzschild (1954) [4], Hide (1955) [5], Chandrasekhar (1955) [6], Joseph (1976) [7], and Drazin and Reid (1981) [8] to include various parameters. Rayleigh-Taylor instability is mainly used to analyze the frequency of gravity waves in deep oceans, liquid vapour/globe, to extract oil from the earth to eliminate water drops, lazer and inertial confinement fusion etc. Quantum plasma can be composed of electrons, ions, positrons, holes, and (or) grains, which plays an important role in ultra-small electronic devices which have been given by Dutta and McLennan (1990) [9], dense astrophysical plasmas system has been given by Madappa et al. (2001) [10], intense laser-matter experiments has been investigated by Remington (1999) [11], and non-linear quantum optics has been given by Brambilla et al. (1995) [12]. The pressure term in such plasmas is divided to two terms 𝑝𝑝 = 𝑝𝑝𝐶𝐶 + 𝑝𝑝𝑄𝑄 (classical (𝑝𝑝𝐶𝐶) and quantum (𝑝𝑝𝑄𝑄) pressure) and has been investigated by Gardner (1994) [13] for the quantum hydrodynamic model. In the momentum equation, the classical pressure rises in the form (−∇𝑝𝑝), while the quantum pressure rises in the form 𝑄𝑄 2𝑚𝑚𝑒𝑒 𝑖𝑖 , where ℎ~ is the Plank constant, 𝑚𝑚𝑒𝑒 is the mass of electron and 𝑚𝑚𝑖𝑖 is the mass of ion. The linear quantum growth rate of a finite layer plasma, in which the density is continuously stratified exponentially along the vertical, was studied by Goldston and Rutherford (1997) [14]. Nuclear fusion, which is plasma based, is one of the most promising candidates for the energy needs of the future when fossil fuels finally run out. It is well known that quantum effects become important in the behavior of charged plasma particles when the de Broglie wavelength of charge carriers become equal to or greater than the dimension of the quantum plasma system, which has been investigated by Manfredi and Haas (2001) [15]. Two models are used to study quantum plasmas systems. The first one is the Wigner-Poisson and the other is the Schrodinger-Poisson approaches (2001, 2005) [15-17] they have been widely used to describe the statistical and hydrodynamic behavior of the plasma particles at quantum scales in quantum plasma. The quantum hydrodynamic model was introduced in semiconductor physics to describe the transport of charge, momentum and energy in plasma (1994) [13]. A magnetohydrodynamic model for semiconductor devices was investigated by Haas (2005) [16], which is an important model in astrophysics, space physics and dusty plasmas. The effect of quantum term on Rayleigh-Taylor instability in the presence of vertical and horizontal magnetic field, separately, has been studied by Hoshoudy (2009) [18, 19]. The Rayleigh-Taylor instability in a non-uniform dense quantum magneto-plasma has been studied by Ali et al. (2009) [20]. Hoshoudy (2010) [21] studied quantum effects on RayleighTaylor instability of incompressible plasma in a vertical magnetic field. Rayleigh-Taylor instability in quantum magnetized viscous plasma has been studied by Hoshoudy (2011) [22]. External magnetic field effects on the Rayleigh-Taylor instability in an inhomogeneous rotating quantum plasma has been studied by Hoshoudy (2012) [23]. In all the above studies, the plasma/fluids have been considered to be Newtonian. With the growing importance of the nonNewtonian fluids in modern technology and industries, the investigations of such fluids are desirable. There are many elastico-viscous constitutive relation or Oldroyd constitutive relation. We are interested there in Rivlin-Ericksen Model. Rivlin-Ericksen Model (1955) [24] proposed a theoretical model for such elastic-viscous fluid. Molten plastics, petroleum oil additives and whipped cream are examples of incompressible Copyright © 2021, Scholarly Research Journal for Interdisciplinary Studies

