On Instability of a Composite Rotating Stellar Atmosphere
PARDEEP KUMAR
Department of Mathematics, ICDEOL, Himachal Pradesh University,
Summer-Hill, Shimla-171005 (HP) INDIA
Abstract: The aim of the present work was to study 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. Following the linearized stability theory,
Boussinesq approximation and normal mode analysis, the dispersion is obtained in each case. The
criteria for monotonic instability in each case have been obtained which generalize the Defouw’s
criterion derived for thermal-convective instability in the absence of above effects.
Key-words: Convection, Medium Permeability, Rotation, Solute Gradient, Variable Magnetic Field
Received: June 25, 2021. Revised: March 19, 2022. Accepted: April 15, 2022. Published: June 2, 2022.
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 form adiabatic motion and
has shown that a thermally unstable atmosphere
is also convectively unstable, irrespective of the
atmospheric temperature gradient.
Defouw [1] has found that an inviscid stellar
atmosphere becomes unstable if
where is the heat-loss 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
specific heat at constant pressure, the wave
number of perturbation, the partial derivatives of
with respect to ; both evaluated in the
equilibrium state. In general, the instability due
to inequality (1) may be either oscillatory or
monotonic. The effects of a uniform rotation and
a uniform magnetic field on thermal-convective
instability of a stellar atmosphere have been
studied, separately by Defouw [1] and
simultaneously by Bhatia [2].
Quite frequently it happens that the plasma is not
fully ionized but, instead, may be permeated
with neutral atoms. Stromgren [3] has reported
that ionized hydrogen is limited to certain rather
sharply bounded regions in space surroundings,
for example, O-type stars and clusters of such
stars, and that the gas outside these regions is
essentially non-ionized. As a reasonably simple
approximation, the plasma may be idealized as a
composite mixture of a hydromagnetic (ionized)
component and a neutral component, the two
interacting through mutual collisional
(frictional) effects. Hans [4] made this
simplified approximation and found that these
collisions have a stabilizing effect on the
Rayleigh-Taylor instability. The thermal
hydromagnetic instability of a partially-ionized
plasma, for incompressible and compressible
cases, has been studied by Sharma [5] and
Sharma and Misra [6]. Usually the magnetic
field has a stabilizing effect on the instability.
However, Kent [7] has studied the effect of a
horizontal magnetic field which 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 stable.
The thermal and convective heat transfer in flat
solar collectors have been studied by
Amirgaliyev et al. [8]. In stellar interiors and
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Pardeep Kumar
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atmospheres, the magnetic field may be variable
and may altogether alter the nature of the
instability. The Coriolis force also plays an
important role on the stability of stellar
atmospheres.
A detailed account of thermal convection, under
varying assumptions of hydrodynamics and
hydromagnetics, has been given by
Chandrasekhar [9]. Veronis [10] 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 Veronis [10]
thermohaline configuration in that helium acts
like salt in raising the density and in diffusing
more slowly than heat. In thermosolutal-
convective instability problem, buoyancy forces
can arise not only from density differences due
to variations in temperature 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. Marcu and Ballai
[11] have studied the thermosolutal linear
stability of a composite two-component plasma
in the presence of Coriolis forces, finite Larmor
radius, taking into account the collisions
between neutral and ionized particles. The
thermosolutal instability appears due to a
material convection (thermosolutal convection)
in a two component fluid with different
molecular diffusivities which contribute in an
opposing sense to a locally vertical density
gradient. Jamwal and Rana [12] have studied the
magnetohydrodynamic Veronis’s thermohaline
convection.
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 study of
the onset of convection in a porous medium has
attracted considerable interest because if its
natural occurrence and of its intrinsic
importance in many industrial problems,
particularly in petroleum exploration, chemical,
and nuclear industries. The derivation of the
basic equations of a layer of fluid heated from
below in porous medium, using the Boussinesq
approximation, has been given by Joseph [13].
The study of a layer of fluid heated from below
in porous media is motivated both theoretically
and by its practical applications in engineering
disciplines. Among the applications in
engineering disciplines one can find the food
process industry, chemical process industry,
solidification and centrifugal casting of metals.
The development of geothermal power
resources has increased general interest in the
properties of convection in porous medium.
When a fluid permeates an isotropic and
homogeneous porous medium, the gross effect
is represented by Darcy’s law. A great number
of applications in geophysics may be found in
the books by Phillips [14], Ingham and Pop [15],
and Nield and Bejan [16]. 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 importance of porosity in
astrophysical context (see McDonnel [17]).
Purkayastha and Choudhury [18] have studied
the Hall current and thermal radiation effect on
MHD convection flow of an elastico-viscous
fluid in a rotating porous channel. The porosity
is important in several geophysical situations.
Effects of permeability on double diffusive
MHD mixed convective flow past an inclined
porous plate have been studied by Uddin et al.
[19]. Kumar and Singh [20] have considered the
thermal convection of a plasma in porous
medium to include simultaneously the effect of
rotation and the finiteness of the ion Larmor
radius (FLR) in the presence of a vertical
magnetic field.
