Triple-Diffusive Convection in Rivlin-Ericksen
Fluid under varying gravity field in Porous Medium
S. K.
Kango1*, Sanjay Sharma2 and Vikram
Singh3
1Department of
Mathematics, Govt. College, Haripur (Manali) Distt. Kullu
(HP)-175 136
2Govt. College, Bassa, Dist. Mandi (HP)
3Jwalaji Degree
College, Jwalamukhi, Distt. Kangra
(HP)
*Corresponding Author: skkango72@gmail.com
ABSTRACT:
The Triple-Diffusive
convection in Rivlin-Ericksen fluid with varying
gravity field is considered in the presence of uniform vertical magnetic field
in porous medium. For the case of stationary convection, the magnetic field,
varying gravity field and the stable solute gradients have stabilizing effects
whereas the medium permeability has destabilizing (or stabilizing) effect on
the system under certain conditions. A linear stability analysis theory and
normal mode analysis method have been carried out to study the onset
convection. The kinematic viscoelasticity has no
effect on the stationary convection. The solute gradients, magnetic field,
varying gravity field, porosity and kinematic viscoelasticity
introduce oscillatory modes in the system, which were non-existent in their
absence. The sufficient conditions for the non-existence of overstability
are also obtained. The results are also shown graphically.
KEYWORDS: Triple-Diffusive Convection; Rivlin-Ericksen Fluid; Solute Gradients, Vertical Magnetic
Field; Varying Gravity Field; Porous Medium.
2010 Mathematics Subject
Classification: 76A10, 76D50, 76E25, 76S05.
INTRODUCTION:
The instability of the plane interface
separating two Newtonian fluids when one is superposed over the other, under
varying assumptions of hydrodynamics and hydromagnetics,
has been studied by several researchers and a comprehensive account of these
investigations has been given by Chandrasekhar [10]. The problem of thermohaline convection in a layer of fluid heated from
below and subjected to a stable salinity gradient has been considered by Veronis [7].
In classical thermal instability problems, it
has been assumed that the deriving density differences are produced by the
spatial variation of single diffusing property i.e. Kivu are strongly
stratified by temperature and a salinity which is the sum of comparable
concentrations of many salts, while the oceans contain many salts in
concentrations less than a few percent of the sodium chloride concentration. In
laboratory experiments on double-diffusive convection, dyes or small
temperature anomalies introduce a third property which affects the density of
the fluid.
With the growing importance of non-Newtonian
fluids in geophysical fluid dynamics, chemical technology and petroleum
industry, the investigations on such fluids are desirable. The Rivlin-Ericksen fluid [8] is one such fluid.
The idealization of uniform gravity field can
be hardly justified in the presence of large scale convection phenomenon
occurring in atmosphere, the ocean or the mantle of the earth. Pradhan et al [6] studied the thermal instability of the
fluid layer under variable gravitational field.
In the physical world, the investigation of
flow of Rivlin-Ericksen fluid through porous medium
has become an important topic due to the recovery of crude oil from the pores
of reservoir rocks. Flows in porous region are a creeping flow. When a fluid
permeates a porous material, the actual path of the individual particles cannot
be followed analytically. When the density of a stratified layer of a
single-component fluid decreases upwards, the configuration is stable. This is not
necessarily so far a fluid consisting of two or more components which can
diffuse relative to each other. The reason lies in the fact that the
diffusivity of heat is usually much greater than the diffusivity of a solute. A
displaced particle of fluid thus, loses any excess heat more rapidly than any
excess solute. The resulting buoyancy force may tend to increase the
displacement of the particle from its original position and thus cause
instability.
The flow through porous medium has been of
considerable interest in recent years, particularly in geophysical fluid
dynamics. A porous medium is a solid with holes in it, and is characterized by
the manner in which the holes are imbedded, how they are interconnected and the
description of their location, shape and interconnection. However, the flow of
a fluid through a homogeneous and isotropic porous medium is governed by
Darcy’s law which states that the usual viscous term in the equations of Rivlin-Ericksen fluid
motion is replaced by the resistance term
Out of large published work in pure fluid, the thermosolutal
convection in porous medium has received only attention, because of its
engineering applications. A comprehensive review of the literature concerning thermosolutal convection in a fluid-saturated porous medium
may be found in the book written by Nield and Bejan [4]. The thermosolutal
instability in Walters’ B' fluid in the presence of Hall currents
in porous medium in hydromagnetics has been studied
by Kango et al [11]. S. Chand
[9] studied triple-diffusive convection in Walters’ (Model B') fluid in porous
medium in hydromagnetics. Oldenburg and Pruess [3] have developed a model for convection in a
Darcy’s porous medium, where the mechanism involves temperature, NaCl, CaCl
In the standard Benard
problem, the instability is driven by a density difference caused by a
temperature difference between the upper and lower planes bounding the fluid.
If the fluid additionally has salt dissolved in it, then there are potentially
two destabilizing sources for the density difference, the temperature field and
the salt field. The solution behaviour in the double-diffusive convection
problem is more interesting than that of the single component situation in so
much as new instability phenomena may occur which is not present in the
classical Benard problem. When
temperature and two or more agencies, or two different salts, are present the
physical and mathematical situation becomes increasingly richer. Very
interesting results in triply-diffusive convection have been obtained by
Pearlstein et al., [1].. They demonstrate that for
triple diffusive convection linear instability can occur in discrete sections
of the Rayleigh number domain with the fluid being linearly stable in a region
in between the linear instability ones. This is because for certain parameters
the neutral curve has a finite isolated oscillatory instability curve lying
below the usual unbounded stationary convection one. Straughan
and Walker [2] derive the equation for non-Boussinesq
convection in a multi-component fluid and investigate the situation analogous
to that of Pearlstein et al., but allowing for a density non-linear in a
temperature field. In reality the density of a fluid is never a linear function
of temperature, and so the work of Straughan and
Walker applies to the general situation where the equation of state is one of
the density quadric in temperature. This is important,
since they find that departure from the linear Boussinesq
equation of state changes the perfect symmetry of the heart shaped neutral
curve of Pearlstein et al.
There are many technological important alloys
that contain significant mass fractions of three or more metallic elements.
Among these are a number of nickel-based super alloys used in turbine blades
and other high-strength applications and another application stems from
advances in instrumentation and data reduction techniques which have led to
renewed interest among physical chemists in the measurement of multi-component
diffusion coefficients.
Keeping in mind the importance of triple
diffusion in lake Kivu, Gulf Stream eddies,
solidification of molten alloys and magmas nickel-based super alloys used in
turbine blades and other high-strength application and that of ground-water
rotation in hydrology and chemical engineering etc. Motivated by the above
mentioned applications of triple diffusion, we set out to study the
triple-diffusive convection in Rivlin-Ericksen fluid
under varying gravity field saturating a porous medium.
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Received on 22.01.2013 Accepted
on 07.02.2013
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