Urmil Kumari & Prakash Chand Chopra

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viscoelastic fluids. Such types of polymers are used in agriculture, communication appliances and in bio-medical applications. Previous work on the effects of incompressible quantum plasma on Rayleigh-Taylor instability of Oldroyd model through a porous medium has been investigated by Hoshoudy (2011) [25], where the author has shown that both maximum 𝑘𝑘𝑚𝑚𝑚𝑚𝑥𝑥∗ and critical 𝑘𝑘𝑐𝑐∗ point for the instability are unchanged by the addition of the strain retardation and the stress relaxation. All growth rates are reduced in the presence of porosity of the medium, the medium permeability, the strain retardation time and the stress relaxation time. This paper aims at numerical analysis of the effect of the quantum mechanism on Rayleigh-Taylor instability for a finite thickness layer of incompressible viscoelastic plasma in a porous medium. Hoshoudy (2013) [26] has studied Quantum effects on Rayleigh-Taylor instability of a plasma-vacuum. Hoshoudy (2014) [27] studied RayleighTaylor instability of Magnetized plasma through Darcy porous medium. Sharma et al. (2014) [28] has investigated the Rayleigh-Taylor instability of two superposed compressible fluids in un- magnetized plasma. The present paper deals with quantum effects on the Rayleigh –Taylor instability in an infinitely electrically conducting inhomogeneous stratified incompressible, viscoelastic fluid/plasma through a porous medium. The solution of the linearized equations of the system together with the appropriate boundary conditions leads to the dispersion relation (the relation between the normalized growth rate and square normalized wavenumber). The behavior of growth rate with respect to quantum effect and kinematic viscoelasticity are examined in the presence of porous medium, medium permeability and kinematic viscoelasticity. Formulation of the problem and perturbation equations We consider the initial stationary state whose stability is that of an incompressible, heterogeneous infinitely conducting viscoelastic Rivlin–Ericksen (Model) [24] fluid of thickness h bounded by the planes 𝑧𝑧 = 0 and 𝑧𝑧 = 𝑑𝑑. The variable density, kinematic viscosity, kinematic viscoelasticity and quantum pressure are arranged in horizontal strata electrons and immobile ions in a homogenous, saturated, isotropic porous medium with the Oberbeck– Boussinesq approximation for density variation are considered, so that the free surface behaves almost horizontal. The fluid is acted on by gravity force = (0,0, −𝑔𝑔). 𝑧 𝑧 g = (0,0, - g) 𝑧=𝑑 𝑧 𝑑

Incompressible heterogeneous infinitely conducting Rivlin-Ericksen fluid 𝑜 𝑜 𝑦 𝑦

𝑥 𝑥 𝑧=0 𝑧

Fig. 1. Diagram of finite quantum plasma layer. Following Hoshoudy (2009) [18, 19], the equations of motion, continuity (conservation of mass), incompressibility, Gauss divergence equation and Magnetic induction equations are taken as 𝜌𝜌𝒈𝒈 −𝜇𝜇 + 𝜇𝜇 Copyright © 2021, Scholarly Research Journal for Interdisciplinary Studies

Urmil Kumari & Prakash Chand Chopra

(Pg. 16020-16030) 𝜌𝜌𝜀𝜀

𝜕𝜕𝜕𝜕𝜕𝜕

𝜀𝜀 (1)

∇. 𝒒𝒒 = 0, 𝜀𝜀 𝜕𝜕

𝜕𝜕𝜕𝜕𝜌𝜌 +

16023

𝑘𝑘11′ 𝜕𝜕𝜕𝜕𝜕𝜕 𝒒𝒒 + 𝑸𝑸,

(𝒒𝒒. ∇)𝜌 = 0, (2, 3)

where 𝒒𝒒, 𝜌𝜌, 𝑝𝑝, 𝜇𝜇, 𝜇𝜇′, 𝑘𝑘1, 𝜀𝜀, 𝑸𝑸 represent velocity, density, pressure, viscosity, viscoelasticity, medium permeability, medium porosity and Bohr vector potential, respectively. Equation (3) ensures that the density of a particle remains unchanged as we follow with its motion. Then equilibrium profiles are expressed in the form 𝒖𝒖𝟎𝟎 = (0,0,0), 𝜌𝜌0 = 𝜌𝜌0(𝑧𝑧), 𝑝𝑝 = 𝑝𝑝0(𝑧𝑧) and 𝑸𝑸 = 𝑸𝑸0(𝑧𝑧). To investigate the stability of hydromagnetic motion, it is necessary to see how the motion responds to a small fluctuation in the value of any flow of the variables. Let the infinitesimal perturbations in fluid velocity, density, pressure, magnetic field and quantum pressure be taken by 𝑞𝑞 = (𝑢𝑢, 𝑣𝑣, 𝑤𝑤), 𝜌𝜌 = 𝜌𝜌0 + 𝛿𝛿𝜌𝜌, 𝑝𝑝 = 𝑝𝑝0 + 𝛿𝛿𝑝𝑝 and 𝑄𝑄 = 𝑄𝑄0 + 𝑄𝑄1𝑄𝑄𝑥 , 𝑄𝑄𝑦𝑦 , 𝑄𝑄𝑧𝑧. (4) Using these perturbations and linear theory (neglecting the products of higher order perturbations because their contributions are infinitesimally very small), equations (1) - (3) in the linearized perturbation form become 𝜌𝜌𝜀𝜀0 𝜕𝜕𝜕𝜕

𝜕𝜕𝜕𝜕 = −𝛻𝛻𝛿𝛿𝑝𝑝 + 𝑔𝑔𝛿𝛿𝜌𝜌 − 𝑘𝑘11 𝜇𝜇 + 𝜇𝜇′ 𝜕𝜕𝜕𝜕𝜕𝜕 𝑞𝑞 + 𝑄𝑄1, (5)

𝛻𝛻. 𝑞𝑞 = 0, 𝜀𝜀

𝜕𝜕𝜕𝜕𝜕𝜕 𝛿𝛿𝜌𝜌

+ 𝑤𝑤 𝑑𝑑𝜌𝜌𝑑𝑑𝑧𝑧0 = 0, (6, 7)

