INTRODUCTION:
Micropolar
theory was introduced by Eringen [1] in order to describe some physical systems
which do not satisfy the Navier Stokes
equations. These fluids are able to describe the behaviour of colloidal solutions,
liquid crystals; animal blood etc. The equations governing the flow of micropolar
fluid theory involve a spin vector and a microinertia tensor in addition to velocity
vector. A generalization of the theory including thermal effects has been developed
by Kazakia and Ariman [2] and Eringen [3]. Micropolar fluid stabilities have become
an important field of research these days. A particular stability problem is the
Rayleigh-Bénard instability in a horizontal thin layer of fluid heated from below.
A detailed account of thermal convection in a horizontal thin layer of Newtonian
fluid heated from below has been given by Chandrasekhar [4]. Ahmadi [5] and Pérez-Garcia
et al. [6] have studied the effects of the microstructures on the thermal convection
and have found that in the absence of coupling between thermal and micropolar effects,
the principle of exchange of stabilities may not be fulfilled and consequently micropolar
fluids introduce oscillatory motions. The existence of oscillatory motions in micropolar
fluids has been depicted by Lekkerkerker in liquid crystals [7,8], Bradley in dielectric fluids [9] and Laidlaw in
binary mixture [10]. In the study of problems of thermal convection, it is frequent
practice to simplify the basic equations by introducing an approximation which is
attributed to Boussinesq [11]. In geophysical situations, the fluid is often not
pure but contains suspended particles. Saffman [12] has considered the stability
of laminar flow of a dusty gas. Scanlon and Segel [13] have considered the effects
of suspended particles on the onset of Bénard convection. The suspended particles
were thus found to destabilize the layer. Palaniswami and Purushotham [14] have
studied the stability of shear flow of stratified fluids with the fine dust and
found that the presence of dust particles increases the region of instability. The
theoretical and experimental results of the onset of themal instability (Bénard
convection) in a fluid layer under varying assumptions of hydromagnetics, has been
depicted in a treatise by Chandrasekhar [4]. Lapwood [15] has studied the convective
flow in porous medium using linearized stability theory. The Rayleigh instability
in flow through a porous medium has been considered by Wooding [16]. The problem
of thermal convection in a fluid in porous medium is of importance in geophysics,
soil–science, ground–water, hydrology and astrophysics. The physical property of
comets, meteororites and inter–planetary dust strongly suggests the importance of
porosity in the astrophysical context (McDonnel [17]). Moreover, Saffman and Taylor
[18] have shown that the motion in a Hele–Shaw cell is mathematically analogous
to two-dimensional flow in porous medium. In recent years, there has been a considerable
interest in the study of breakdown of the stability of a layer of a fluid subjected
to a vertical temperature gradient in a porous medium and also in the possibility
of convective flow.
When a fluid permeates a porous material, the gross effect is represented
by Darcy’s law. As a result of this macroscopic law, the usual viscous term in the
equations of motion of microscopic fluid is replaced by the resistance term, where
and
are
viscosity and dynamic microrotation viscosity respectively, 
is
the medium permeability and 
is
the Darcian (filter) velocity of the fluid. The heat and solute being two diffusing
components, thermosolutal convection is the general term dealing with such phenomena.
