On Instability of a Dusty Stellar Atmosphere in Stern’s Type
Configuration
PARDEEP KUMAR
Department of Mathematics,
ICDEOL,
Himachal Pradesh University,
Summerhill, Shimla-171005,
INDIA
Abstract: - The thermal-convective instability of a stellar atmosphere in the presence of a stable solute gradient in
Stern’s type configuration is studied in the presence of suspended particles. The criteria for monotonic instability
are derived which are found to hold well in the presence of uniform rotation and uniform magnetic field,
separately, on the thermosolutal-convective instability of a stellar atmosphere in the presence of suspended
particles.
Key-Words: - Convection, suspended particles, solute gradient, uniform rotation, uniform magnetic field
Received: August 22, 2022. Revised: February 17, 2023. Accepted: March 16, 2023. Published: April 5, 2023.
1 Introduction
Defouw, [1], has termed ‘thermal-convective
instability’ as the instability in which motions are
driven by buoyancy forces of a thermally unstable
atmosphere. He has generalized the Schwarzschild
criterion for convection to include departures from
adiabatic motion and has shown that a thermally
unstable atmosphere is also convectively unstable,
irrespective of the atmospheric temperature gradient.
[1], has shown that an inviscid stellar atmosphere is
unstable if
(1)
where is the energy-lost function (the energy lost
minus the energy gained per gram per second) and
denote, respectively, the
density, the coefficient of thermal expansion, the
coefficient of thermometric conductivity, the wave
number of the perturbation, the specific heat at
constant pressure, the partial derivative of with
respect to temperature and the partial derivative of
concerning density both evaluated in the
equilibrium state. In general, the instability due to
inequality (1) may be either oscillatory or monotonic.
[1], has also shown that inequality (1) is a sufficient
condition for monotonic instability in the presence of
a magnetic field and rotation on thermal convective
instability.
A detailed account of thermal convection, under
varying assumptions of hydrodynamics and
hydromagnetics, has been given by [2]. [3], has
considered the problem of thermohaline convection
in a layer of fluid heated from below and subjected to
a stable salinity gradient. In the stellar case, the
physics is quite similar to [3], thermohaline
configuration in that helium acts like salt in raising
the density and in diffusing more slowly than heat.
The thermohaline convection in a horizontal layer of
viscous fluid heated from below and salted from
above has been studied by [4]. In the thermohaline-
convective instability problem, buoyancy forces can
arise not only from density differences due to
temperature variations but also from those due to
variations in solute concentrations. The conditions
under which convective motions are important in
stellar atmospheres are usually far removed from the
consideration of a single component fluid and rigid
boundaries and, therefore, it is desirable to consider
one gas component acted on by solute concentration
gradient and free boundaries. Keeping such situations
in mind, [5], have considered the thermosolutal-
convective instability in a stellar atmosphere and
have also studied the effects of uniform rotation and
International Journal on Applied Physics and Engineering
DOI: 10.37394/232030.2023.2.2
Pardeep Kumar
E-ISSN: 2945-0489
7
Volume 2, 2023
variable/uniform magnetic field on the instability.
The criteria for monotonic instability are derived and
are found to hold well also in the presence of the
above effects. The onset of double-diffusive reaction-
convection in a fluid layer with viscous fluid, heated
and salted from below subject to chemical
equilibrium on the boundaries has been investigated
by [6]. [7], have studied the magnetohydrodynamic
Veronis’ thermohaline convection. The thermal-
convective instability of a stellar atmosphere in the
presence of a stable solute gradient in Stern’s type
configuration has been studied in the presence of
radiative transfer effect by [8]. [9], has studied the
thermal-convective instability of a composite rotating
stellar atmosphere in the presence of a variable
horizontal magnetic field to include, separately, the
effects of medium permeability and solute gradient.
In geophysical situations, more often than not,
the fluid is not pure but may instead be permeated
with suspended (or dust) particles. The suspended
particles are present in the stellar atmospheres and
many astro-physical situations. Recent spacecraft
observations have confirmed that dust particles play
an important role in the dynamics of the Martian
atmosphere as well as in the diurnal and surface
variations in the temperature of the Martian weather,
[10]. [11], have considered the effect of suspended
particles on the onset of Benard convection and
found that the critical Rayleigh number was reduced
solely because the heat capacity of the pure gas was
supplemented by that of the particles. The effect of
suspended particles was found to destabilize the layer
whereas the effect of a magnetic field was stabilizing.
