Stability of Stratified viscoelastic Rivlin-Ericksen Fluid/Plasma in the presence of variable Horizontal Magnetic Field

Veena Sharma, Anukampa Thakur, Priti Bala

Department of Mathematics & Statistics, H.P. University, Shimla-171 005

ABSTRACT:

A study has been made of the instability of an electrically conducting stratified viscoelastic Rivlin-Ericksen fluid/plasma arranged in a horizontal strata to include a variable horizontal magnetic field. Applying the linear stability theory and normal mode technique to a set of partial differential equations, the solutions of an eigen value problem have been obtained by assuming exponential vertical stratifications in density, neutral gas density, viscosity, viscoelasticity and magnetic field. Using these solutions, the dispersion relation so obtained has been analysed numerically. It has been found that viscosity, viscoelasticity , collisional frequency and magnetic field have stabilizing effects on the growth rates for unstable configuration under certain wave-number band.

INTRODUCTION:

Since the pioneering work of Rayleigh, the problem of the instability of a semi-infinite layer of a stratified fluid has been studied by several authors under various operative forces and a detailed account of these investigations has been given by Chandrasckhar [1961], Drazin and Reid [1981] in their monographs. Among others, Lehnert [1959], Wobig [1972], Srivastava [1974], Bhatia and Chhoakar [1985], Gupta and Bhatia [1991] have studied this problem under varying assumptions. The medium has been considered to be fully ionized.

A partially ionized plasma represents a state which often exists in the Universe and there are several situations in which the interaction between the ionized and neutral gas components becomes important in cosmic physics. Stromgren [1939] has reported that ionized hydrogen is limited to certain rather sharply bounded regions in space surrounding, for example, O-type stars and clusters of such stars and that the gas outside these regions is essentially non-ionized. Other examples of the existence of such situations are given by Alfvén’s [1954] theory on the origin of the planetary system, in which a high ionization rate is suggested to appear from collisions between a plasma and a neutral-gas cloud and by the absorption of plasma waves due to ion-neutral collisions such as in the solar photosphere and chromospheres and in cool interstellar Clouds. According to Hans [1968] and Bhatia [1970], the medium may be idealized as a composite mixture of a hydromagnetic (ionized) component and a neutral component, the two interacting through mutual collisional (frictional) effects. They have shown that the collisions have a stabilizing effect on the Rayleigh-Taylor instability. Lehnert [1972] has found that both ion viscosity and neutral gas friction have a stabilizing influence on cosmical plasma interacting with a neutral gas.

Uberoi and Selack [1992] have investigated the Alfvén waves in two-dimensional variable magnetic-field, the magnetic field varying along the perpendicular to the plane in which the magnetic field lie. The instability of a stratified plasma in variable one-dimensional horizontal magnetic field along x-axis (stratified along z-axis) has been studied by Sharma et al. [1993], for the longitudinal mode of propagation.

There are several astrophysical situations where the plasma is not fully lonized and instead may be permeated with neutrals. Attention to such situations has been drawn by Alfvén and Piddington. The instability of such partially ionized plasmas has been carried out by several authors in the past. Generally the magnetic field has a stabilizing effects on the instability but there are a few exceptions also. For example, Kent [1966] has studied the effect of horizontal magnetic field which varies in the vertical direction on the stability of parallel flows and has shown that the system is unstable under certain conditions, while in the absence of magnetic field the system is known to be stable. In all the above studies the fluid has been assumed to be newtonian. The Rayleigh-Taylor instability of the ionosphere and astrophysical situations e.g. sunspots, solar corona and stellar atmospheres are well known in literature.

There is growing importance of non-newtonian viscoelastic fluids in chemical technology and petroleum industry etc. Another important class of viscoelastic fluids is given by Rivlin Ericksen [1955]. Joshi [1976] has discussed the viscoelastic Rivlin-Ericksen incompressible fluid under time-dependent pressure gradient. Srivastava and Singh [1988] have studied the unsteady flow of a dusty elastic-viscous Rivlin-Ericksen fluid through channel of different cross-section in the presence of the time-dependent pressure gradient. In another study Garg et al. [1994] have studied the rectilinear oscillations, of a sphere along its diameter in a conducing dusty Rivlin-Ericksen fluid in the presence of a uniform magnetic field. Sharma et al. [2001] have also studied the Rivlin-Eriksen fluids in porous medium. Sharma and Kishor are [2010] have studied the effect of uniform of magnetic field on the stability of stratified rotating Rivlin-Ericksen fluid. Such polymers are used in agriculture, communication appliances and in biomedical applications.

Keeping in mind the above mentioned applications, the present paper deals with the stability of electrically conducting stratified viscoelastic Rivlin-Ericksen fluid/plasma in the presence of a variable horizontal magnetic field. To the best of our knowledge, the problem has not been investigated so far.

Formulation of the Problem and Perturbation Equations

Consider an incompressible composite layer consisting of an infinitely electrically conducting Rivlin-Ericksen fluid of density , permeated with neutrals of density , arranged in horizontal strata and acted on by the gravity force and variable horizontal magnetic field . Assume that both the ionized fluid and the neutral gas behave like continuum fluids and that effects on the neutral component resulting from the presence of magnetic field, pressure and gravity are neglected.

Let and denote, respectively, viscosity, viscoelasticity, the pressure, collision frequency between the two components of the plasma and the velocity of fluid. Then the equations expressing conservation of momentum, mass, incompressibility and Maxwell’s equations for the visco-elastic Rivlin- Ericksen fluid/plasma are

where , the magnetic permeability is, assumed to be constant. denotes the velocity of the neutral gas.

Eliminating from equation (1) by using equation (3), we get

Since the equilibrium state under consideration is static one, it is clearly characterized by following equations

The character of equilibrium of this stationary state can be determined by disturbing the system slightly and then, following its further evolution.