⎡ =

𝑸𝑸1

𝑒 𝑒

𝑖 𝑖

⎤ ( ) ( ∇ ∇𝜌0 ∇𝛿𝜌 + ∇ ∇𝜌0 ∇𝜌0 ∇ 𝜌0 − 0 0 0 𝑚𝑚ℎ2𝑚𝑚 ⎢⎢2𝛿𝛿𝜌𝜌𝜌𝜌2 2 𝜌 𝜌 𝜌 𝛿 𝜌 𝜌2 2 21𝜌𝜌 4𝛿𝛿𝜌𝜌𝜌𝜌 )2 +⎥⎥. ⎢⎣ 1 2∇𝛿𝛿 1 ∇𝜌𝜌 ∇𝛿𝛿

The Cartesian form of equations (5) - (7) yield 𝜌𝜌0 𝜕𝜕𝜕𝜕𝜕𝜕 1 ′ 𝜕𝜕 , ∇𝜌𝜌 𝛿𝛿𝜌𝜌 ∇𝜌𝜌 3 ⎥⎦ ∇𝜌𝜌 𝜀𝜀 𝜕𝜕𝜕𝜕 = −𝜕𝜕𝑥𝑥 𝛿𝛿𝑝𝑝 − 𝑘𝑘1 𝜇𝜇 + 𝜇𝜇 𝜕𝜕𝜕𝜕𝑢𝑢 + 𝑄𝑄𝑥𝑥 𝜌𝜌𝜀𝜀0 𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕 𝜌𝜌𝜀𝜀0

𝜕𝜕𝜕𝜕𝑦𝑦 𝑘𝑘11 ′ 𝜕𝜕𝜕𝜕𝜕𝜕 𝑣𝑣 + 𝑄𝑄𝑦𝑦,

𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕

(8)

(9) = − 𝛿𝛿𝑝𝑝 − 𝜇𝜇 + 𝜇𝜇

−𝜕𝜕𝜕𝜕𝑧𝑧 𝛿𝛿𝑝𝑝 − 𝑔𝑔𝛿𝛿𝜌𝜌 − 𝑘𝑘11 𝜇𝜇 𝑄𝑄𝑧𝑧,

′

𝜕𝜕𝜕𝜕𝜕𝜕 𝑤𝑤 + (10) + 𝜇𝜇

(11)

Urmil Kumari & Prakash Chand Chopra

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16024

𝜀𝜀 𝛿𝛿𝜌𝜌 = −𝑤𝑤 𝜕𝜕𝜕𝜕 𝜕𝜕 𝜕𝜕 1

ℎ2

𝜕

𝑄𝑥 = 2𝑚 𝑚 𝜕𝜕𝑥 𝑒 𝑖 𝑄𝑥 𝑚 𝑒 𝑚 𝑖 𝜕𝑥 ℎ2

𝜕

𝐷2 𝛿𝜌 − 2𝜌 𝐷𝐷0 𝐷𝛿𝛿 + 0 , 2 𝛿𝜌 1 𝜌 𝜌𝜌 𝐷𝜌𝜌 1 𝜕2 𝜕𝐷 2 2 𝜕 2 + 𝜕𝜕𝑦 𝜕 2 − 2𝜌 0 𝐷 𝜌0 + 2𝜌 2 (𝐷𝐷0 ) 𝛿𝛿 𝜌 𝜕𝜕 0 𝜌 𝑥𝑥 𝑦 𝜌 𝐷 𝜌 𝜌 𝜌𝜌 1 2 1 𝐷 𝛿𝜌 − 2𝜌 𝐷𝐷0 𝐷𝛿𝛿 + 2 0 , 2 𝛿𝜌 1 𝜌 𝜌𝜌 𝐷𝜌𝜌 1 𝜕2 𝜕𝐷 2 2 ( ) 𝐷 𝜌0 + 2𝜌 2 𝐷𝐷0 𝛿𝛿 𝜌 𝜕 + 𝜕 − 0 0 𝐷 𝜌 𝜌 𝜌 𝜌𝜌 2

1 2

𝑄𝑦 = 2𝑚 𝑚 𝜕𝜕𝑦 1 𝑒 𝑖 𝑄𝑦 𝑚 𝑒 𝑚 𝑖 𝜕𝑦 𝜕𝜕 𝜕𝜕 𝜕𝜕𝑥𝑥 + 𝜕𝜕𝑦𝑦 + 𝜕𝜕𝑧𝑧 = 0, (12)

(13)

