Stability
of Stratified viscoelastic RivlinEricksen
Fluid/Plasma in the presence of variable Horizontal Magnetic Field
Veena Sharma,
Anukampa Thakur, Priti Bala
Department of
Mathematics & Statistics, H.P. University, Shimla171 005
ABSTRACT:
A study has been made of the instability of
an electrically conducting stratified viscoelastic RivlinEricksen 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
wavenumber band.
INTRODUCTION:
Since the pioneering work of Rayleigh, the problem of
the instability of a semiinfinite 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, Otype stars and clusters of such
stars and that the gas outside these regions is essentially
nonionized. 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 neutralgas
cloud and by the absorption of plasma waves due to ionneutral 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 RayleighTaylor 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 twodimensional
variable magneticfield, the magnetic field varying along the perpendicular to
the plane in which the magnetic field lie.
The instability of a stratified plasma in variable onedimensional
horizontal magnetic field along xaxis
(stratified along zaxis) 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
RayleighTaylor 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 nonnewtonian
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 RivlinEricksen
incompressible fluid under timedependent pressure gradient. Srivastava and
Singh [1988] have studied the unsteady flow of a dusty elasticviscous RivlinEricksen fluid through channel of different
crosssection in the presence of the timedependent pressure gradient. In another study Garg
et al. [1994] have studied the rectilinear oscillations, of a sphere along its
diameter in a conducing dusty RivlinEricksen fluid
in the presence of a uniform magnetic field.
Sharma et al. [2001] have also studied the RivlinEriksen
fluids in porous medium. Sharma and Kishor are [2010] have studied the effect of uniform of
magnetic field on the stability of stratified rotating RivlinEricksen
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 RivlinEricksen
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 RivlinEricksen
fluid of density
Let
where
Eliminating
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
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
Multiplying equation (27) and (27) by
Substituting value of
Multiplying equations (27) and (31) by the
Eliminating u and
Last twoterms 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 RivlinEricksen
fluid/plasma.
The Case of Exponentially
Varying Stratifications
In order to obtain the solution of the stability
problem of a layer of RivilinEricksen fluid/plasma,
we suppose that the fluid density
where and
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 RivilinEricksen fluid/plasma.
By taking kinematic viscoelasticity
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
wavenumbers.
(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
wavenumbers satisfying the inequality
where
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
Figure 1 shows the variation of growth rate
Figure 2 shows the variation of growth rate
cm/sec^{2}, for three values of, respectively.
Figure1: Variation of
Figure 2: Variation of
fixed wave number, the growth rate increases for
Figure 3 shows the variation of growth rate
Figure 3: Variation of
Figure 4 shows the variation of growth rate
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
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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 200206 DOI: 10.5958/2321581X.2015.00030.6 