Keeping such astrophysical and geophysical
situations in mind, thermal-convective
instability of a composite rotating stellar
atmosphere in the presence of a variable
horizontal magnetic field is considered in the
present paper to include, separately, the effects
of medium permeability and solute gradient.
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2 Presence of Porous Medium
2A: Formulation of the Problem and
Perturbation Equations
Consider an infinite horizontal composite layer
consisting of a finitely conducting
hydromagnetic fluid of density and a neutral
gas of density , which is in a state of uniform
rotation
, acted on by a variable
horizontal magnetic field
and
gravity force through a porous
medium of porosity and medium permeability
. This layer is heated such that a steady
temperature gradient
is
maintained. Regard the model under
consideration so that both the ionized fluid and
the neutral gas behave like continuum fluids and
that the effects on the neutral component
resulting from the presence of gravity and
pressure are neglected. The magnetic field
interacts with the ionized component only.
Let and
denote the perturbations in density , pressure ,
filter velocity, and magnetic field
,
respectively;
, and denote,
respectively, the gravitational acceleration, the
kinematic viscosity, the resistivity, the velocity
of neutral gas, and the collisional frequency
between the two components of the composite
medium. Then the linearized perturbation
equations governing the motion of the mixture
of the hydromagnetic fluid and a neutral gas are
The first law of thermodynamics can be written
as
where and denote, respectively, the
thermal conductivity, the specific heat at
constant volume, the temperature, and the time.
is the convective derivative.
Following Defouw [1], the linearized
perturbation form of equation (7) is
where is the perturbation in temperature . We
have employed the Boussinesq approximation
modified modified so as to apply to thin layers
of compressible fluids (cf. Spiegel and Veronis
[21]) and used the Boussinesq equation of state
where we consider the case in which both
boundaries are free and the medium adjoining
the fluid is non-conducting. The case of two free
boundaries is the most appropriate for stellar
atmospheres (Spiegel [22]).
The boundary conditions for the problem are (cf.
Chandrasekhar [9]; Lapwood [23]):
and are continuous with an external
vacuum field, where and denote the z-
components of vorticity and current density,
respectively.
2B: The Dispersion Relation
Analyzing in terms of normal modes, we
seek solutions whose dependence on space- and
time-coordinates is of the form
where is the growth rate and
being any integer and is the thickness of the
layer and
is the wave
number of the perturbation.
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If we eliminate
between equations (2) and
(6), Equations (8) and (2)-(6), using expression
(10), we find that
where
If we eliminate and from
equations (11) – (14) and using (10), we obtain
the dispersion relation
where
and
to having a large number of terms and
being not needed in the discussion on instability,
have not been written here.
2C: Discussion
Theorem 1: A criteria for thermal-convective
instability of a composite stellar atmosphere in
the presence of a variable horizontal magnetic
field, rotation and collisional effects through
porous medium to be unstable if
Proof: Taking the dispersion relation (15), when
the constant term in equation (15) is negative.
Equation (15), therefore, involves at least one
change of sign and, hence, contains one positive
real root. The occurrence of positive root implies
monotonic instability. We thus obtain the
criteria for thermal-convective instability of a
composite stellar atmosphere in the presence of
a variable horizontal magnetic field, rotation and
collisional effects through porous medium to be
unstable if
Hence the result.
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3 Presence of Solute Gradient
3A: Formulation of the Problem and
Dispersion Relation
Here we consider the same configuration as in
previous section except that the medium is non-
porous and that the system is acted on by a stable
solute gradient
The linearized
perturbation equations governing the motion of
the mixture of the hydromagnetic fluid and a
neutral gas are
together with the Boussinesq equation of state
Here again we consider the case of two free
boundaries and the medium adjoining the fluid
to be non-conducting. Eliminating
between
equations (18) and (22), equations (18) – (23),
using expression (10), give
If we eliminate and from
equations (24) – (28) and using (10), we obtain
the dispersion relation
where
and
Coefficients to having a large number of
terms and being not needed in the discussion on
stability, have not been written here.
3B: Discussion
Theorem 2: A criteria for thermosolutal-
convective instability of a composite stellar
atmosphere in the presence of rotation, variable
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horizontal magnetic field, stable solute gradient
and collisional effects to be unstable if
Proof: Taking the dispersion relation (29), when
the constant term in equation (29) is negative.
Equation (29), therefore, involves one change of
sign and, hence, contains one positive real root,
meaning thereby monotonic instability. We
have, therefore, obtained the criteria for
thermosolutal-convective instability of a
composite stellar atmosphere in the presence of
rotation, variable horizontal magnetic field,
stable solute gradient and collisional effects to
be unstable if
Hence the result.
4 Conclusions
The convective stability of a star has customarily
been determined by Schwarzschild criterion and
one of the fundamental assumptions used in
deriving this criterion is that the motion is
adiabatic. The Schwarzschild criterion in the
interior of a star, where the photon mean free
path is small, the assumption that the motion is
adiabatic is justified. The departure from
adiabatic motion may be significant in the outer
layers of a stellar atmosphere, where the
effective heat transfer is no longer prevented by
opacity. The Schwarzschild criterion for
convection has been generalized to include
departures from adiabatic motion by Defouw
[1].