(14)

where 2 𝜕𝜕𝑥𝑥2 𝜕𝜕𝑦𝑦2 2𝜌𝜌 12𝐷𝐷3𝛿𝛿𝜌𝜌 − 𝜌𝜌1 𝐷𝐷𝜌𝜌0𝐷𝐷2𝛿𝛿𝜌𝜌 + ⎤ 0 ℎ2 ⎢ 1 𝜕𝜕2 𝜕𝜕2 1 2𝜌𝜌 3 ⎥ 𝑄𝑄𝑧𝑧 = 2 𝑚𝑚𝑒𝑒𝑚𝑚𝑖𝑖 ⎢⎢ 2𝜕𝜕𝑥𝑥2 + 𝜕𝜕𝑦𝑦2− 𝜌𝜌0 𝐷𝐷0 + 2 𝜌 𝜌𝜌02 (𝐷𝐷𝜌𝜌0)3𝐷𝐷𝛿𝛿𝜌𝜌 + ⎥⎥. (15) ⎢− 1 𝐷𝐷𝜌𝜌0 𝜕𝜕𝑥𝑥𝜕𝜕22 + 𝜕𝜕𝑦𝑦𝜕𝜕22 + 2𝜌𝜌102 𝐷𝐷𝜌𝜌0𝐷𝐷2𝜌𝜌0 − 𝜌𝜌 103 (𝐷𝐷𝜌𝜌0)3𝛿𝛿𝜌𝜌⎦⎥ ⎣ 2𝜌𝜌0 ⎡

Since the boundaries are assumed to be rigid. Therefore the boundary conditions appropriate to the problem are 𝑤𝑤 = 0, 𝐷𝐷𝑤𝑤 = 0 at 𝑧𝑧 = 0 and 𝑧𝑧 = 𝑑𝑑, on a rigid surface.

(16)

To investigate the stability of the system, we analyze an arbitrary perturbation into a complex set of normal modes individually. For the present problem, analysis is made in terms of twodimensional periodic waves of assigned wavenumber. Thus to all quantities are ascribed describing the perturbation dependence on 𝑥𝑥, 𝑦𝑦 and 𝑡𝑡 of the forms 𝑓𝑓1(𝑥𝑥, 𝑦𝑦, 𝑧𝑧, 𝑡𝑡) = 𝑓𝑓(𝑧𝑧)𝑒𝑒𝑥𝑥𝑝𝑝𝑒𝑒𝑘𝑘𝑥𝑥𝑥𝑥 + 𝑘𝑘𝑦𝑦𝑦𝑦 −𝑛𝑛𝑡𝑡, (17) where 𝑘𝑘𝑥𝑥 and 𝑘𝑘𝑦𝑦 are wavenumbers along 𝑥𝑥 and 𝑦𝑦 directions, 𝑘𝑘 = 2 2 𝑘𝑘𝑥𝑥 + 𝑘𝑘𝑦𝑦 is the resultant wavenumber and 𝑛𝑛 is the growth rate which is, in general a complex constant. Using (17) in (8)-(11) and after some simplification, we obtain the characteristic equation: (−𝑒𝑒𝑛𝑛) − A ( 𝐷𝐷𝜌𝜌𝜌𝜌00)2𝐷𝐷2w + (−𝑖𝑖𝑖𝑖)(𝐷𝐷𝜌𝜌0) − A (𝐷𝐷𝜌𝜌03)3 − 2A ( 𝐷𝐷𝜌𝜌0𝜌𝜌)0𝐷𝐷22𝜌𝜌0𝐷𝐷𝑤𝑤 + Copyright © 2021, Scholarly Research Journal for Interdisciplinary Studies

Urmil Kumari & Prakash Chand Chopra

(Pg. 16020-16030) 𝜌𝜌0

2 −(−𝑒𝑒𝑛𝑛)𝑘𝑘2 − 𝜌𝜌 𝐷𝐷𝜌𝜌𝜌𝜌002)2𝑤𝑤 = 0,

𝜌𝑔𝑔𝑘𝑘𝑖𝑖𝑖𝑖2

16025

𝜌𝜌0

(𝐷𝐷𝜌𝜌0) − 𝜌𝜌 (18)

𝜌𝑘𝑘02𝑘𝑘𝜀𝜀1

𝜇𝜇 + 𝜇𝜇′(−𝑒𝑒𝑛𝑛) + A𝑘𝑘2

(

0 𝑘𝑘1(𝑧𝑧) = 𝑘𝑘10(0)𝑒𝑒𝑥𝑥𝑝𝑝𝐿𝐿𝑧𝑧𝐷𝐷 , 𝑛𝑛𝑞𝑞(𝑧𝑧) = 𝑛𝑛𝑞𝑞0(0)𝑒𝑒𝑥𝑥𝑝𝑝𝐿𝐿𝑧𝑧𝐷𝐷 , 𝜀𝜀(𝑧𝑧) = (19) 𝜀𝜀0(0)𝑒𝑒𝑥𝑥𝑝𝑝𝐿𝐿𝑧𝑧𝐷𝐷, where 𝜌0(0), 𝜇𝜇0(0), 𝜇𝜇0′ (0), 𝑛𝑛𝑞𝑞0(0), 𝑘𝑘10(0), 𝜀𝜀0(0) and LD are constants. Making use of (19) in (18), yield (−𝑒𝑒𝑛𝑛) − 𝐴𝐴 12𝐷𝐷𝐷𝐷2𝑤𝑤 + ( 𝐿𝐿 2 𝑔𝑔𝑘𝑘2𝑘𝑘2𝜀𝜀