[12], have studied the stability of the shear flow of
stratified fluids with fine dust and found the effect of
fine dust to increase the region of instability. [13],
have studied the onset of double-diffusive convection
in a horizontal Brinkman cavity and analysis made on
the linear stability of the quiescent state within a
horizontal porous cavity subject to vertical gradients
of temperature and solute. [14], has investigated the
boundary roughness of a mounted obstacle that is
inserted into an incompressible, external, and viscous
flow field of a Newtonian fluid. [15], have
investigated the instability of the plane interface
between two viscoelastic superposed conducting
fluids in the presence of suspended particles and
variable horizontal magnetic fields through a porous
medium. Coupled parallel flow of fluid with
pressure-dependent viscosity through an inclined
channel underlain by a porous layer of variable
permeability and variable thickness has been studied,
[16]. It is, therefore, of interest to study the presence
of suspended particles in astrophysical situations.
It is, therefore, the motivation of the present
study to re-examine the thermosolutal-convective
instability in Stern’s type configuration of a stellar
atmosphere (the thermal-convective instability in the
presence of stable solute gradient) in the presence of
suspended particles and to seek the modification, if
any, in the criteria for instability. The Coriolis forces
and magnetic field play important roles in
astrophysical situations. The effects of rotation and
magnetic field (separately) are, therefore, also studied
on the thermal-convective instability of a stellar
atmosphere in the presence of a stable solute gradient
in Stern’s type configuration in the presence of
suspended particles. These aspects form the subject
matter of the present paper.
2 Descriptions of the Instability and
Perturbation Equations
Consider an infinite horizontal fluid layer of
thickness heated from above and subjected to a
stable solute concentration gradient so that the
temperatures and solute concentrations at the bottom
surface are and and at the upper surface
are and respectively, the -axis being
taken as vertical. This layer is acted on by a gravity
force . Let
de
note, respectively, the density, pressure, temperature,
solute concentration, fluid velocity, particle velocity,
gravitational acceleration, thermal coefficient of
expansion, an analogous solvent coefficient of
expansion, kinematic viscosity, thermal diffusivity,
and solute diffusivity. The suffix zero refers to values
at the reference level . Let stands for
the number density of the particles ,
where is the particle radius, denotes the Stokes’
drag, and
. Then, following
the Boussinesq approximation, which states that the
inertial effects produced by density variation are
negligible in comparison to its gravitational effects
i.e. ρ can be taken as constant everywhere in the
equations of motion except in the term with external
force, the equations expressing the conservation of
International Journal on Applied Physics and Engineering
DOI: 10.37394/232030.2023.2.2
Pardeep Kumar
E-ISSN: 2945-0489
8
Volume 2, 2023
momentum, mass, solute mass concentration, and
equation of state are
In the equations of motion for the gas, the
presence of particles adds an extra force term,
proportional to the velocity difference between
particles and gas. Since the force exerted by the gas
on the particles is equal and opposite to that exerted
by the particles on the gas, there must be an extra
force term equal in magnitude but opposite in sign in
the equations of motion for the particles. Interparticle
reactions are not considered for we assume that the
distances between particles are quite large compared
with their diameter. Because of the small size and
large distances between particles, the effects of
gravity, pressure, etc., are negligible. If is the
mass of particles per unit volume, then the equations
of motion and continuity for the particles, under the
above assumptions, are
Let and denote, respectively, the
gas-specific heat at constant volume, particles-
specific heat, the temperature, and the ‘effective’
thermal conductivity, which is the conductivity of the
pure gas. Since the volume fraction of the particles is
assumed extremely small, the effective properties of
the suspension are taken to be those of the clean gas.
If we assume that the particles and the gas are in
thermal equilibrium, the first law of thermodynamics
may be written in the form
The steady-state solution is
where
are the magnitudes of uniform temperature and
concentration gradients.
We now consider a small perturbation on the
steady state solution and let
and N
denote, respectively, the perturbations in density,
pressure, temperature, solute concentration , the
velocity of the gas, the velocity of particles, and the
number density of the particles .
Then equations (2) - (4) and (6)-(7) on linearization
give
where
and
In writing
equation (10), use has been made of the equation of
state (5) wherefrom the change in density , caused
by the perturbations and in temperature and
concentration is given by
International Journal on Applied Physics and Engineering
DOI: 10.37394/232030.2023.2.2
Pardeep Kumar
E-ISSN: 2945-0489
9
Volume 2, 2023
Following, [1], the linearized perturbation form of
equation (8) is
where
and is the
gas-specific heat at constant pressure.
