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Global Journal of Pure and Applied Mathematics.
ISSN 0973-1768 Volume 13, Number 6 (2017), pp. 1777-1798
© Research India Publications
http://www.ripublication.com
Mathematical Analysis for Optically Thin Radiating/
Chemically Reacting Fluid in a Darcian Porous Regime
Nava Jyoti Hazarika1 and Sahin Ahmed2
1Department of Mathematics, Tyagbir Hem Baruah College,
Jamugurihat, Sonitpur-784189, Assam, India.
2Department of Mathematics, Rajiv Gandhi University, Rono Hills,
Itanagar, Arunachal Pradesh-791112, India.
Abstract
In this paper, we analyzed an unsteady MHD flow of
two-dimensional,
laminar, incompressible, Newtonian, electrically-conducting and
radiating
fluid along a semi-infinite vertical permeable moving plate with
periodic heat
and mass transfer by taking into account the effect of viscous
dissipation in
presence of chemical reaction. A uniform magnetic field is
applied
transversely to the porous plate. The plate moves with a
constant velocity in
the direction of the fluid flow while the free stream velocity
follows an
exponentially increasing small perturbation law subject to a
constant suction
velocity to the plate. The dimensionless governing equations for
this
investigation are solved analytically using two-term harmonic
and non-
harmonic functions. Numerical evaluation of the analytical
results are
performed and graphical results for velocity, temperature and
concentration
profiles within the boundary layer and the tabulated results for
the Skin-
friction co-efficient, Nusselt number and Sherwood number are
presented and
discussed. It is seen that, an increase in chemical reaction
parameter leads to
decrease both fluid velocity as well as concentration. Moreover,
the skin-
friction has been depressed by the influence of chemical
reaction parameter,
where as the rate of heat transfer is escalated. The present
model has several
important applications such as dispersion of chemicals
contaminants,
superconvecting geothermics, geothermal energy extractions and
plasma
physics.
Keywords: Thin gray gas; Dispersion of chemicals contaminants;
Viscous
dissipation; MHD; Darcian regime; skin-friction.
Corresponding author: Sahin Ahmed
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1778 Nava Jyoti Hazarika and Sahin Ahmed
1. INTRODUCTION
The study of heat and mass transfer to chemical reacting MHD
free convection flow
with radiation effects on a vertical plate has received a
growing interest during the last
decades. Accurate knowledge of the overall convection heat
transfer has vital
importance in several fields such as thermal insulation, dying
of porous solid
materials, heat exchangers, stream pipes, water heaters,
refrigerators, electrical
conductors and industrial, geophysical and astrophysical
applications such as polymer
production, manufacturing of ceramic, packed-bed catalytic
reactor, food processing,
cooling of nuclear reactor, enhanced oil recovery, underground
energy transport,
magnetized plasma flow, high speed plasma wind, cosmic jets and
stellar system. For
some industrial application such as glass production, furnace
design, propulsion
systems, plasma physics and spacecraft re-entry
aerothermodynamics which operate
at higher temperatures and radiation effect can also be
significant. Consolidated
effects of heat and mass transfer problems are of importance in
many chemical
formulations and reactive chemicals. Therefore, considerable
attention had been paid
in recent years to study the influence of the participating
parameters on the velocity
fields. More such engineering application can be seeing in
electrical power generation
system when the electrical energy is extracted directly from a
moving conducting
fluid.
There has been a renewed interest in studying
Magnetohydrodynamic (MHD) flow
and heat transfer in porous and non-porous media due to the
effect of magnetic fields
on the boundary layer flow control and on the performance of
many systems using
electrically conducting fluids. In addition, this type of flow
finds applications in many
engineering problems such as MHD generators, plasma studies,
nuclear reactors and
geothermal energy extractors. Chamkha [1] presented an unsteady
MHD convective
heat and mass transfer past a semi-infinite vertical permeable
moving plate with heat
absorption. An analysis of an unsteady MHD convective flow past
a vertical moving
plate embedded in a porous medium in the presence of transverse
magnetic field a
reported by Kim [2]. Singh [3] studied the effects of mass
transfer on free convection
in MHD flow of viscous fluid. Ahmed [4] looked the effects of
unsteady free
convective MHD flow through a porous medium bounded by an
infinite vertical
porous plate. Raptis [5] studied mathematically the case of
unsteady two-dimensional
natural convective heat transfer of an incompressible,
electrically conducting viscous
fluid in a highly porous medium bound by an infinite vertical
porous plate.
Soundalgekar [6] obtained approximate solutions for the
two-dimensional flow an
incompressible, viscous fluid past an infinite porous vertical
plate with constant
suction velocity normal to the plate, the difference between the
temperature of the
plate and the free stream is moderately large causing the free
convection currents.
Recently, free convective fluctuating MHD flow through porous
media past a vertical
porous plate with variable temperature and heat source was
studied by Acharya et al. [7]. Rao et al. [8] was discussed the
heat transfer on steady MHD rotating flow through porous medium in
a parallel plate channel. Pattnaik and Biswal [9] studied
the analytical solution of MHD free convective flow through
porous media with time
dependent temperature and concentration. More recently, Hazarika
and Ahmed [10]
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Mathematical analysis for optically thin Radiating/ Chemically
reacting fluid 1779
have investigated the analytical study of unsteady MHD
chemically reacting fluid
over a vertical porous plate in a Darcian porous Regime.
Chemical reaction effects on
MHD free convective flow through porous medium with constant
suction and heat
flux has discussed by Seshaiah and Varma [11].
All the above investigations are restricted to MHD flow and heat
transfer problems
only. However of the late the effects of radiation on MHD flow,
heat and mass
transfer have becomes more important industrially. The radiation
flows of an
electrically conducting fluid with high temperature, in the
presence of magnetic fields,
are encountered in electrical power generation, astrophysical
flows, solar power
technology, space vehicle re-entry, nuclear engineering
applications and other
industrial areas. Radiative heat and mass transfer play an
important role in
manufacturing industries for the design of fins, steel rolling,
nuclear power plants, gas
turbines and various propulsion devices for aircraft, missiles,
satellites and space
vehicles are examples of such engineering applications.
Radiation effects on mixed
convection along an isothermal vertical plate were studied by
Hossain and Takhar
[12]. Prasad et al. [13] studied the radiation and mass transfer
effects on unsteady MHD free convection flow past a vertical porous
plate embedded in porous medium.