The buoyancy forces can arise not only from density differences due to variations
in temperature, but also from those due to variations in solute concentration. Brakke
[19] explained a double–diffusive instability that occurs when solution of a slowly
diffusing protein is layered over a dense solution of more diffusing sucrose. Convection
that is dominated by the presence of two components is very common in geophysical
systems. The problem of thermosolutal
convection (double–diffusive convection) in a layer of fluid heated from below and
subjected to a stable solute gradient has been studied by Veronis [20]. Thermosolutal
convection problems arise in oceanography, limnology and engineering. The case of
fluids with uniform salinity gradients when the fluxes are driven by other mechanisms
has been looked at by McDougall [21] who assumed that the fluxes were proportional
to the salinity difference between the convective layers and independent of the
layer thickness and by Holyer [22], who assumed that the fluxes were driven by molecular
diffusivities. In all these cases where an unbounded fluid has uniform horizontal
and vertical compositional gradients, the fluid is always unstable and so considerations
of marginal stability are inappropriate. Sharma and Gupta [23] have studied the
effect of rotation on thermal convection in micropolar fluids in the presence of
suspended particles. Sharma et al. [24] have studied the thermal convection
of micropolar fluids in the presence of suspended particles in hydromagnetics in
porous medium. Sharma
and Gupta [25] have studied the thermosolutal convection of micropolar fluids in
the presence of suspended particles. Lata and Gupta [26] have studied the effect of rotation and suspended
particles on micropolar fluid heated and soluted from below saturating porous medium.
Keeping
in mind the importance and relevance of porosity, solute parameter magnetic field
and rotation in chemical engineering, geophysics and biomechanics, thermal instability
of micropolar fluids in the presence of rotation and magnetic field to include the
effect of solute parameter and suspended particles (dust particles) in porous medium
has been considered in the present paper.
MATERIAL AND
METHODS:
Consider
an infinite, horizontal layer of an incompressible electrically conducting micropolar
fluid of thickness 
permeated
with suspended particles (or fine dust) in an isotropic and homogeneous medium of
porosity 
and
medium permeability .
A uniform vertical magnetic field
pervades
the system. This fluid-particles layer is heated and soluted from below but convection
sets in when the temperature gradient between the lower and upper boundaries exceeds
a certain critical value. A uniform vertical rotation pervades the system. This
is the Rayleigh-Bénard instability problem in presence of salinity gradient and
fine dust in micropolar fluids. Both the boundaries are taken to be free and perfect
conductor of heat. The mass, momentum, internal angular momentum, internal energy
balance equations and analogous solute equation using the Boussinesq approximation
are
where
denote the filter (seepage) velocity, the spin, the pressure, the fluid density,
the acceleration due to gravity, the reference density, magnetic permeability and
velocity of the suspended particles, respectively. denotes the number density of
dust particles and 
is
the dynamic microrotation viscosity.., being the particle radius, is the Stokes
drag coefficient and, denote, respectively, the thermal conductivity, the solute
conductivity, the specific heat at constant volume, the heat capacity of solid matrix
, the heat capacity of particles, the coefficient giving account of coupling between
spin flux with heat flux , spin flux with solute flux and microinertial constant.
are the coefficients of angular viscosity.
Assuming
dust particles of uniform size, spherical shape and small relative velocities between
the two phases (fluid and particles), the net effect of the particles on the fluid
is equivalent to an extra body force term per unit volume, as has been taken in
equation [2]. In the equations (2), the term 
represent
the Coriolis acceleration and the term represents the centrifugal force (which
is of very small magnitude). We also use the Boussinesq approximation by allowing
the density to change only in the gravitational body force term.
The
density equation of the state is
where
are
reference density, reference temperature at the lower boundary and 
,
is
the coefficient of thermal expansion and analogous solvent coefficient, respectively.
Since
the force exerted by the fluid on the particles is equal and opposite to that exerted
by the particles on the fluid. The distance between the particles is assumed to
be so large compared with their diameter that interparticle reactions are ignored.
The buoyancy force on the particles is also neglected. If 
is
the mass of suspended particles per unit volume, then the equations of motion and
continuity for the particles, under the above assumptions, are
where
is called resistivity and 
is
electrical conductivity.
In
the quiescent state, the solution of equations (1) – (10) is
(constant),
where
are their respective reference values at 
and
is the magnitude of uniform temperature gradient.