−𝐿𝐿𝑖𝑖𝑖𝑖𝐷𝐷 ) − 𝐿𝐿13𝐷𝐷𝐷𝐷𝑤𝑤 + ′

𝑘𝑘2

, (20)

where 𝐴𝐴 = 4 (𝑖𝑖𝑖𝑖ℎ)2𝑚𝑚𝑘𝑘2𝑒𝑒𝑚𝑚𝑖𝑖. For the case of incompressible continuously stratified viscoelastic plasma layer considered in a porous medium, the density, viscosity, viscoelasticity and quantum pressure are taken as 𝜌𝜌0(𝑧𝑧) = 𝜌𝜌0(0)𝑒𝑒𝑥𝑥𝑝𝑝𝐿𝐿𝑧𝑧𝐷𝐷 , 𝜇𝜇(𝑧𝑧) = 𝜇𝜇0𝑒𝑒𝑥𝑥𝑝𝑝𝐿𝐿𝑧𝑧𝐷𝐷 , 𝜇𝜇′(𝑧𝑧) = 𝜇𝜇0′ (0)𝑒𝑒𝑥𝑥𝑝𝑝𝐿𝐿𝑧𝑧𝐷𝐷, 𝐿𝐿𝐷𝐷𝑖𝑖𝑖𝑖 𝑘𝑘1 𝐿𝐿2𝐷𝐷 and 𝑖𝑖𝑞𝑞22𝑤𝑤 + (−𝑖𝑖𝑖𝑖) − 𝑖𝑖𝑞𝑞2 𝐷𝐷𝑤𝑤 + (−𝑒𝑒𝑛𝑛) −𝐷𝐷 (𝑖𝑖𝑖𝑖) 𝐿𝐿𝐷𝐷(𝑖𝑖𝑖𝑖)𝐿𝐿𝐷𝐷 2 𝑔𝑔𝑘𝑘2𝑘𝑘2𝜀𝜀

′

𝑘𝑘2𝑖𝑖𝑞𝑞2

(21)

−(−𝑒𝑒𝑛𝑛)𝑘𝑘 − − 𝜈𝜈 + 𝜈𝜈 (−𝑒𝑒𝑛𝑛) + A 𝑤𝑤 = 0 −(−𝑒𝑒𝑛𝑛)𝑘𝑘 − − 𝜈𝜈 + 𝜈𝜈 (−𝑒𝑒𝑛𝑛) + 𝑤𝑤 = 0 𝐿𝐿𝐷𝐷 𝑘𝑘1 (𝑖𝑖𝑖𝑖) 2 2 where 𝑛𝑞𝑞2 = 4 𝑚𝑚ℎ𝑒𝑒 𝑚𝑚𝑘𝑘 𝑖𝑖𝐿𝐿2𝐷𝐷 represents quantum effect. In addition to the boundary conditions given by (16), we also have

𝐷𝐷2𝑤𝑤 = 0 at 𝑧𝑧 = 0 and 𝑧𝑧 = 𝑑𝑑.

(22)

Making use of (21) in (16) and (22) and assuming 𝑤𝑤 = 𝑠𝑠𝑒𝑒𝑛(𝑛𝑛𝑧𝑧)𝑒𝑒𝑥𝑥𝑝𝑝(𝜆𝜆𝑧𝑧), where 𝑛𝑛 = 𝑖𝑖ℎ1𝜋𝜋, we obtain 2 (𝜆𝜆2 − 𝑛𝑛2) (−𝑒𝑒𝑛𝑛) −

2 𝑖𝑖𝑞𝑞 + 𝜆𝜆(

−𝑖𝑖𝑖𝑖) − 𝑖𝑖𝑞𝑞 +

Urmil Kumari & Prakash Chand Chopra

𝐿𝐿𝐷𝐷(𝑖𝑖𝑖𝑖)𝐿𝐿𝐷𝐷

(𝑖𝑖𝑖𝑖)

′

2 𝑔𝑔𝑘𝑘2 𝑘𝑘2𝜀𝜀 (𝑖𝑖𝑖𝑖)𝐿𝐿𝐷𝐷 𝑘𝑘1 𝑖𝑖1𝜋𝜋

𝑖𝑖𝑞𝑞2

ℎ

(𝑖𝑖𝑖𝑖)

𝑘𝑘2𝑖𝑖𝑞𝑞2

16026

(23)

(𝑖𝑖𝑖𝑖) and

𝑖𝑖1𝜋𝜋 (−𝑖𝑖𝑖𝑖) 𝑖𝑖𝑞𝑞2 ℎ

(Pg. 16020-16030)

(24)

𝐿𝐿𝐷𝐷 (𝑖𝑖𝑖𝑖)𝐿𝐿𝐷𝐷

In equation (24), implies that λ=− 2L1D .