Zueco and Ahmed [14] proposed the mixed convection MHD flow
along a porous
plate with chemical reaction in presence of heat source. The
transient MHD free
convective flow of a viscous, incompressible, electrically
conducting, gray,
absorbing-emitting, but not scattering, optically thick fluid
medium which occupies a
semi-infinite porous region adjacent to an infinite hot vertical
plate moving with
constant velocity was presented by Ahmed and Kalita [15]. The
effects of chemical
reaction as well as magnetic field on the heat and mass transfer
of Newtonian two-
dimensional flow over an infinite vertical oscillating plate
with variable mass
diffusion investigated by Ahmed and Kalita [16]. Recently, Ahmed
[17] presented the
effects of conduction-radiation, porosity and chemical reaction
on unsteady
hydromagnetic free convection flow past an impulsively started
semi-infinite vertical
plate embedded in a porous medium in presence of thermal
radiation. The thermal
radiation and Darcian drag force MHD unsteady thermal-convection
flow past a semi-
infinite vertical plate immersed in a semi-infinite saturated
porous regime with
variable surface temperature in the presence of transversal
uniform magnetic field
have been discussed by Ahmed et al. [18]. Radiation and mass
transfer on unsteady MHD convective flow past an infinite vertical
plate in presence of Dufour and Soret
effects studied by Vedavathi et al. [19]. Ahmed et al. [20]
investigated the effects of chemical reaction and viscous
dissipation on MHD heat and mass transfer flow
through Perturbation method.
In all these investigations, the viscous dissipation is
neglected. Gebhart [21] had
shown the importance of viscous dissipative heat in free
convection flow in the case
of isothermal and constant heat flux at the plate. Soundalgekar
[22] analyzed the
viscous dissipative heat on the two-dimensional unsteady free
convective flow past an
infinite vertical porous plate when the temperature oscillates
in time and there is
constant suction at the plate. Prasad and Reddy [23] had
discussed about the Radiation
and Mass transfer effects on an unsteady MHD convection flow
with viscous
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1780 Nava Jyoti Hazarika and Sahin Ahmed
dissipation. Cookey et al. [24] had investigated the influence
of viscous dissipation and radiation on unsteady MHD free
convection flow past an infinite heated vertical
plate in a porous medium with time dependent suction. Recently,
radiation effects on
an unsteady MHD convective flow past a vertical plate in porous
medium with
viscous dissipation analyzed by Gudagani et al. [25]. In this
paper the effects of chemical reaction and thermal radiation of
optically thin
gray gas on a mixed convective boundary layer flow of an
electrically conducting
fluid over an semi-infinite porous surface embedded in a Darcian
porous regime in
presence of viscous dissipative heat is investigated. The
governing equations are
solved by using a regular perturbation theory.
2. MATHEMATICAL ANALYSES
In this flow model, we consider two-dimensional unsteady
hydromagnetic laminar
mixed convective boundary layer flow of a viscous,
incompressible, electrically
conducting and radiating fluid in an optically thin environment,
past a semi-infinite
vertical permeable moving plate embedded in a Darcian porous
medium, in presents
of thermal and concentration buoyancy effects with chemical
reaction of first order.
The 𝑥-axis is taken in the upward direction along the plate and
𝑦-axis normal to it. A uniform magnetic field is applied in the
direction perpendicular to the plate. The
transverse applied magnetic field and magnetic Reynolds number
are assumed to be
very small, so that the induced magnetic field is negligible.
Also, it is assumed that
there is no applied voltage, so that the electric field is
absent. The concentration of the
diffusing species in the binary mixture is assumed to be very
small in comparison
with the other chemical species which are present, and hence the
Soret and Dufour
effects are negligible. Further, due to semi-infinite plane
surface assumption, the flow
variables are functions of normal distance 𝑦 and 𝑡 only. Now,
under the usual Boussinesq’s approximation, the governing boundary
layer equations are:
𝜕𝑣
𝜕𝑦= 0 (1)
𝜕𝑢
𝜕𝑡+ 𝑣
𝜕𝑣
𝜕𝑦= −
1
𝜌
𝜕𝑝
𝜕𝑥+ 𝜈
𝜕2𝑢
𝜕𝑦2 + 𝑔𝛽𝑇(𝑇 − 𝑇∞) + 𝑔𝛽𝐶(𝐶 − 𝐶∞) − (
𝜈
𝜅+𝜎𝐵0
2
𝜌)𝑢 (2)
𝜕𝑇
𝜕𝑡+ 𝑣
𝜕𝑇
𝜕𝑦=
𝑘
𝜌𝑐𝑝[𝜕2𝑇
𝜕𝑦2 −
1
𝑘
𝜕𝑞
𝜕𝑦] +
𝜈
𝑐𝑝(𝜕𝑢
𝜕𝑦)
2
(3)
𝜕2𝑞
𝜕𝑦2 − 3𝛼
2𝑞 − 16𝜎∗𝛼𝑇∞3 𝜕𝑇
𝜕𝑦= 0 (4)
𝜕𝐶
𝜕𝑡+ 𝑣
𝜕𝐶
𝜕𝑦= 𝐷
𝜕2𝐶
𝜕𝑦2 − 𝐶𝑟(𝐶 − 𝐶∞) (5)
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Mathematical analysis for optically thin Radiating/ Chemically
reacting fluid 1781
The third and fourth terms on the right hand side of momentum
Eq. (2) denote the
thermal and concentration buoyancy effects respectively. The
second and third terms
on right hand side of energy Eq. (3) represent the radiative
heat flux and viscous
dissipation respectively. Also the second term on right hand of
concentration Eq. (5)
represents the chemical reaction effect.
The permeable plate moves with a constant velocity in the
direction of fluid flow and
the free steam velocity follows the exponentially increasing
small perturbation law. In
addition, it is assumed that the temperature and concentration
at the wall as well as the
suction velocity are exponentially varying with time. Eq. (4) is
the differential
approximation for radiation and the radiative heat flux 𝑞
satisfies this non-linear differential equation.
