Convection in Compressible Dusty Fluids
PARDEEP KUMAR
Department of Mathematics, ICDEOL, Himachal Pradesh University,
Summer-Hill, Shimla-171005 (HP) INDIA
Abstract: The aim of the present research was to study the thermosolutal convection in compressible
fluids with suspended particles in permeable media. Following the linearized stability theory,
Boussinesq approximation and normal mode analysis, it is found that that stable solute gradient
introduces oscillatory modes which were non-existent in its absence. For the case of stationary
convection, it is found that medium permeability and suspended particles have destabilizing effects
whereas the stable solute gradient has a stabilizing effect on the system. This problem was further
extended to include uniform rotation. In this case for stationary convection, the suspended particles
are found to have destabilizing effect whereas stable solute gradient, rotation and compressibility have
stabilizing effect on the system. The medium permeability has a destabilizing effect in the absence of
rotation but has both stabilizing and destabilizing effects in the presence of rotation.
Key-words: Convection, Porous Medium, Rotation, Suspended Particles
Received: August 21, 2021. Revised: April 12, 2022. Accepted: May 15, 2022. Published: July 1, 2022.
1 Introduction
The theoretical and experimental results on
thermal convection in a fluid layer, under
varying assumptions of hydrodynamics have
been discussed in a treatise by Chandrasekhar
[1]. The use of Boussinesq approximation has
been made throughout which states that the
density changes are disregarded in all other terms
in the equations of motion except the external
force term. The approximation is well justified in
the case of incompressible fluids. Chandra [2]
observed that in an air layer, convection occurred
at much lower gradients than predicted if the
layer depth was less than 7 mm and called this
motion “columnar instability”. However, for
layers deeper than 10 mm, a Be'nard-type
cellular convection was observed. Thus there is a
contradiction between the theory and experiment.
In geophysical situations, the fluid is often not
pure but contains suspended particles. The effect
of particle mass and heat capacity on the onset of
Be′nard convection has been considered by
Scanlon and Segel [3]. They found that the
critical Rayleigh number was reduced solely
because the heat capacity of the pure fluid was
supplemented by that of the particles. The effect
of suspended particles was found to destabilize
the layer i.e. to lower the critical temperature
gradient. Palaniswamy and Purushotham [4]
have considered the stability of shear flow of
stratified fluids with fine dust and have found the
effect of fine dust to increase the region of
instability. Venetis [5] has investigated the
boundary roughness of a mounted obstacle
which is inserted into an incompressible, external
and viscous flow field of a Newtonian fluid.
When the fluids are compressible, the equations
governing the system become quite complicated.
To simplify them, Boussinesq tried to justify the
approximation for compressible fluids when the
density variations arise principally from thermal
effects. Spiegel and Venonis [6] have simplified
the set of equations governing the flow of
compressible fluids under the following
assumptions:
(a) the depth of the fluid layer is much less than
the scale height, as defined by them; and
(b) the fluctuations in temperature, density, and
pressure, introduced due to motion, do not
exceed their total static variations.
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Under the above approximations, the flow
equations are the same as those for
incompressible fluids, except that the static
temperature gradient is replaced by its excess
over the adiabatic one and is replaced by .
Using these approximations, Sharma [7] has
studied the thermal instability in compressible
fluids in the presence of rotation and a magnetic
field. Hoshoudy and Kumar [8] have studied the
Rayleigh-Taylor instability of a heavy fluid
supported by a lighter one with suspended dust
particles and small uniform general rotation.
Compressibility effects on Rayleigh-Taylor
instability of two plasmas layers are investigated
by Hoshoudy et al. [9].
The investigation of double-diffusive convection
is motivated by its interesting complexities as a
double-diffusion phenomena as well as its direct
relevance to geophysics and astrophysics. The
conditions under which convective motion in
double-diffusive convection are important (e.g.
in lower parts of the Earth’s atmosphere,
astrophysics, and several geophysical situations)
are usually far removed from the consideration of
a single component fluid and rigid boundaries
and therefore it is desirable to consider a fluid
acted on by a solute gradient and free boundaries.
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 [10]. The physics is quite similar in the
stellar case in that helium acts like salt in raising
the density and in diffusing more slowly than
heat. The problem is of great importance because
of its application to atmospheric physics and
astrophysics, especially in the case of the
ionosphere and the outer layer of the atmosphere.
The thermosolutal convection problems also
arise in oceanography, limnology and
engineering. The onset of double-diffusive
reaction-convection in fluid layer with viscous
fluid, heated and salted from below subject to
chemical equilibrium on the boundaries has been
investigated by Gupta and Singh [11].
