Rayleigh Taylor
Instability of Two Rotating Maxwellian
Superposed Fluid with Variable Magnetic Field in Porous Medium
Chander Bhan Mehta, Susheel Kumar
Associate Professor, Department of Mathematics, Centre of Excellence, Sanjauli Shimla (H.P)-171006,
India, Assistant Professor, Department of Mathematics, St. Bede’s College, Shimla (H.P)171002, India.
E-mail:-
*Corresponding Author: chanderbmehta@gmail.com,
susheel04gupt@yahoo.com
ABSTRACT:
The instability of
the plane interface between two viscoelastic (Maxwellian) superposed fluids in porous medium in the
presence of uniform rotation and variable magnetic field is considered. The
magnetic field, the viscosity and the density are assumed to be exponentially
varying. For stable density stratification, the system is found to be stable
for disturbances of all wave numbers. The magnetic field stabilizes the
potentially unstable stratification for small wave-length perturbations which
are otherwise unstable. The long wave-length perturbations remain unstable and
are not stabilized by magnetic field. Rotation does not affect the stability or
instability, as such, of stratification.
KEYWORDS: Rayleigh-Taylor
instability, Maxwellian viscoelastic
fluids, variable magnetic field, uniform rotation, porous medium.
INTRODUCTION:
A detailed account
of the instability of the plane interface between two Newtonian fluids under
varying assumptions of hydrodynamics and hydromagnetics, has been given by Chandrasekhar [4]. Bhatia [1] has
considered the Rayleigh–Taylor instability of two viscous superposed conducting
fluids in the presence of a uniform horizontal magnetic field. Bhatia and
Steiner [3] have studied the problem of thermal instability of a Maxwellian viscoelastic fluid in
the presence of rotation and have found that the rotation has a destabilizing
effect in contrast to the stabilizing effect on Newtonian fluid.
The problem in
porous medium is of importance in soil, ground water hydrology and in
atmosphere. When the fluid slowly percolates through the pores of a
macroscopically homogeneous and isotropic porous medium, the gross effect is
represented by Darcy’s law according to which the usual viscous term in the
equations of fluid motion is replaced by the resistance term -
Vest and Arpaci
[10] have studied the stability of a horizontal layer of Maxwell’s viscoelastic fluid heated from below. Sharma [9] has
studied the instability of the plane interface between two Oldroydian
viscoelastic superposed conducting fluids in the
presence of a uniform magnetic field. Generally, the magnetic field has a
stabilizing effect on the stability, but there are a few exceptions. For
example, Kent [6] has studied the effect of a horizontal magnetic field that
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 unstable. Lapwood [7] has studied the stability of convective flow in
hydromagnetics in a porous medium using Rayleigh’s
procedure. Bhatia and Mathur [2] have studied the
Rayleigh–Taylor instability of two superposed Oldroydian
fluids in a uniform vertical magnetic field through porous medium. The magnetic
field stabilizes the unstable configuration for wave number band k>k* in
which system is unstable in the absence of magnetic field. It is also found
that the viscosity, viscoelasticity and medium
porosity have stabilizing influence while elasticity and medium permeability
have destabilizing influence.
Since viscoelastic fluids play
an important role in polymers and in the electro–chemical industry, the studies
of waves and stability in different viscoelastic
fluids dynamical configuration has been carried out by several researchers in
the recent past. Jukes [5] investigated the Rayleigh–Taylor instability problem
in MHD with finite conductivity and found that finite conductivity introduces
new and unexpected modes. Sengupta and Basak [8] have studied the instability of the plane
interface separating two superposed viscoelastic
(Maxwell) conducting fluids in a uniform vertical magnetic field. Numerically
it is found that both viscosity and viscoelasticity
of fluid have stabilizing influence while permeability of porous medium has
mostly destabilizing effect on the growth rate of unstable mode of disturbance.
Keeping in mind the importance of
non–Newtonian fluids in modern technology and industries and owing to the
importance of variable magnetic field, rotation and porous medium in chemical
engineering and geophysics, the present paper attempts to study the
+instability of the plane interface separating two incompressible superposed
rotating Maxwellian viscoelastic
fluids in porous medium in presence of a variable magnetic field.
1. Bhatia, P.K., Rayleigh-Taylor Instability
of Two Viscous Superposed Conducting Fluids, Nuovo Cimento, 19B (1974),
161 .
2. Bhatia, P.K and Mathur,
R. P., Instability of Viscoelastic Superposed
Fluids in a Vertical Magnetic Field Through Porous Medium, Ganita Sandesh, 17(2)
(2003), 21.
3. Bhatia, P.K. and Steiner, J.M., Thermal
Instability of a Viscoelastic Fluid Layer in Hydromagnetics, J. Math. Anal.
Appl., 41 (1973), 271.
4. Chandrasekhar, S., Hydrodynamic and Hydromagnetic Stability, Dover publication, New York,
1981.
5. Jukes, I.D., On The
Rayleigh-Taylor Problem in Magnetohydrodynamics with
Finite Resistivity, J. Fluid Mech., 16 (1963), 177.
6. Kent, A., Instability of Laminar Flow of
a Magneto Fluid, Phys. Fluid, 9 (1966), 1286.
7. Lapwood, E.R., Convection
of a Fluid in a Porous Medium, Proc. Camb. Phil. Soc., 44 (1948), 508.
8. Sengupta, P.R and Basak, P., Stability of Two Superposed Viscoelastic (Maxwell) Fluids in a Vertical Magnetic
Field, Indian J. Pure Appl. Math., 35(7) (2004), 905.
9. Sharma, R.C., Instability of the Plane
Interface Between Two Viscoelastic SuperposedConducting Fluids, J.Math.Phys.Sci.,12 (1978), 603.
10. Vest, C.M. and Arpaci,
V.S., Overstability of a Viscoelastic Fluid Layer Heated from
Below, J.
Fluid Mech., 36 (1969), 613.
Received on 20.01.2013 Accepted
on 10.02.2013
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