The boundary conditions for the velocity, temperature and
concentration fields are:
{𝑢 = 𝑢𝑝, 𝑇 = 𝑇𝑤 + 𝜀(𝑇𝑤 − 𝑇∞)𝑒
𝑛𝑡, 𝐶 = 𝐶𝑤 + 𝜀(𝐶𝑤 − 𝐶∞)𝑒𝑛𝑡 𝑎𝑡 𝑦 = 0
𝑢 = 𝑈∞ = 𝑈0(1 + 𝜀𝑒𝑛𝑡), 𝑇 ⟶ 𝑇∞ , 𝐶 ⟶ 𝐶∞ 𝑎𝑠 𝑦 ⟶ ∞
} (6)
It is clear from the equation (1) that the suction velocity at
the plate is either a
constant or function of time only. Hence, the suction velocity
normal to the plate is
assumed in the form:
𝑣 = −𝑉0(1 + 𝜀𝐴𝑒𝑛𝑡) (7)
The negative sign indicates that the suction is towards the
plate.
Outside the boundary layer, Eq. (2) gives:
−1
𝜌
𝜕𝑝
𝜕𝑥= 𝑑𝑈∞
𝑑𝑡+𝜈
𝜅 𝑈∞ +
𝜎
𝜌 𝐵0
2𝑈∞ (8)
Since the medium is optically thin with relatively low density
and 𝛼 ≪ 1, the radiative heat flux given by Eq. (3), in the spirit
of Cogley et al. [22] becomes:
𝜕𝑞
𝜕𝑦= 4𝛼2 (𝑇 − 𝑇∞) where 𝛼
2 = ∫ 𝛿𝜆𝜕𝐵
𝜕𝑇
∞
0
, (9)
where B is Planck’s function.
In order to write the governing equations and boundary
conditions in dimensionless
form, the following non-dimensional quantities are
introduced.
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1782 Nava Jyoti Hazarika and Sahin Ahmed
{
𝑢 =
𝑢
𝑈0 , 𝑣 =
𝑣
𝑉0, 𝑦 =
𝑉0𝑦
𝜈 , 𝑈∞ =
𝑈∞𝑈0
,
𝑈𝑝 =𝑢𝑝
𝑈0, 𝑡 =
𝑡 𝑉02
𝜈 ,
𝜃 =𝑇 − 𝑇∞
𝑇𝑤 − 𝑇∞ , 𝜙 =
𝐶 − 𝐶∞
𝐶𝑤 − 𝐶∞ , 𝑛 =
𝑛 𝜈
𝑉02 , 𝐾 =
𝐾 𝑉02
𝜈2 , 𝐶𝑟 =
𝜈𝐶𝑟
𝑉02
𝑃𝑟 = 𝜈𝜌𝐶𝑝
𝑘, 𝑆𝑐 =
𝜈
𝐷 , 𝑀 =
𝜎𝐵02𝜈
𝜌𝑉02 , 𝐺𝑟 =
𝜈𝛽𝑇𝑔(𝑇𝑤 − 𝑇∞)
𝑈0𝑉02
,
𝐺𝑚 =𝜈𝛽𝐶𝑔(𝐶𝑤 − 𝐶∞)
𝑈0𝑉02 , 𝐸𝑐 =
𝑈02
𝐶𝑝(𝑇𝑤 − 𝑇∞), 𝑅2 =
𝛼2(𝑇𝑤 − 𝑇∞)
𝜌𝐶𝑝𝑘𝑈02
,
}
(10)
In view of Eqs. (4) and (7) –(10), Eqs. (2), (3) and (5) reduce
to the following
dimensionless form:
𝜕𝑢
𝜕𝑡− (1 + 𝜀𝐴𝑒𝑛𝑡)
𝜕𝑢
𝜕𝑦=𝑑𝑈∞𝑑𝑡
+𝜕2𝑢
𝜕𝑦2+ 𝐺𝑟𝜃 + 𝐺𝑚𝜙 + 𝑁(𝑈∞ − 𝑢) (11)
𝜕𝜃
𝜕𝑡− (1 + 𝜀𝐴𝑒𝑛𝑡)
𝜕𝜃
𝜕𝑦=
1
𝑃𝑟[𝜕2𝜃
𝜕𝑦2− 𝑅2𝜃] + 𝐸𝑐 (
𝜕𝑢
𝜕𝑦)2
(12)
𝜕𝜙
𝜕𝑡− (1 + 𝜀𝐴𝑒𝑛𝑡)
𝜕𝜙
𝜕𝑦=1
𝑆𝑐
𝜕2𝜙
𝜕𝑦2− 𝐶𝑟𝜙 (13)
where 𝑁 = 𝑀 + 𝐾−1
The corresponding dimensionless boundary conditions are:
{𝑢 = 𝑈𝑝 , 𝜃 = 1 + 𝜀𝑒
𝑛𝑡, 𝜙 = 1 + 𝜀𝑒𝑛𝑡, 𝑎𝑡 𝑦 = 0
𝑢 = 𝑈∞ = 1 + 𝜀𝑒𝑛𝑡, 𝜃 ⟶ 0, 𝜙 ⟶ 0 𝑎𝑠 𝑦 ⟶ ∞
} (14)
SOLUTION OF THE PROBLEM
The Eqs. (11-13) are coupled, non-linear partial differential
equations and these
cannot be solved in closed-form. However, these equations can be
reduced to a set of
ordinary differential equations, which can be solved
analytically. This can be done by
representing the velocity, temperature and concentration of the
fluid in the
neighbourhood of the plate as:
{
𝑢(𝑦, 𝑡) = 𝑢0(𝑦) + 𝜀𝑒𝑛𝑡𝑢1(𝑦) + 0(𝜀
2) + ⋯
𝜃(𝑦, 𝑡) = 𝜃0(𝑦) + 𝜀𝑒𝑛𝑡𝜃1(𝑦) + 0(𝜀
2) + ⋯
𝜙(𝑦, 𝑡) = 𝜙0(𝑦) + 𝜀𝑒𝑛𝑡𝜙1(𝑦) + 0(𝜀
2) + ⋯
} (15)
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Mathematical analysis for optically thin Radiating/ Chemically
reacting fluid 1783
Substituting Eq. (15) in Eqs. (11-13) and equating the harmonic
and non-harmonic
terms, and neglecting the higher order terms of 0(𝜀2), we
obtain:
𝑢0″(𝑦) + 𝑢0
′ (𝑦) − 𝑁𝑢0(𝑦) = −𝑁 − 𝐺𝑟𝜃0(𝑦) − 𝐺𝑚𝜙0(𝑦) (16)
𝑢1″(𝑦) + 𝑢1
′ (𝑦) − (𝑁 + 𝑛)𝑢1(𝑦)
= −(𝑁 + 𝑛) − 𝐴𝑢0′ (𝑦) − 𝐺𝑟𝜃1(𝑦) − 𝐺𝑚𝜙1(𝑦) (17)
𝜃0″(𝑦) + 𝑃𝑟 𝜃0
′(𝑦) − 𝑅2 𝜃0(𝑦) = −𝑃𝑟𝐸𝑐 [𝑢0′ (𝑦)]2 (18)
𝜃1″(𝑦) + 𝑃𝑟 𝜃1
′(𝑦) − (𝑅2 + 𝑛𝑃𝑟)𝜃1(𝑦) = −𝑃𝑟𝐴 𝜃0′(𝑦) − 2𝑃𝑟𝐸𝑐 𝑢0
′ (𝑦)𝑢1′(𝑦) (19)
𝜙0″(𝑦) + 𝑆𝑐 𝜙0
′ (𝑦) − 𝑆𝑐 𝐶𝑟𝜙0(𝑦) = 0 (20)
𝜙1″(𝑦) + 𝑆𝑐 𝜙1
′ (𝑦) − 𝑆𝑐(𝑛 + 𝐶𝑟)𝜙1(𝑦) = −𝐴𝑆𝑐 𝜙0′ (𝑦) (21)
where prime denotes ordinary differentiation with respect to
y.