In recent years, the investigations of flow of
fluids through porous media have become an
important topic due to the recovery of crude oil
from the pores of reservoir rocks. The problem of
thermosolutal convection in fluids in a porous
medium is of importance in geophysics, soil
sciences, ground-water hydrology and
astrophysics. The development of geothermal
power resources holds increased general interest
in the study of the properties of convection in
porous media. The scientific importance of the
field has also increased because hydrothermal
circulation is the dominant heat transfer
mechanism in the development of young oceanic
crust (Lister [12]). Generally it is accepted that
comets consist of a dusty “snowball” of a
mixture of frozen gases which, in the process of
their journey, changes from solid to gas and vice-
versa. The physical properties of comets,
meteorites and interplanetary dust strongly
suggest the importance of porosity in the
astrophysical context. A mounting evidence, both
theoretical and experimental, suggests that
Darcy’s equation provides an unsatisfactory
description of the hydrodynamic conditions,
particularly near the boundaries of a porous
medium. Beavers et al. [13] have experimentally
demonstrated the existence of shear within the
porous medium near surface, where the porous
medium is exposed to a freely flowing fluid, thus
forming a zone of shear-induced flow field. The
Darcy’s equation however, cannot predict the
existence of such a boundary zone, since no
macroscopic shear term is included in this
equation (Joseph and Tao [14]). To be
mathematically compatible with the Navier-
Stokes equations and physically consistent with
the experimentally observed boundary shear zone
mentioned above, Brinkman proposed the
introduction of the term
in addition to
in the equations of fluid motion. The
elaborate statistical justification of the Brinkman
equations has been presented by Saffman [15]
and Lundgren [16]. Stommel and Fedorov [17]
and Linden [18] have remarked that the length
scales characteristics of double-diffusive
convecting layers in the ocean could be
sufficiently large for Earth’s rotation to become
important in their formation. Moreover, the
rotation of the Earth distorts the boundaries of a
hexagonal convection cell in a fluid flowing
through a porous medium, and the distortion
plays an important role in the extraction of
energy in the geothermal regions. Brakke [19]
explained a double-diffusive instability that
occurs when a solution of a slowly diffusing
protein is laid over a denser solution of more
rapidly diffusing sucrose. Nason et al. [20] found
that this instability, which is deleterious to
certain biochemical separations, can be
suppressed by rotation in the ultracentrifuge.
Sharma and Sharma [21] and Sharma and
Kumari [22] have considered the thermosolutal
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convection in porous medium under varying
assumptions of hydrodynamics and
hydromagnetics. Misra et al. [23] have studied
the numerical simulation of double-diffusive
laminar mixed convection flow in a lid-driven
porous cavity. Thermosolutal instability of
magneto-hydrodynamic flow through porous
medium has been studied by Choudhary [24].
Choudhary and Bhattacharjee [25] presents the
study of three-dimensional flow and the
injection/suction on an oscillatory flow of a
visco-elastic incompressible fluid through a
highly porous medium bounded between two
infinite horizontal porous plates. Harfash and
Alshara [26] have studied the problem of double
diffusive convective movement of a reacting
solute in a viscous incompressible occupying a
plane layer in a saturated porous medium and
subjected to a vertical magnetic field. Sriveni and
Ratnam [27] have considered the double
diffusive mixed convective heat and mass
transfer flow of a viscous fluid through a porous
medium in a rectangular duct. Coupled parallel
flow of fluid with pressure-dependent viscosity
through an inclined channel underlain by a
porous layer of a variable permeability and
variable thickness has been studied by Zaytoon
and Hamdan [28]. Kumar and Gupta [29] have
investigated the instability of the plane interface
between two viscoelastic superposed conducting
fluids in the presence of suspended particles and
variable horizontal magnetic field through porous
medium.
Keeping in mind the importance in geophysics,
astrophysics and various applications mentioned
above, the thermosolutal convection in
compressible fluids with suspended particles in a
porous medium, in the absence and presence of a
uniform rotation, separately, has been considered
in the present paper.
2 Formulation of the Problem and
Basic Equations
Here we consider an infinite horizontal,
compressible fluid-particle layer of thickness
bounded by the planes and in a
porous medium of porosity and permeability
. This layer is heated from below and
subjected to a stable solute gradient such that
steady adverse temperature gradient
and a solute concentration gradient
are maintained.
Let and
denote respectively the
density, viscosity, pressure and filter velocity of
the pure fluid:
and denote filter
velocity and number density of the particles,
respectively. If is acceleration due to gravity,
where is the particle radius,
and
, then the
equation of motion and continuity for the fluid
are
Since the distances between particles are
assumed to be quite large compared with their
diameter, the interparticle relations, buoyancy
force, Darcian force and pressure force on the
particles are ignored. Therefore the equations of
motion and continuity for the particles are
Let and denote respectively the
heat capacity of fluid at constant volume, heat
capacity of fluid at constant pressure, heat
capacity of particles, temperature, solute
concentration and “effective thermal
conductivity” of the clean water. Let
and
denote the analogous solute coefficients.
When particles and the fluid are in thermal and
solute equilibrium, the equations of heat and
solute conduction give
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where are the density and heat capacity of
the solid matrix, respectively.