Let and denote, respectively, the perturbations in density , pressure p(z), velocity and variable horizontal magnetic field . Then the equations (3) – (7), after perturbations and by using linear theory, become in cartesian form

Dispersion Relation

Analyzing the disturbances into normal modes, we seek solutions whose dependence on x, y, z and time t is given by

Now substituting the values of and from equations (24) – (26) in equations (18) – (20), we get

Multiplying equation (27) and (27) by and , respectively and adding we get

Substituting value of in equation (28), it reduces to

Multiplying equations (27) and (31) by the and , respectively and then adding and using equation (20), we obtain

Eliminating u and from equations (29) and (30) using equations (22) and (31) after little algebra, we get

Last two-terms in equation (33), appearing with the factor, represents the effect of the heterogeneity of the fluid on the intertia; neglected in comparision with the effect on potential energy.

Equation (33) can also be written as

Equation (34) is the characteristic equation formulating the effect of variable horizontal magnetic field, collisional frequency, viscosity and viscoelasticity on the stability of stratified Rivlin-Ericksen fluid/plasma.

The Case of Exponentially Varying Stratifications

In order to obtain the solution of the stability problem of a layer of Rivilin-Ericksen fluid/plasma, we suppose that the fluid density , neutral gas density , viscosity , viscoelasticity and magnetic field vary exponentially along the vertical direction i.e.

where and are constants and so the kinematic viscosity, , the kinematic viscoelasticity and the Alfvén velocity are constant everywhere.

Using stratifications of the form (35), equation (34) transforms to

Here the case of two free boundaries is considered. The boundary conditions are

The appropriate solution of equation (35) satisfying above boundary conditions is

where m is an integer and A is a constant. Substituting the value of w from equation (38) in equation (36), we obtain

Equation (39) is a cubic equation in n and is the dispersion relation governing the effects of variable horizontal magnetic field, collisional frequency, kinematic viscosity and kinematic viscoelasticity on the stability of stratified Rivilin-Ericksen fluid/plasma.

By taking kinematic viscoelasticity , equation (39) reduces to

Results and Discussion

(a) Case of Stable Stratification (i.e.) . Equation (39) does not admit of any positive real root or complex root with positive real part using Routh–Hurwitz criterion; therefore, the system is always stable for disturbances of all wave-numbers.

(b) Case of Unstable Stratifications (i.e.). If , equation (39) has at least one root with positive real part or complex root with positive real part using Routh–Hurwitz criterion; so the system is unstable for all wave-numbers satisfying the inequality

where is the angle between and i.e. .

If and, equation (39) does not admit of any positive real root or complex root with positive real part, therefore, the system is stable. The system is clearly unstable in the absence of variable magnetic field. However, the system can be completely stabilized by a large enough magnetic field.

Numerical Model Example

The dispersion relation (39) has been computed numerically for for different values of the physical parameters and square of velocity , using the s oftware Mathematica version (2007).

Figure 1 shows the variation of growth rate (positive real part of n) with respect to k satisfying equation (39) for fixed permissible values of , g = 980. These values are the permissible values for the respective parameters and are in good agreement with corresponding values used by Chandrasekhar [1961] while describing various hydrodynamic and hydromagnetic stability problems and many others. The graph shows that for fixed wave number, the growth rate increases for with the increase in kinematic viscosity , which depicts the destabilizing effect of kinematic viscosity whereas the growth rate decreases for , implying thereby the stabilizing effect of kinematic viscosity on the system.

Figure 2 shows the variation of growth rate with respect to the wave number k for fixed permissible values of , cm/sec², , for three values of , respectively. The graph shows that for

cm/sec², for three values of, respectively.

Figure1: Variation of (real part of n) with positive wave number k.

Figure 2: Variation of (real part of n) with positive wave number k.

fixed wave number, the growth rate increases for , with the increase in kinematic viscoelasticity which indicates the destabilizing influence of kinematic viscoelasticity, whereas growth rate decreases for , implying thereby the stabilizing effect of kinematic viscoelasticity on the system.

Figure 3 shows the variation of growth rate with respect to the wave number k for fixed permissible values of , cm/sec², , for three values of , respectively. The graph shows that for fixed wave number, the growth rate increases for , with the increase in collision frequency between two components of plasma which indicates the destabilizing influence of , whereas the growth rate decreases for , implying thereby the stabilizing effect of collision frequency.

Figure 3: Variation of (real part of n) with Figure 4: Variation of (real part of n) with wave number k. wave number k.

Figure 4 shows the variation of growth rate with respect to the wave number k for fixed permissible values of , cm/sec², for three values of respectively. The graph shows that for fixed wave-number, the growth rate increases with the increase in the square of Alfvén velocity for which indicates the destabilizing influence of the square of the Alfvén velocity, whereas growth rate decreases for implying thereby the stabilizing effect of the square of the Alfvén velocity on the system.

CONCLUSIONS:

We may thus conclude the whole analysis with the following statements:

(i) The criteria determining stability or instability are independent of the effects of viscosity, viscoelasticity, collisional frequency.

(ii) The magnetic field stabilizes the system which is otherwise unstable in the absence of the magnetic field.

(ii) The kinematic viscosity , kinematic viscoelasticity , collisional frequency and variable horizontal magnetic field have a stabilizing influence on the instability of stratified Rivlin-Ericksen fluid/plasma for , which have been computed numerically.

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Received on 08.01.2015 Accepted on 05.02.2015

©A&V Publications all right reserved

Research J. Engineering and Tech. 6(1): Jan.-Mar. 2015 page 200-206

DOI: 10.5958/2321-581X.2015.00030.6