The corresponding boundary conditions can be written as:
{𝑢0 = 𝑈𝑝, 𝑢1 = 0, 𝜃0 = 1, 𝜃1 = 1, 𝜙0 = 1, 𝜙1 = 1 𝑎𝑡 𝑦 = 0
𝑢0 = 1, 𝑢1 = 1, 𝜃0 ⟶ 0, 𝜃1 ⟶ 0, 𝜙0 ⟶ 0, 𝜙1 ⟶ 0 𝑎𝑠 𝑦 ⟶ ∞ }
(22)
The Eqs. (16) – (21) are still coupled and non-linear, whose
exact solutions are not
possible. So we expand 𝑢0 , 𝑢1 , 𝜃0 , 𝜃1 , 𝜙0 , 𝜙1 in terms of
𝐸𝑐 in the following form, as the Eckert number is very small for
incompressible flows.
𝐹(𝑦) = 𝐹0(𝑦) + 𝐸𝑐 𝐹1(𝑦) + 0(𝐸𝑐2) (23)
where 𝐹 stands for any 𝑢0 , 𝑢1 , 𝜃0 , 𝜃1 , 𝜙0 , 𝜙1 .
Substituting Eq. (23) in Eqs. (16) – (21), equating the
co-efficient of 𝐸𝑐 to zero and neglecting the terms in 𝐸𝑐2 and
higher order, we get the following equations:
The zeroth order equations are:
𝑢01″ (𝑦) + 𝑢01
′ (𝑦) − 𝑁𝑢01(𝑦) = −𝑁 − 𝐺𝑟 𝜃01(𝑦) − 𝐺𝑚 𝜙01(𝑦) (24)
𝑢02″ (𝑦) + 𝑢02
′ (𝑦) − 𝑁𝑢02(𝑦) = −𝐺𝑟 𝜃02(𝑦) − 𝐺𝑚 𝜙02(𝑦) (25)
𝜃01″ (𝑦) + 𝑃𝑟 𝜃01
′ (𝑦) − 𝑅2 𝜃01(𝑦) = 0 (26)
𝜃02″ (𝑦) + 𝑃𝑟 𝜃02
′ (𝑦) − 𝑅2 𝜃02(𝑦) = −𝑃𝑟[𝑢01′ (𝑦)]2 (27)
𝜙01″ (𝑦) + 𝑆𝑐 𝜙01
′ (𝑦) − 𝑆𝑐 𝐶𝑟 𝜙01(𝑦) = 0 (28)
𝜙02″ (𝑦) + 𝑆𝑐 𝜙02
′ (𝑦) − 𝑆𝑐 𝐶𝑟 𝜙02(𝑦) = 0 (29)
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1784 Nava Jyoti Hazarika and Sahin Ahmed
and the respective boundary conditions are:
{𝑢01 = 𝑈𝑝, 𝑢02 = 0, 𝜃01 = 1, 𝜃02 = 0, 𝜙01 = 1, 𝜙02 = 0 𝑎𝑡 𝑦 =
0
𝑢01 ⟶ 1, 𝑢02 ⟶ 0, 𝜃01 ⟶ 0, 𝜃02 ⟶ 0,𝜙01 ⟶ 0, 𝜙02 ⟶ 0 𝑎𝑡 𝑦 ⟶ ∞}
(30)
The first order equations are:
𝑢11″ (𝑦) + 𝑢11
′ (𝑦) − (𝑁 + 𝑛)𝑢11(𝑦) = {−(𝑁 + 𝑛) − 𝐺𝑟 𝜃11(𝑦)
−𝐺𝑚 𝜙11(𝑦) − 𝐴 𝑢01′ (𝑦)
} (31)
𝑢12″ (𝑦) + 𝑢12
′ (𝑦) − (𝑁 + 𝑛)𝑢12(𝑦) = −𝐺𝑟 𝜃12(𝑦) − 𝐺𝑚 𝜙12(𝑦) − 𝐴 𝑢02′ (𝑦)
(32)
𝜃11″ (𝑦) + 𝑃𝑟 𝜃11
′ (𝑦) − 𝑁1𝜃11(𝑦) = −𝑃𝑟𝐴 𝜃01′ (𝑦) (33)
𝜃12″ (𝑦) + 𝑃𝑟 𝜃12
′ (𝑦) − 𝑁1𝜃12(𝑦) = −𝑃𝑟𝐴 𝜃02′ (𝑦) − 2𝑃𝑟 𝑢01
′ (𝑦)𝑢11′ (𝑦) (34)
𝜙11″ (𝑦) + 𝑆𝑐 𝜙11
′ (𝑦) − 𝑆𝑐(𝑛 + 𝐶𝑟)𝜙11(𝑦) = −𝐴𝑆𝑐 𝜙01′ (𝑦) (35)
𝜙12″ (𝑦) + 𝑆𝑐 𝜙12
′ (𝑦) − 𝑆𝑐(𝑛 + 𝐶𝑟)𝜙12(𝑦) = −𝐴𝑆𝑐 𝜙02′ (𝑦) (36)
where 𝑁1 = 𝑅2 + 𝑛𝑃𝑟.