(25)

(𝑒𝑒𝑛𝑛)𝑘𝑘 − − 𝜈𝜈 + 𝜈𝜈 (−𝑒𝑒𝑛𝑛) + =0 2𝜆𝜆 (−𝑒𝑒𝑛𝑛) − + − =0 Eq. no. (23) with the aid of (25) takes the form 4

𝐿𝐿12 − 𝑛𝑛2(−𝑒𝑒𝑛𝑛) − (𝑖𝑖𝑖𝑖𝑖𝑖𝑞𝑞2) − 2𝐿𝐿1𝐷𝐷 (−𝐿𝐿𝑖𝑖𝑖𝑖𝐷𝐷 ) − (𝑖𝑖𝑖𝑖𝑖𝑖)𝑞𝑞2𝐿𝐿𝐷𝐷 + 𝐷𝐷

−

(𝑒𝑒𝑛𝑛)𝑘𝑘2 𝑘𝑘(2𝑖𝑖𝑖𝑖𝑖𝑖)𝑞𝑞2 = 0.

𝑘𝑘𝑘𝑘21𝜀𝜀′(−𝑒𝑒𝑛𝑛)

(𝑖𝑖𝑖𝑖𝑔𝑔𝑘𝑘)𝐿2𝐷𝐷

(26) −

𝜈𝜈 + 𝜈𝜈

To facilitate the problem, we introduce the non-dimensional quantities as 𝑛𝑛∗2 = 𝑖𝑖 𝑖𝑖𝑝𝑝22𝑒 , 𝑛𝑛𝑞𝑞∗2 = 𝑘𝑘 ∗𝑖𝑖2𝑖𝑖𝑞𝑞2𝑝𝑝2𝑒𝑒 , 𝑛𝑛𝜀𝜀∗ = 𝑖𝑖 𝜀𝜀𝑝𝑝𝑒𝑒 , 𝑛𝑛𝜈𝜈∗ = 𝑖𝑖𝜈𝜈𝑝𝑝𝑒𝑒 , 𝑛𝑛𝜕𝜕∗′ = 𝑣𝑣′, 𝑛𝑛𝑘𝑘∗1 𝜌 𝑖𝑖𝑘𝑘𝑝𝑝1𝑒𝑒 ∗2𝐿𝐿ℎ𝐷𝐷2 ∗2 𝑘𝑘2𝐿𝐿2𝐷𝐷, 𝑔𝑔∗ 𝑖𝑖 𝑔𝑔𝐿 , where 𝑛𝑛𝑝𝑝𝑒𝑒 = differential equation 𝑝𝑝𝑒𝑒 𝐷𝐷 given by (23) in (25) yield

𝑚𝑚

𝑒𝑒2𝜀𝜀0 is the plasma frequency, then using the

14 − 𝑛𝑛∗2−𝑒𝑒𝑛𝑛∗ − 𝑖𝑖𝑞𝑞∗𝑖𝑖𝑖𝑖2 ∗𝑘𝑘∗2− 12−𝑒𝑒𝑛𝑛∗ − 𝑖𝑖 ∗ ∗2

𝑞𝑞∗𝑖𝑖𝑖𝑖2 ∗𝑘𝑘∗2 +

∗2 ∗ (27)

Let 𝑛𝑛 𝑒𝑒𝑖𝑖 and in the case of 𝑛𝑛 and 𝑖𝑖 ≠ 0 (stable oscillations), the square normalized growth rate may be determined from equations (27) as 14 − 𝑛𝑛∗2𝑖𝑖 + 𝑖𝑖𝑞𝑞∗𝛾𝛾2𝑘𝑘∗2− 12𝑖𝑖 + 𝑖𝑖𝑞𝑞∗𝛾𝛾2𝑘𝑘∗2 + −𝑖𝑖𝑘𝑘∗2 + 𝑔𝑔 𝑘𝑘𝑖𝑖∗2𝑘𝑘∗𝑖𝑖1𝜀𝜀∗ 𝑛𝑛𝜈𝜈∗ + 𝑖𝑖 𝑛𝑛𝜕𝜕∗′ = 0, (28)

∗𝛾𝛾𝑘𝑘∗2 −

Urmil Kumari & Prakash Chand Chopra

(Pg. 16020-16030)

16027

𝑖𝑖 𝑘𝑘1∗ 14 𝑖𝑖𝑖𝑖𝜀𝜀∗𝑘𝑘𝑖𝑖1∗′𝑖𝑖𝑖𝑖∗𝑘𝑘𝑖𝑖1∗ 1 𝑖𝑖2∗𝜋𝜋2𝑛𝑛𝑞𝑞∗2 − 𝑔𝑔∗ = 0, 𝑎𝑎1𝑖𝑖2 + 𝑎𝑎2𝑖𝑖 + 𝑎𝑎3 = 0,