We now consider the case in which both
boundaries are free as well as perfect conductors of
both heat and solute concentration. The density
changes arise principally from thermal effects. The
case of two free boundaries is the most appropriate
for stellar atmospheres as pointed out by [17]. The
boundary conditions appropriate for the problem are
3 The Dispersion Relation
We shall now analyze an arbitrary perturbation into a
complete set of normal modes and by seeking
solutions whose dependence on space and time
coordinates is of the form
where is the growth rate,
being
any integer and is the thickness of the layer) and
is the wave number of the
perturbation.
Eliminating
and from equations (10) – (12), we
obtain
Now, eliminating from equations (19), (14), and
(16), and using (18), we obtain the dispersion relation
where
4 Discussion and Further Extensions
Theorem 1: A criterion that the thermosolutal-
convective instability of a stellar atmosphere in the
presence of suspended (or dust) particles is unstable
if
Proof: Taking the dispersion relation (20), when
the constant term in relation (20) is negative.
Equation (20), therefore, involves one change of sign
and hence contains one positive real root. The
occurrence of a positive root implies monotonic
instability.
We thus obtain a criterion that the thermosolutal-
convective instability of a stellar atmosphere in the
International Journal on Applied Physics and Engineering
DOI: 10.37394/232030.2023.2.2
Pardeep Kumar
E-ISSN: 2945-0489
10
Volume 2, 2023
presence of suspended (or dust) particles is unstable
if
Hence the result.
Further Extension-1: Here we consider the same
problem as described above except that the system is
in a state of uniform rotation. Since the volume
fraction of the particles is assumed extremely small
and the particles are assumed to be far apart from one
another, the Coriolis force on the particles is also
negligible. On the right-hand side of the equation of
motion (10) of the gas, the Coriolis force term
is added and equations (11) – (16)
remain unaltered.
Theorem 2: The criterion for monotonic instability
(23) also holds well in the presence of rotation and
suspended particles on thermosolutal-convective
instability in Stern’s type configuration in a stellar
atmosphere.
Proof: Here we consider an infinite horizontal gas-
particle layer of thickness heated from above, solute
concentrated from below and acted on by a uniform
rotation
and gravity force . The
linearized perturbation equations, for the problem
under consideration, then become
together with equations (11), (14), and (16).
Eliminating from equations (24) – (26) and using
(11), we obtain
where
denotes the z-component of vorticity.
Equations (24) and (25) yield
Eliminating from equations (27), (28),
(14), (16) and using expression (18), we obtain the
dispersion relation
International Journal on Applied Physics and Engineering
DOI: 10.37394/232030.2023.2.2
Pardeep Kumar
E-ISSN: 2945-0489
11
Volume 2, 2023
When (23) is satisfied, the constant term in equation
(29) is negative. The product of the roots must be
negative. Therefore at least one root of equation (29)
is positive and one root is negative. The occurrence
of a positive root implies monotonic instability. The
criterion for monotonic instability (23) thus holds
well in the presence of rotation and suspended
particles on thermosolutal-convective instability in a
stellar atmosphere.
Further Extension-2: Further we consider an
infinite horizontal viscous and finitely conducting
gas-particle layer subjected to a stable solute
concentration gradient and acted on by a uniform
vertical magnetic field
and gravity force
This layer is heated from above such
that a steady temperature gradient
is
maintained.
Now we prove the following Theorem:
Theorem 3: The criterion for monotonic instability
(23) derived for thermosolutal-convective instability
of a stellar atmosphere in the presence of suspended
particles also holds good in the presence of uniform
magnetic field and suspended particles on
thermosolutal-convective instability in Stern’s type
configuration in a stellar atmosphere.
Proof: Let
denote the perturbation in
a magnetic field
. Then the linearized perturbation
equations appropriate to the problem are
where
( being electrical conductivity) is
the electrical resistivity. Equations (11), (12), (14),
and (16) remain unchanged.
From equations (11), (12), and (30)-(32), we obtain
International Journal on Applied Physics and Engineering
DOI: 10.37394/232030.2023.2.2
Pardeep Kumar
E-ISSN: 2945-0489
12
Volume 2, 2023
Eliminating and from equations (14), (16),
(33), (34) and using expression (18), we obtain the
dispersion relation
where
where
denotes the square of the Alfv’en
velocity.
If the criteria (23) are satisfied i.e. when
the constant term in equation (36) is negative.
Equation (36), therefore, involves one change of sign
and hence contains one positive real root. The
occurrence of a positive root implies monotonic
instability.