and respective boundary conditions are:
{ 𝑢11 = 0, 𝑢12 = 0, 𝜃11 = 1, 𝜃12 = 0, 𝜙11 = 1, 𝜙12 = 0 𝑎𝑡 𝑦 =
0
𝑢11 ⟶ 1, 𝑢12 ⟶ 0, 𝜃11 ⟶ 0, 𝜃12 ⟶ 0, 𝜙11 ⟶ 0, 𝜙12 ⟶ 0 𝑎𝑡 𝑦 ⟶ ∞ }
(37)
Solving Eqs. (24) – (29) under the boundary conditions in Eq.
(30) and Eqs. (31) -
(36) under the boundary conditions in Eq. (37) and using Eqs.
(15) and (23), we
obtain the Velocity, Temperature and Concentration distributions
in the boundary
layer as:
𝑢(𝑦, 𝑡) =
{
𝑃3𝑒−𝑚3𝑦 + 𝑃1𝑒
−𝑚2𝑦 + 𝑃2𝑒−𝑚1𝑦 + 1
+𝐸𝑐 { 𝐽8𝑒
−𝑚3𝑦 + 𝐽1𝑒−𝑚2𝑦 + 𝐽2𝑒
−2𝑚3𝑦 + 𝐽3𝑒−2𝑚2𝑦 + 𝐽4𝑒
−2𝑚1𝑦
+𝐽5𝑒−(𝑚2+𝑚3)𝑦 + 𝐽6𝑒
−(𝑚1+𝑚2)𝑦 + 𝐽7𝑒−(𝑚1+𝑚3)𝑦
}
+ 𝜀𝑒𝑛𝑡
[ {
𝐺6𝑒−𝑚6𝑦 + 𝐺1𝑒
−𝑚5𝑦 + 𝐺2𝑒−𝑚2𝑦 + 𝐺3𝑒
−𝑚4𝑦
+𝐺4𝑒−𝑚1𝑦 + 𝐺5𝑒
−𝑚3𝑦 + 1}
+𝐸𝑐
{
𝐿19𝑒−𝑚6𝑦 + 𝐿1𝑒
−𝑚5𝑦 + 𝐿2𝑒−𝑚2𝑦 + 𝐿3𝑒
−2𝑚3𝑦
+𝐿4𝑒−2𝑚2𝑦 + 𝐿5𝑒
−2𝑚1𝑦 + 𝐿6𝑒−(𝑚2+𝑚3)𝑦 + 𝐿7𝑒
−(𝑚1+𝑚2)𝑦
+𝐿8𝑒−(𝑚1+𝑚3)𝑦 + 𝐿9𝑒
−(𝑚3+𝑚6)𝑦 + 𝐿10𝑒−(𝑚3+𝑚5)𝑦
+𝐿11𝑒−(𝑚3+𝑚4)𝑦 + 𝐿12𝑒
−(𝑚2+𝑚6)𝑦 + 𝐿13𝑒−(𝑚2+𝑚5)𝑦
+𝐿14𝑒−(𝑚2+𝑚4)𝑦 + 𝐿15𝑒
−(𝑚1+𝑚6)𝑦 + 𝐿16𝑒−(𝑚1+𝑚5)𝑦
+𝐿17𝑒−(𝑚1+𝑚4)𝑦 + 𝐿18𝑒
−𝑚3𝑦 }
]
}
-
Mathematical analysis for optically thin Radiating/ Chemically
reacting fluid 1785
𝜃(𝑦, 𝑡) =
{
𝑒−𝑚2𝑦 + 𝐸𝑐 {
𝑆7𝑒−𝑚2𝑦 + 𝑆1𝑒
−2𝑚3𝑦 + 𝑆2𝑒−2𝑚2𝑦 + 𝑆3𝑒
−2𝑚1𝑦
+𝑆4𝑒−(𝑚2+𝑚3)𝑦 + 𝑆5𝑒
−(𝑚1+𝑚2)𝑦 + 𝑆6𝑒−(𝑚1+𝑚3)𝑦
}
+𝜀𝑒𝑛𝑡
[
{𝐷2𝑒−𝑚5𝑦 + 𝐷1𝑒
−𝑚2𝑦} +
𝐸𝑐
{
𝑅17𝑒−𝑚5𝑦 + 𝑅1𝑒
−𝑚2𝑦 + 𝑅2𝑒−2𝑚3𝑦 + 𝑅3𝑒
−2𝑚2𝑦
+𝑅4𝑒−2𝑚1𝑦 + 𝑅5𝑒
−(𝑚2+𝑚3)𝑦 + 𝑅6𝑒−(𝑚1+𝑚2)𝑦
+𝑅7𝑒−(𝑚1+𝑚3)𝑦 + 𝑅8𝑒
−(𝑚3+𝑚6)𝑦 + 𝑅9𝑒−(𝑚3+𝑚5)𝑦
+𝑅10𝑒−(𝑚3+𝑚4)𝑦 + 𝑅11𝑒
−(𝑚2+𝑚6)𝑦 +
𝑅12𝑒−(𝑚2+𝑚5)𝑦 + 𝑅13𝑒
−(𝑚2+𝑚4)𝑦 + 𝑅14𝑒−(𝑚1+𝑚6)𝑦
+𝑅15𝑒−(𝑚1+𝑚5)𝑦 + 𝑅16𝑒
−(𝑚1+𝑚4)𝑦 }
]
}
𝜙(𝑦, 𝑡) = 𝑒−𝑚1𝑦 + 𝜀𝑒𝑛𝑡{𝑍2𝑒−𝑚4𝑦 + 𝑍1𝑒
−𝑚1𝑦}
The Skin-friction, Nusselt number and Sherwood number are
important physical
parameters for this type of boundary layer flow.