∗2 (29) (30)

where 𝑛𝑛∗𝜀𝜀𝑛𝑛∗𝑣𝑣′ 𝑛𝑛𝜀𝜀∗∗𝑛𝑛𝜈𝜈∗ ∗ 𝑎𝑎 1+ℎ4ℎ∗ ∗+𝑛𝑛𝑛𝑛 𝑘𝑘∗1𝜋𝜋2 ℎ∗ 𝑘1 𝑘1 𝑛𝑛+2𝑛𝑛1∗𝜋𝜋22 𝑘𝑘𝜋𝜋∗22. , 𝑎𝑞𝑞 , 𝑎3𝑔𝑔∗2ℎ+∗𝑖𝑖212 = 2 𝑘2 2 𝑘 2 − 4ℎ 1 =1+ 2 =2 𝑘𝑘∗2 (31) 𝑛𝑛 ∗ 𝑎 𝑎 2 ∗2 4ℎ∗ 𝑘𝑘 Case (i). When 𝑛 , in Eq. (29) we find that a1 =1,a2 = 0 and 4 𝑔𝑔∗ℎ∗2 𝑘𝑘∗2 𝑎𝑎3 = − ℎ∗2+𝑖𝑖12𝜋𝜋2 and we obtain the classical normalized growth rate ( 𝑖𝑐𝑐) in the absence of quantum physics as 4 𝑔𝑔∗ℎ∗2 𝑘𝑘∗2 𝑖𝑖𝐶𝐶 = ℎ∗2+𝑖𝑖12𝜋𝜋2 .

(32)

In the absence of viscoelastic parameter 𝑛𝑛𝜕𝜕∗′ = 0, in (29), we obtain the normal growth ratewhich is similar as given by Goldston and Rutherford (1997) [14]. ∗, we have a1 =1,a2 = 0 while a3 as in Case (ii). When 𝑛𝑛 equation (31) and the quantum normalized growth rate is given by

2 𝑖𝑖𝑞𝑞

2 ℎ +𝑖𝑖1𝜋𝜋 −

𝑛𝑛𝑞𝑞∗2𝑘𝑘∗2, (33)

which is in good agreement with the earlier result obtained by Hoshoudy (2009) [18, 19]. It is clear from the comparison of expressions (31) and (33) that the quantum term stabilize the effect on Rayleigh-Taylor instability problem. 2. Results and discussion We shall now analyze the effect of various parameters on the instability of the system under consideration. For this we solve equation (30) using the software Mathematica 5.2. For the role of porosity of the porous medium, the medium permeability, kinematic viscosity with quantum term one may be referred to (Hoshoudy 2009, [18, 19]). So, we shall confine our attention on numerical results to study the role of simultaneous presence of kinematic viscoelasticity and quantum effect. For numerical computation we taken following values of the relevant parameters 𝑛𝑛

Urmil Kumari & Prakash Chand Chopra

16028

(Pg. 16020-16030)

∗,

respectively. Figures 1 and 2 correspond to the variation of the square of the normalized growth rate 2 𝑖𝑖 w.r.t the square normalized wave number 𝑘𝑘∗2 for four different values ofkinematic viscoelasticity 𝑛𝑛𝜈𝜈∗′ = 0.1, 0.3, 0.5, 0.9 and kinematic viscosity 𝑛𝑛 = 0.2, 0.4, 0.6, 0.8, respectively. It is clear from the graphs that with the increase in kinematic viscosity and kinematic viscoelasticity, the growth rate of the unstable perturbation decreases; thereby stabilizing the system, however the critical wavenumber kc∗2 remains the same i.e. 1.6. 0.7

0.6

i 0.6 0.5 2

0.4

∋

ν ∗ ∋ ν ∗ ∋ ν ∗ ∋ ν

ii n = 0.3 iii n = 0.5

0.5

iii ii

iv n = 0.9

iii n∗= 0.6 ν iv n∗= 0.8

0.4

∗

n = 0.2 ν ii n∗= 0.4

n∗ = 0.1

γ 0.3

0.2

0.1

0.0 0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

k∗2

0.0 0.0

0.2

0.4

0.6

iii

0.8

1.0

1.2

1.4

1.6

k∗2

different values of kinematic viscoelasticity Fig. 1. Variation of ∗ different values of kinematic viscosity Fig. 2. Variation of

0.3

𝑖𝑖2

𝑖2 with

𝑘𝑘∗2 for

with 𝑘∗2 for 𝑛𝑛𝜈𝜈∗.

𝑛𝑛𝜈𝜈′.

Figures 3 and 4 correspond to the variation of the square of the normalized growth rate 𝑖𝑖2 w.r.t the square normalized wave number 𝑘𝑘∗2 for three different values of medium porosity 𝑛𝑛𝜀𝜀∗ = 0.1, 0.3, 0.7 and quantum plasma 𝑛𝑛

, respectively. It is

clear from the graphs that in the presence of medium porosity 𝑛𝑛𝜀𝜀∗ has a slight stabilizing effect, whereas the critical wavenumber remains the same. i.e. 1.6. It is clear from the figure that in the presence of quantum plasma 𝑛𝑛𝑞𝑞∗ square of the normalized growth rate 𝑖𝑖2 increases with the increasing 𝑘𝑘∗2 until arrives at the maximum instability, then decrease with the increasing 𝑘𝑘∗2 until arrives at the complete stability, where the maximum instability appears at 𝑘𝑘𝑚𝑚𝑚𝑚∗2 𝑥𝑥=0.7 and the complete stability appears at 𝑘𝑘𝑐𝑐∗2=1.1. This graph shows that quantum effect play a major role in securing a complete stability.