Assume small perturbations
around the basic state, and let and denote, respectively, the perturbations on
fluid velocity 
,
particles velocity 
spin 
,
pressure 
density
,
temperature 
and
magnetic field so that the change in density 
caused
mainly by the perturbations 
and
in
temperature and solute concentration, is given by
Then
the linearized perturbation equations of the microplar fluid become
Using
the non–dimensional numbers
Equations
[13] - [20] in the non-dimensional form are
where
the following non-dimensional parameters are ,
Where
is
known as dimensionless Rayleigh number, 
is
analogous solute number, 
is
thermal Prandtl number , 
is
magnetic Prandtl number and 
is
analogous Schmidt number.
Eliminating
from
equation (26), (27) with the help of (28), yields
Eliminating
between
equations (24) and (28) and on linearizing, we obtain 
where
and
Applying
the curl operator to equations (24), (25), (29) and taking 
–component,
we get
where,
are the 
components
of current density and vorticity, respectively. and account for coupling between
vorticity and spin effects and spin diffusion, respectively.
Applying
the curl operator twice to equations [24] and taking –component,
we get
where
The
boundaries are considered to be free. The case of two free boundaries is little
artificial except in astrophysical situations but it enables us to find analytical
solutions.
Thus
the boundary conditions appropriate to problem are
at
and
Now
we analyze the perturbations into a complete set of normal modes and then examine
the stability of each of these modes individually. We ascribe to all quantities
describing the perturbation dependence on 
and
of
the form, where, are the wave numbers along the 
-
and 
-
directions, respectively, is the resultant wave number, 
is
the stability parameter which can be, complex, in general. The solution of the stability
problem requires the specifications of the state for each 
.
The above considerations allow us to suppose that the perturbation quantities have
the form
where
The
boundary conditions [40] transform to
Using
boundary conditions [49], equations [42]–[48] transform to
.The
proper solution of 
belonging
to the lowest mode is
where
is
a constant.
Eliminating
from equations [42]–[48] and substituting the solution given by equation [51],
we obtain the dispersion relation
where.
The
case of oscillatory modes
Here
we examine the possibility of oscillatory modes, if any, in the stability problem
due to the presence of salinity gradient, magnetic field intensity, rotation parameter
and suspended particles number density. Equating the imaginary parts of equation
[52], we have
It
is evident from equation [53] that 
may
be either zero or non-zero, meaning thereby that the modes may be non-oscillatory
or oscillatory. In the absence of suspended particles number density, magnetic field
intensity, rotation parameter, magnetic permeability and solute parameter, equation
[53] reduces to
and
term within the brackets is definitely positive, which implies that 
0.
Therefore, the oscillatory modes are not allowed and principal of exchange of stabilities
is satisfied for porous medium in the absence of suspended particles, solute parameter,
rotation parameter, and magnetic field. The presence of the suspended particles
number density, the magnetic field intensity, medium permeability and solute parameter
bring oscillatory modes (as 
may
not be zero) which were non–existent in their absence.
The
case of overstability:
The
present section is devoted to the possibility that instability may occur as overstability.
Since we wish to determine the Rayleigh number for onset of instability via a state
of pure oscillations, it suffices to find the conditions for which equation [52]
will admit of solutions with 
real.
Substituting in equation [52], and then equating the real and imaginary parts of
equation [52] we obtain
and
Eliminating
between
equation (55) and (56), we get
The
case of stationary convection:
When
the instability sets in as stationary convection, the marginal state is characterized
by. Putting in equation (55), we obtain
In
the absence of stable solute parameters and rotation parameter
equation [58] reduces to
a result
in good agreement with Gupta and Sharma [27].
RESULTS:
Equation [57] has been examined numerically using the Newton–Raphson method
through the software Fortran 90. We have plotted the variation of Rayleigh number
with respect to wave-number using equation [57] satisfying [56] for overstable case
and equation [58] for stationary case, for the fixed permissible values of the dimensionless
parameters,.Figures 1–3 correspond to three values of rotation parameter 
10,
16 and 20 rev/min, respectively. The graphs show that Rayleigh number increases
with increase in rotation parameter depicting thereby the stabilizing effect of
rotation parameter. Moreover, the rotation parameter introduces the oscillatory
modes in the system.