THE SKIN FRICTION
Knowing the velocity field, the Skin-friction at the plate can
be obtained, which in
non-dimensional form is given by:
𝐶𝑓 =𝜏𝑤
𝜌𝑈0𝑉0= (
𝜕𝑢
𝜕𝑦 )𝑦=0
= ( 𝜕𝑢0𝜕𝑦
+ 𝜀𝑒𝑛𝑡𝜕𝑢1𝜕𝑦 )𝑦=0
=
[ −𝑚3𝑃3 −𝑚2𝑃1 −𝑚1𝑃2 + 𝐸𝑐 {
–𝑚3𝐽8 −𝑚2𝐽1 − 2𝑚3𝐽2 − 2𝑚2𝐽3 − 2𝑚1𝐽4 −(𝑚2 +𝑚3)𝐽5 − (𝑚1 +𝑚2)𝐽6 −
(𝑚1 +𝑚3)𝐽7
}
+𝜀𝑒𝑛𝑡
[
(−𝑚6𝐺6 −𝑚5𝐺1 −𝑚2𝐺2 −𝑚4𝐺3 −𝑚1𝐺4 −𝑚3𝐺5)
+𝐸𝑐
{
−𝑚6𝐿19 −𝑚5𝐿1 −𝑚2𝐿2 − 2𝑚3𝐿3 − 2𝑚2𝐿4 − 2𝑚1𝐿5 −(𝑚2 +𝑚3)𝐿6 − (𝑚1
+𝑚2)𝐿7 − (𝑚1 +𝑚3)𝐿8
−(𝑚3 +𝑚6)𝐿9 − (𝑚3 +𝑚5)𝐿10 − (𝑚3 +𝑚4)𝐿11−(𝑚2 +𝑚6)𝐿12 − (𝑚2
+𝑚5)𝐿13 − (𝑚2 +𝑚4)𝐿14
−(𝑚1 +𝑚6)𝐿15 − (𝑚1 +𝑚5)𝐿16 − (𝑚1 +𝑚4)𝐿17 −𝑚3𝐿18}
]
]
RATE OF HEAT TRANSFER
Knowing the temperature field, the rate of heat transfer
co-efficient can be obtained,
which in the non-dimensional form, in terms of the Nusselt
number is given by:
𝑁𝑢 = −𝑥
(𝜕𝑇𝜕𝑦⁄)𝑦=0
𝑇𝑤 − 𝑇∞= 𝑁𝑢𝑅𝑒𝑥
−1 = −( 𝜕𝜃
𝜕𝑦 )𝑦=0
= −( 𝜕𝜃0𝜕𝑦
+ 𝜀𝑒𝑛𝑡𝜕𝜃1𝜕𝑦 )𝑦=0
-
1786 Nava Jyoti Hazarika and Sahin Ahmed
= −
[ −𝑚2 + 𝐸𝑐 {
−𝑚2𝑆7 − 2𝑚3𝑆1 − 2𝑚2𝑆2 − 2𝑚1𝑆3 − (𝑚2 +𝑚3)𝑆4−(𝑚1 +𝑚2)𝑆5 − (𝑚1
+𝑚3)𝑆6
}
+𝜀𝑒𝑛𝑡
[
(−𝑚5𝐷2 −𝑚2𝐷1)
+𝐸𝑐
{
−𝑚5𝑅17 −𝑚2𝑅1 − 2𝑚3𝑅2 − 2𝑚2𝑅3 − 2𝑚1𝑅4−(𝑚2 +𝑚3)𝑅5 − (𝑚1 +𝑚2)𝑅6 −
(𝑚1 +𝑚3)𝑅7−(𝑚3 +𝑚6)𝑅8 − (𝑚3 +𝑚5)𝑅9 − (𝑚3 +𝑚4)𝑅10−(𝑚2 +𝑚6)𝑅11 − (𝑚2
+𝑚5)𝑅12 − (𝑚2 +𝑚4)𝑅13−(𝑚1 +𝑚6)𝑅14 − (𝑚1 +𝑚5)𝑅15 − (𝑚1 +𝑚4)𝑅16}
]
]
where 𝑅𝑒𝑥 =𝑉0𝑥
𝜈 is the local Reynolds number.
RATE OF MASS TRANSFER
Knowing the concentration field, the rate of mass transfer
co-efficient can be
obtained, which in the non-dimensional form, in terms of the
Sherwood number is
given by:
𝑆ℎ = −𝑥
(𝜕𝐶𝜕𝑦⁄ )
𝑦=0
𝐶𝑤 − 𝐶∞ ,
𝑆ℎ𝑅𝑒𝑥−1 = −(
𝜕𝐶
𝜕𝑦 )𝑦=0
= −( 𝜕𝐶0𝜕𝑦
+ 𝜀𝑒𝑛𝑡𝜕𝐶1𝜕𝑦 )𝑦=0
= −[−𝑚1 + 𝜀𝑒𝑛𝑡(−𝑚4𝑍2 −𝑚1𝑍1)]
VALIDITY
When Cr = 0, the present paper reduces to the work which was
done by Prasad and Reddy [23].
Table 1: Comparison of the present results with those of Prasad
and Reddy [23] with
effects of Gr and Gm on Cf when Gr=2.0, Gm=1.0, Pr=0.71, Sc=0.6,
M=1.0, R=0.5, K=0.5, n=0.1, Up=0.5, A=0.5, Cr=0.2, t=1.0, Ec=0.001,
Ԑ=0.001.
Gr Gm Prasad and Reddy [23] Present work
Effects of
Gr on Cf Effect of Gm
on Cf Effects of Gr
on Cf Effects of
Gm on Cf
0
1
2
3
4
0
1
2
3
4
1.6877
2.0974
2.5123
2.9345
3.3660
1.9741
2.5123
3.0515
3.5918
4.1331
1.60691
2.04773
2.48857
2.92944
3.37035
2.03578
2.48857
2.94137
3.39418
3.84699
-
Mathematical analysis for optically thin Radiating/ Chemically
reacting fluid 1787
The Table 1 shows that the accuracy of the present model in
comparison with the
previous model studied by Prasad and Reddy [23] and this
comparison is validated the
present study.
RESULTS AND DISCUSSION
The formulation of the problem that accounts for the effects of
radiation and viscous
dissipation on the flow of an incompressible viscous chemically
reacting fluid along a
semi-infinite, vertically moving porous plate embedded in a
porous medium in the
presence of transverse magnetic field was accomplished.