Urmil Kumari & Prakash Chand Chopra

(Pg. 16020-16030)

16029

3. Conclusions The effect of quantum term on the Rayleigh-Taylor instability of stratified viscoelastic Rivlin –Ericksen (Model) fluid /plasma saturating a porous media has been studied. The principal conclusions of the present analysis are as follows: 1. The kinematic viscoelasticity stabilizing effect on the system and the critical wavenumber is 𝑘𝑘𝑐𝑐∗2=1.6. 2. The kinematic viscosity has a slight stabilizing effect on the system. 3. The medium porosity has a large stabilizing effect on the system. 4. Quantum plasma plays a major role in approaching a complete stability implying thereby the large enough stabilizing effect on the system. 0.8 1.2 0.7

n∗ = 0.1 ε ∗ ε ∗ ε

ii n = 0.3 iii n = 0.7

0.6 0.5

1.0

γ 0.4

γ 0.6

iii

0.3

nq = 0.0 ii n*q = 0.4 iii n*q = 0.6

0.8

i ii

iv n*q = 0.9

0.4

iii

0.2 0.2

0.1 0.0 0.0

0.2

0.4

0.6

0.8

k∗2

1.0

1.2

1.4

1.6

0.0 0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

k∗2

Fig. 3. Variation of 𝑖𝑖2 with 𝑘𝑘∗2 for different Fig. 4. Variation of 𝑖𝑖2 with 𝑘∗2 for different values of medium porosity 𝑛𝑛𝜀𝜀∗. values of quantum plasma 𝑛𝑛𝑞𝑞∗. References Lord Rayleigh Scientific papers 2 (1900) 200. G.I. Taylor Proceedings of Royal Society of London A 201(1065) (1950) 192. D.J. Lewis Proceedings of Royal Society of London A 202(1068) (1950) 81. M. Kruskal, M. Schwarzschild Proceedings of Royal Society of London A 223(1154) (1954) 348. R. Hide Proceedings of Royal Society of London A 233(1194) (1955) 376. S. Chandrasekhar Mathematical Proceedings of the Cambridge Philosophical Society 51 (1955) 162. D.D. Joseph, Stability of fluids motions II (Springer Verlag, New York, 1976). P.G. Drazin, W.H. Reid, Hydrodynamic Stability (Cambridge University Press, Cambridge, 1981). S. Dutta, M.J. McLennan Reports on Progress in Physics 53 (1990) 1003. P. Madappa, James M. Lattimer, Raymond F. Sawyer, Raymond R. Volkas Annual Review of Nuclear and Particle Science 51 (2001) 295. B.A. Remington, In: 41st Annual Meeting of the Division of Plasma Physics. Session AR1.01 (Seattle, Washington, November 15-19, 1999).

Urmil Kumari & Prakash Chand Chopra

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16030

M. Brambilla, F. Castelli, A. Gatti, L. A. Lugiato, G. L. Oppo, G. Grynberg II Nuovo Cimento 110 (1995) 635. C.L. Gardner SIAM Journal on Applied Mathematics 54 (1994) 409. R.J. Goldston, P.H. Rutherford, Introduction to plasma physics (Institute of Physics, London, 1997). G. Manfredi, F. Haas Physical Review B 64 (2001) 7. F. Haas Physics of Plasmas 12 (2005) 062117. G. Manfredi, In: Topics in Kinetic Theory, ed. by T. Passot, C. Sulem, P.-L. Sulem (Fields Institute Communications, 2005), vol. 46, p. 263. G.A. Hoshoudy Physics of Plasmas 16 (2009) 024501. G.A. Hoshoudy Physics of Plasmas 16 (2009) 046501. S. Ali, Z. Ahmed, Arshad M. Mirza, I. Ahmad Physics Letters 373 (2009) 2940. G.A. Hoshoudy Chinese Physics Letters 27 (2010) 125201. G.A. Hoshoudy Plasma Physics Reports 37 (2011) 775. G.A. Hoshoudy Journal of Modern Physics 3 (2012) 1792. R.S. Rivlin, J.L. Ericksen Journal of Rational Mechanics and Analysis 4 (1955) 323. G.A. Hoshoudy Journal of Modern Physics 2 (2011) 1146. G.A. Hoshoudy Physical Review and Research International 3 (2013) 256. G.A. Hoshoudy Journal of Modern Physics 5 (2014) 186. P.K. Sharma, A. Tiwari, S. Argal, R.K. Chhajlani International Conference on Recent Trends in Physics 534 (2014) 1742.