Spiegel and Venonis [6] have expressed any state
variable (pressure, density or temperature), say
, in the form
where stands for the constant space
distribution of is the variation in in the
absence of motion, and stands for
the fluctuations in due to the motion of the
fluid. Following Spiegel and Veronis [6], we
have
Thus stand for the constant space
distribution of and and stand for the
temperature and density of the fluid at the lower
boundary (and in the absence of motion
Since density variations are mainly due to
variations in temperature and solute
concentration, equations (1) – (6) must be
supplemented by the equation of state
Let
and denote the
perturbations in fluid density , pressure ,
temperature , solute concentration , fluid
velocity and particle number density ,
respectively. Then the linearized perturbation
equations, under the Spiegel and Veronis [6]
assumptions, are
Here
Using
and to
denote the length, time, velocity, pressure,
temperature and solute concentration scale
factors, respectively, the linearized
dimensionless perturbation equations become
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where
and starred quantities are expressed in
dimensionless form. Hereafter, we suppress the
stars for convenience.
Eliminating
from equation (14) with the help
of equation (16) and then eliminating
from the three scalar equations of (14), and using
equation (15), we obtain
where
Analyzing the perturbations into normal modes
by seeking solutions in the form of functions of
and
where is, in general, complex, and
is the wave number of disturbance.
Eliminating between equations (20) – (22)
and using expression (23), we obtain
where
3 Some Important Theorems
Theorem I: The stable solute gradient introduces
oscillatory modes in the system while in its
absence principle of exchange of stabilities is
satisfied.
Proof: Let
In terms of , the equation satisfied by is
Consider the case of two free surfaces having
uniform temperature and solute concentration.
The boundary conditions appropriate for the
problem are
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Multiplying equation (26) by , the complex
conjugate of , integrating over the range of
and using the boundary conditions (27), we
obtain
where
The integrals are all positive definite.
Putting , where is real, into equation
(28) and equating imaginary parts, we obtain
or
In the absence of stable solute gradient,
equations (30) and (31) become
or
Since the integrals are positive definite and is
real. It follows that and the principle of
exchange of stabilities is satisfied, in the
absence of stable solute gradient. In the
presence of stable solute gradient, the
principle of exchange of stabilities is not
satisfied and oscillatory modes come into play.
The stable solute gradient, thus, introduces
oscillatory modes which were non-existent in
its absence.
Theorem II: For the case of stationary
convection, the medium permeability and
suspended particles have destabilizing effects,
whereas the stable solute gradient has a
stabilizing effect on the system.
Proof: When instability sets in as stationary
convection, the marginal state will be
characterized by and equation (24)
reduces to
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Considering the case of two free boundaries, it
can be shown that all the even order derivatives
of vanish on the boundaries and hence the
proper solution of equation (34) characterizing
the lowest mode is
where is a constant. Substituting the solution
(35) in equation (34), we obtain
If denotes the critical Rayleigh number in the
absence of compressibility and stands for the
critical Rayleigh number in the presence of
compressibility, then we find that
Since critical Rayleigh number is positive and
finite, so and we obtain a stabilizing effect
of compressibility as its result is to postpone the
onset of double-diffusive convection in a fluid-
particle layer of porous medium.
It is evident from equation (36) that
and
The medium permeability and suspended
particles have thus destabilizing effects,
whereas the stable solute gradient has a
stabilizing effect on the thermosolutal
convection in compressible fluids with
suspended particles in a porous medium.
4 Effect of Rotation
Formulation of the Problem: In this section, we
consider the same problem as that studied above
except that the system is in a state of uniform
rotation
. The Coriolis force acting on
the particles is also neglected under the
assumptions made in the problem. The linearized
non-dimensional perturbation equations of
motion for the fluid are
where
is the non-dimensional number
accounting for rotation, and equations (14) – (19)
remain unaltered.
Eliminating
with the help of (16) and
then eliminating between equations (40)
– (42), we obtain
Eliminating and between equations (21), (22)
and (43) and using expression (23), we get
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Theorem III: For the case of stationary
convection, the suspended particles have a
destabilizing effect, whereas the rotation and
stable solute gradient have stabilizing effects on
the system under consideration. The medium
permeability has both stabilizing and
destabilizing effects, depending on the rotation
parameter.
Proof: For the stationary convection and
equation (44) reduces to
Considering again the case of two free
boundaries with constant temperature and solute
concentration and using the proper solution (35),
we obtain from equation (45)
It is evident from equation (46) that
Therefore the suspended particles have a
destabilizing effect, whereas the rotation and
stable solute gradient have stabilizing effects
on the system under consideration.
Equation (46) also yields
If
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then
is positive.
If
then
is negative.
Thus the medium permeability has both
stabilizing and destabilizing effects, depending
on the rotation parameter, whereas in the
absence of rotation as concluded earlier from
equation (36) that medium permeability always
has destabilizing effect.
Acknowledgements
The authors are grateful to all the three learned
referees for their useful technical comments and
valuable suggestions, which led to a significant
improvement of the paper.
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