Following Cogley et al. [22] approximation for the radiative heat
flux in the optically thin environment, the
governing equations on the flow field were solved analytically,
using a perturbation
method and the expressions for the velocity, temperature,
concentration, Skin-friction,
Nusselt number and Sherwood number were obtained. In order to
get a physical
insight of the problem, the above physical quantities are
computed numerically for
different values of the governing parameters viz. Thermal
Grashof number Gr, the Solutal Grashof number Gm, Radiation
parameter R, Magnetic parameter M, Permeability parameter K, Plate
velocity Up, Prandtl number Pr, Schmidt number Sc, Eckert number Ec
and Chemical reaction Cr. Figure 1 shows the typical velocity
profiles in the boundary layer for various values
of the thermal Grashof number. It is observed that an increase
in Gr, leads to a rise in the values of the velocity due to
enhancement in the buoyancy force. Here, the
positive values of Gr correspond to cooling of the plate. In
addit0ion, it is observed that the velocity increases rapidly near
the wall of the porous plate as Grashof number
increases and then decays to the free stream velocity. Figure 2
depicts the typical
velocity profiles in the boundary layer for distinct values of
the solutal Grashof
number Gm. The velocity distribution attaints a distinctive
maximum value in the region of the plate surface and then decrease
properly to approach the free stream
value. As expected, the fluid velocity increases and the peak
value becomes more
distinctive due to increase in the buoyancy force represented by
Gm. For different values of thermal radiation parameter R on the
velocity and temperature profiles are shown in Figure 3 and 4. It
is noticed that an increase in the radiation
parameter results a decrease in the velocity and temperature
within the boundary
layer, as well as decreased the thickness of the velocity and
temperature boundary
layers.
The effect of magnetic field on velocity profiles in the
boundary layer is depicted in
Figure 5. It is obvious that the existence of the magnetic field
is to decrease the
velocity in the momentum boundary layer because the application
of the transverse
magnetic field results in a resisting type of force called
Lorentz force, which results in
reducing the velocity of the fluid in the boundary layer. Figure
6 shows the effect of
the permeability of the porous medium parameter K on the
velocity distribution. It is found that the velocity increases with
an increase in K.
-
1788 Nava Jyoti Hazarika and Sahin Ahmed
The velocity distribution across the boundary layer for several
values of plate moving
velocity Up in the direction of the fluid flow is depicted in
Figure 7. Although we have different initial plate moving
velocities, the velocity decreases to a constant
value for given material parameters.
Figure 8 and 9 shows the behaviour velocity and temperature for
different values of
Prandtl number Pr. The numerical results show the effect of
increasing values of Prandtl number results in the decreasing
velocity. From Figure 9, it is observed that
an increase in the Prandtl number results a decrease in the
thermal boundary layer
thickness and in general lower average temperature within the
boundary layer. The
reason is that smaller values of Pr are equivalent to increase
in the thermal conductivity of the fluid and therefore heat is able
to diffuse away from the heated
surface more rapidly for higher values of Pr. Hence in the case
of smaller Prandtl numbers as the thermal boundary layer is thicker
and the rate of heat transfer is
reduced.
Figure 10 and 11 shows the effects of Schmidt number on the
velocity and
concentration respectively. As the Schmidt number increases, the
concentration
decreases. This causes the concentration buoyancy effects to
decrease yielding a
reduction in the fluid velocity. Reductions in the velocity and
concentration
distributions are accompanied by simultaneous reductions in the
velocity and
concentration boundary layers.
The effects of chemical reaction on velocity and concentration
are depicted by Figure
12 and 13. It is noticed that an increase in the chemical
reaction parameter results a
decrease in the velocity and concentration within the boundary
layer.
Table 2-5, represents the effects of Eckert number and Chemical
reaction on the
velocity u, temperature Ө, Skin-friction Cf , Nusselt number Nu
and Sherwood number Sh. The effects of viscous dissipation
parameter i.e. the Eckert number on the velocity
and temperature are shown in Table 2 and 3. It is revealed that
velocity and
temperature profiles scores grow with the increase of the Eckert
number Ec. Eckert number, physically is a measure of frictional
heat in the system. Hence the thermal
regime with large Ec values is subjected to rather more
frictional heating causing a source of rise in the temperature. To
be specific, the Eckert number Ec signifies the relative importance
of viscous heating to thermal diffusion. Viscous heating may
serve as energy source to modify the temperature regime
respectively. It is observed
from Table 4, when Eckert number increases the Skin-friction
increases and Nusselt
number decreases. However, from Table 5, it can be seen that as
the Chemical
reaction increases, the Skin-friction decreases and Sherwood
number increases.
-
Mathematical analysis for optically thin Radiating/ Chemically
reacting fluid 1789
-
1790 Nava Jyoti Hazarika and Sahin Ahmed
-
Mathematical analysis for optically thin Radiating/ Chemically
reacting fluid 1791
Table 2: Effects of Ec on velocity (u) when Gr=2.0, Gm=2.0,
Pr=0.71, Sc=0.6, M=1.0, R=0.5, K=0.5, n=0.1, Up=0.5, A=0.5, Cr=0.2,
t=1.0, Ԑ=0.001.
y Ec=0 Ec=0.1 Ec=0.2 Ec=0.3
0
1
2
3
4
5
0.5
1.33264
1.18374
1.07798
1.0318
1.01321
0.499889
1.33594
1.18577
1.07895
1.03221
1.01337
0.499779
1.33925
1.1878
1.07992
1.03262
1.01354
0.499668
1.34255
1.18983
1.08089
1.03303
1.0137
-
1792 Nava Jyoti Hazarika and Sahin Ahmed
Table 3: Effects of Ec on temperature (Ө) when Gr=2.0, Gm=2.0,
Pr=0.71, Sc=0.6, M=1.0, R=0.5, K=0.5, n=0.1, Up=0.5, A=0.5, Cr=0.2,
t=1.0, Ԑ=0.001.