The criteria for monotonic instability derived for
thermosolutal-convective instability of a stellar
atmosphere in the presence of suspended particles
are, thus, also found to hold good in the presence,
separately, of uniform rotation and uniform magnetic
field on the problem under consideration.
Acknowledgments:
The author is grateful to all three learned referees for
their useful technical comments and valuable
suggestions, which led to a significant improvement
of the paper.
References:
[1] R.J. Defouw, Thermal convective instability,
Astrophys. J., Vol. 160, 1970, pp. 659-669.
[2] S. Chandrasekhar, Hydrodynamic and
Hydromagnetic Stability, Oxford University
Press, London, 1961.
[3] G. Veronis, On the finite amplitude instability
in thermohaline convection, J. Marine Res.,
Vol. 23, 1965, pp. 1-17.
[4] D.A. Nield, The thermohaline convection with
linear gradients, J. Fluid Mech., Vol. 29, 1967,
pp. 545-558.
International Journal on Applied Physics and Engineering
DOI: 10.37394/232030.2023.2.2
Pardeep Kumar
E-ISSN: 2945-0489
13
Volume 2, 2023
[5] R.C. Sharma and J.N. Misra, On thermosolutal
convective instability in a stellar atmosphere I.
Effect of rotation and magnetic field,
Astrophys. Space Sci., Vol. 126, 1986, pp. 21-
28.
[6] V.K. Gupta and A.K. Singh, Double diffusive
reaction-convection in viscous fluid layer, Int.
J. Industrial Maths., Vol. 6(4), 2014, pp. 285-
296.
[7] H.S. Jamwal and G.C. Rana, On
magnetohydrodynamic Veronis’s thermohaline
convection, Int. J. Engng. Appl. Sciences, Vol.
6(4), 2014, pp. 1-9.
[8] P. Kumar and H. Mohan, Thermosolutal-
convective instability in Stern’s type
configuration, WSEAS Trans. on Systems, Vol.
20, 2021, pp. 149-156.
[9] P. Kumar, On instability of a composite
rotating stellar atmosphere, Earth Sciences and
Human Constructions, Vol. 2, 2022, pp. 140-
146.
[10] J.B. Pollack, Mars, Sci. Amer., Vol. 233, 1975,
pp. 106-117.
[11] J.W. Scanlon and L.A. Segel, Some effects of
suspended particles on the onset of Be’nard
convection, Phys. Fluids, Vol. 16, 1973, pp.
1573-1578.
[12] V.I. Palaniswamy and C.M. Purushotham,
Stability of shear flow of stratified fluids with
fine dust, Phys. Fluids, Vol. 24, 1981, pp.
1224-1229.
[13] Z. Alloui, P. Vasseur, L. Robillard and A.
Bahloul, Onset of double-diffusive convection
in a horizontal Brinkman cavity, Che. Eng.
Comm., Vol. 197, 2010, pp. 387-399.
[14] J. Venetis, An analytical simulation of
boundary roughness for incompressible viscous
flows, WSEAS Trans. on Appl. Theoret. Mech.,
Vol. 16, 2021, pp. 9-15.
[15] P. Kumar and S. Gupta, Hydromagnetic
instability of two viscoelastic dusty-fluids in
porous medium, WSEAS Trans. on Heat and
Mass Transfer, Vol. 16, 2021, pp. 137-144.
[16] M.S. Abu Zaytoon and M.H. Hamdan, Fluid
mechanics of the interface between a variable
viscosity fluid layer and a variable permeability
porous medium, WSEAS Trans. on Heat and
Mass Transfer, Vol.16, 2021, pp. 159-169.
[17] E.A. Spiegel, Convective instability in a
compressible atmosphere, Astrophys. J., Vol.
141, 1965, pp. 1068-1070.
Contribution of Individual Authors to the
Creation of a Scientific Article (Ghostwriting
Policy)
The author contributed in the present research, at all
stages from the formulation of the problem to the
final findings and solution.
Sources of Funding for Research Presented in a
Scientific Article or Scientific Article Itself
No funding was received for conducting this study.
Conflict of Interest
The author has no conflict of interest to declare that
is relevant to the content of this article.
Creative Commons Attribution License 4.0
(Attribution 4.0 International, CC BY 4.0)
This article is published under the terms of the
Creative Commons Attribution License 4.0
https://creativecommons.org/licenses/by/4.0/deed.en_
US
International Journal on Applied Physics and Engineering
DOI: 10.37394/232030.2023.2.2
Pardeep Kumar
E-ISSN: 2945-0489
14
Volume 2, 2023