The stellar chromospheres, coronae, and the
interstellar medium may exhibit thermal-
convective instability. For such astrophysical
situations the Coriolis force, the variable
magnetic field, medium permeability, solute
gradient, and collisional effects, play an
important role. The thermal-convective
instability of a composite rotating stellar
atmosphere in the presence of a variable
horizontal magnetic field is considered to
include, separately, the effects of medium
permeability and solute gradient. The criteria for
monotonic instability in each case have been
obtained which generalize the Defouw’s
criterion derived for thermal-convective
instability in the absence of above mentioned
effects.
References
[1] Defouw, R.J., Thermal convective
instability, Astrophys. J., Vol. 160, 1970, pp.
659-669.
[2] Bhatia, P.K., On thermal-convective
instability in a stellar atmosphere, Publ. Astron.
Soc. Japan, Vol. 23, 1971, pp. 181-184.
[3] Stromgren, B., The physical state of
interstellar hydrogen, Astrophys. J., Vol. 89,
1939, pp. 526-546.
[4] Hans, H.K., Larmor radius and collisional
effects on the combined Taylor and Kelvin
instabilities in a composite medium, Nucl.
Fusion, Vol. 8, 1968, pp. 89-92.
[5] Sharma, R.C., Thermal hydromagnetic
instability of a partially ionized medium,
Physica, Vol. 81C, 1976, pp. 199-204.
[6] Sharma, R.C. and Misra, J.N., Thermal
instability of a compressible and partially
ionized plasma, Z. Naturforsch., Vol. 41a, 1986,
pp. 729-732.
[7] Kent, A., Instability of laminar flow of a
perfect magneto fluid, Phys. Fluids, Vol. 9,
1966, pp. 1286-1289.
[8] Amirgaliyev, Y., Kunelbayev, M.,
Kalizhanova, A., Amirgaliyev, B., Kozbakova,
A., Auelbekov, O. and Kataev, N., The study of
thermal and convective heat transfer in flat solar
collectors, WSEAS Transactions on Heat and
Mass Transfers, Vol. 15, 2020, pp. 55-63.
[9] Chandrasekhar, S., Hydrodynamic and
Hydromagnetic Stability, Dover Publication,
New York 1981.
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Volume 2, 2022
[10] Veronis, G., On the finite amplitude
instability in thermohaline convection, J.
Marine Res., Vol. 23, 1965, pp. 1-17.
[11] Marcu, A. and Ballai, I., Thermosolutal
stability of a two-component rotating plasma
with finite Larmor radius, Proc. Rom. Acad.,
Series A, Vol. 8(2), 2007, pp. 111-120.
[12] Jamwal, H.S. and Rana, G.C., On
magnetohydrodynamic Veronis’s thermohaline
convection, Int. J. Engng. Appl. Sciences, Vol.
6(4), 2014, pp. 1-9.
[13] Joseph, D.D., Stability of Fluid Motions,
Springer-Verlag Berlin, Vol. I and II, 1976.
[14] Phillips, O.M., Flow and Reaction in
Permeable Rocks, Cambridge University Press,
Cambridge, UK 1991.
[15] Ingham, D.B. and Pop, I., Transport
Phenomena in Porous Medium, Pergamon
Press, Oxford, UK 1998.
[16] Nield, D.A. and Bejan, A., Convection in
Porous Medium (2nd edition), Springer New
York 1999.
[17] McDonnel, J.A.M., Cosmic Dust, John
Wiley and Sons, Toronto, p. 330, 1978.
[18] Purkayastha, S. and Choudhury, R., Hall
current and thermal radiation effect on MHD
convective flow of an elastico-viscous fluid in a
rotating porous channel, WSEAS Transactions
on Applied and Theoretical Mechanics, Vol. 9,
2014, pp. 196-205.
[19] Uddin, Md. N., Chowdhury, M.M.K. and
Alim, M.A., Effects of permeability on double
diffusive MHD mixed convective flow past an
inclined porous plate, Int. J. Engng. Appl.
Sciences, Vol. 6(3), 2014, pp. 12-20.
[20] Kumar, P. and Singh, G.J., Convection of a
rotating plasma in porous medium, WSEAS
Transactions on Heat and Mass Transfers, Vol.
16, 2021, pp. 68-78.
[21] Spiegel, E.A. and Veronis, G., On the
Boussinesq approximation for a compressible
fluid, Astrophys. J., Vol. 131,1960, pp. 442-446.
[22] Spiegel, E.A., Convective instability in a
compressible atmosphere, Astrophys. J., Vol.
141, 1965, pp. 1068-1070.
[23] Lapwood, E.R., Convection of a fluid in a
porous medium, Proc. Camb. Phil. Soc., Vol.
44, 1948, pp. 508-554
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