Fig.1: The variation of Rayleigh
number
with
wave number 
for
10
rev/min.
Fig. 2: The variation of Rayleigh
number with
wave number for
16
rev/min.
Fig.3:
The variation of Rayleigh number with
wave number for
20
rev/min.
Figures
4–6 correspond to three values of medium permeability 
5,
10 and 30. The graphs show that the Rayleigh number for the stationary convection
and for the case of overstability decreases with the increase in medium permeability
depicting thereby destabilizing effect of medium permeability.
Fig. 4: The variation of Rayleigh
number with
wave number for
Fig..5:
The variation of Rayleigh number with
wave number for
Fig.6:
The variation of Rayleigh number with
wave number for
Figure7–9 correspond to three values of micropolar coefficient 
0.5,
0.7 and 1.0, respectively, accounting for dynamic microrotation viscosity. The graphs
show that the Rayleigh number for the stationary convection and for the case of
overstability decreases with the increase in micropolar coefficient implying
thereby the destabilizing effect of dynamic microrotation viscosity.
Fig.7:
The variation of Rayleigh number
with wave number for
.
Fig.8:
The variation of Rayleigh number with
wave number for
Fig.9:
The variation of Rayleigh number with
wave number for
Figure10–12
correspond to three values of micropolar coefficient 
=1.0,
1.2 and 1.4, respectively. The graphs show that the Rayleigh number for the stationary
convection and for the case of overstability decreases with the increase in micropolar
coefficient implying
thereby the destabilizing effect of coefficient of angular viscosity, therefore
micropolar coefficients have destabilizing effects on the system.
Fig.10:
The variation of Rayleigh number with
wave numberfor
.
Fig.11:
The variation of Rayleigh number with
wave number for
Fig.12:
The variation of Rayleigh number with
wave number for
Figure 13–14 correspond to two different values of the solute parameter
i.e. 
30
and 10, respectively. It is evident from the graphs that Rayleigh number increases
with the increase in stable solute parameter even in the presence of suspended particles
(fine dust) number density depicting the stabilizing effect of solute parameter.
Fig.13:
The variation of Rayleigh numberwave
with wave number for
Fig.14:
The variation of Rayleigh number with
number for
Figures 15–17 correspond to three values of magnetic field intensity 
10,
16 and 20 rev/min, respectively. The graphs show that Rayleigh number increases
with increase in rotation parameter depicting thereby the stabilizing effect of
rotation parameter. Moreover, the rotation parameter introduces the oscillatory
modes in the system.
Fig. 15: The variation of Rayleigh numberwith
wave number for
70
Gauss.
Fig.16: The variation of Rayleigh numberwith
wave number for
100
Gauss.
Fig.
17: The variation of Rayleigh number with
wave number for
120
Gauss.
DISCUSSION
AND CONCLUISION:
There is a competition between the large enough stabilizing effect of rotation
parameter, stable solute parameter and the destabilizing effect of the micropolar
coefficients and medium porosity. The presence of coupling between thermal and micropolar
effects, rotation parameter, solute parameter, medium permeability and suspended
particles number density may bring overstability in the system. It is also noted
from the Figure [3], [4], [7] and [10] that the Rayleigh number for overstability
is always less than the Rayleigh number for stationary convection, for a fixed wave-number.
However, the reverse may also occur for large wave-numbers, as has been depicted
in Figure [1], [2], [5], [6], [8], [9], [11], [12], [15] and [16].
CONFLICT
OF INTEREST:
The
authors declare no conflict of interest.
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Received on 29.09.2020 Modified on 01.12.2020
Accepted on 04.01.2021 ©A and V Publications
All right reserved
Research Journal of Science and Technology.
2021; 13(2):57-69.
DOI: 10.52711/2349-2988.2021.00010
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and
uniform vertical magnetic field
and
coefficient of angular viscosity