y Ec=0 Ec=0.1 Ec=0.2 Ec=0.3
0
1
2
3
4
5
1.00111
0.38005
0.144281
0.0547757
0.0207956
0.0078952
1.00111
0.386705
0.147783
0.0563569
0.0214404
0.00814729
1.00111
0.393361
0.151285
0.0579382
0.0220851
0.00839937
1.00111
0.400016
0.154787
0.0595194
0.0227298
0.00865146
Table 4: Effects of Ec on Cf and NuRex-1. Reference values in
the figure 14 and 15:
Ec Cf NuRex-1
0
0.1
0.2
0.3
2.48849
2.4965
2.50451
2.51252
0.969636
0.881341
0.793046
0.70475
Table 5: Effects of Cr on Cf and NuRex-1. Reference values in
the figure 12 and 13:
Cr Cf NuRex-1
0
0.3
0.6
0.9
2.51189
2.47906
2.45576
2.43759
0.800967
0.956863
1.08227
1.19017
CONCLUSIONS
The governing equations for unsteady MHD convective heat and
mass transfer flow
past a semi-infinite vertical permeable moving plate embedded in
a porous medium
with radiation and viscous dissipation effects were formulated
.Chemical reaction
effects is also included in the present work. The plate velocity
is maintained at
constant value and the flow is subjected to a transverse
magnetic field. The present
investigation brings out the following conclusions of physical
interest on the velocity,
temperature and concentration distribution of the flow
field.
It is found that when thermal and solutal Grashof number is
increased, the thermal and concentration buoyancy effects are
enhanced and thus the fluid
velocity increased.
However, the presence of radiation effects caused reductions in
the fluid temperature, which resulted in decrease in the fluid
velocity.
-
Mathematical analysis for optically thin Radiating/ Chemically
reacting fluid 1793
It is observed that the existence of magnetic body force and
chemical reaction decreases the fluid velocity.
The permeability parameter and plate velocity have the influence
of increasing the fluid velocity.
As Prandtl number increased the velocity and temperature are
both decreased. When Schmidt number increased, the concentration
level decreased resulting
in decreased fluid velocity.
In presence of Eckert number both velocity and temperature
increased.
NOMENCLATURE
𝑢 , 𝑣 Velocity components in 𝑥 , 𝑦 directions respectively,
𝑡 Time,
𝑝 Pressure,
𝑔 Acceleration due to gravity,
𝜅 Permeability of porous medium,
𝑇 Temperature of the fluid in the boundary layer,
𝑇∞ Temperature of the fluid far away from the plate,
𝐶 Species concentration in the boundary layer,
𝐶∞ Species concentration in the fluid far away from the
plate,
𝐵𝑜 Magnetic induction,
𝑐𝑝 Specific heat at constant pressure,
𝑘 Thermal conductivity,
𝑞 Radiative heat flux,
𝜎∗ Stefan-Boltzmann constant,
D Mass diffusivity and
𝐶𝑟 Chemical reaction.
𝑢𝑝 Plate velocity,
𝑇𝑤 Temperature of the plate,
𝐶𝑤 Concentration of the plate,
𝑈∞ Free stream velocity,
𝑈0 Constant,
𝑛 Constant
-
1794 Nava Jyoti Hazarika and Sahin Ahmed
A Real positive constant
𝑉0 Non-zero positive constant
GREEK SYMBOL
𝜌 Density,
𝛽𝑇 Thermal expansion co-efficient,
𝛽𝐶 Concentration expansion co-efficient,
𝜈 Kinematic viscosity,
𝜎 Electrical conductivity of the fluid,
𝛼 Fluid thermal diffusivity,
Ԑ small such that Ԑ ≪ 1
APPENDIX
𝑚1 =𝑆𝑐 + √1 + 4𝑆𝑐𝐶𝑟
2, 𝑚2 =
𝑃𝑟 + √𝑃𝑟2 + 4𝑅2
2, 𝑚3 =
1 + √1 + 4𝑁
2 ,
𝑚4 =𝑆𝑐 + √𝑆𝑐2 + 4𝑆𝑐(𝑛 + 𝐶𝑟)
2 ,𝑚5 =
𝑃𝑟 + √𝑃𝑟2 + 4𝑁12
,𝑚6
=1 + √1 + 4(𝑁 + 𝑛)
2 ,
𝑃1 =−𝐺𝑟
𝑚22 −𝑚2 − 𝑁
, 𝑃2 =−𝐺𝑚
𝑚12 −𝑚1 − 𝑁
, 𝑃3 = 𝑈𝑝 − 1 − 𝑃1 − 𝑃2 ,
𝐽1 =−𝐺𝑟𝑆7
𝑚22 −𝑚2 −𝑁
, 𝐽2 =−𝐺𝑟𝑆1
4𝑚32 − 2𝑚3 − 𝑁
, 𝐽3 =−𝐺𝑟𝑆2
4𝑚22 − 2𝑚2 −𝑁
,
𝐽4 =−𝐺𝑟𝑆3
4𝑚12 − 2𝑚1 − 𝑁
, 𝐽5 =−𝐺𝑟𝑆4
(𝑚2 +𝑚3)2 − (𝑚2 +𝑚3) − 𝑁 ,
𝐽6 =−𝐺𝑟𝑆5
(𝑚1 +𝑚2)2 − (𝑚1 +𝑚2) − 𝑁 , 𝐽7 =
−𝐺𝑟𝑆6(𝑚1 +𝑚3)2 − (𝑚1 +𝑚3) − 𝑁
,
𝐽8 = −(𝐽1 + 𝐽2 + 𝐽3 + 𝐽4 + 𝐽5 + 𝐽6 + 𝐽7), 𝐺1 =−𝐺𝑟𝐷2
𝑚52 −𝑚5 − (𝑁 + 𝑛)
,
𝐺2 =𝐴𝑚2𝑃1 − 𝐺𝑟𝐷1
𝑚22 −𝑚2 − (𝑁 + 𝑛)
, 𝐺3 =−𝐺𝑚𝑍2
𝑚42 −𝑚4 − (𝑁 + 𝑛)
, 𝐺4 =𝐴𝑚1𝑃2 − 𝐺𝑚𝑍1
𝑚12 −𝑚1 − (𝑁 + 𝑛)
,
𝐺5 =𝐴𝑚3𝑃3
𝑚32 −𝑚3 − (𝑁 + 𝑛)
, 𝐺6 = −(1 + 𝐺1 + 𝐺2 + 𝐺3 + 𝐺4 + 𝐺5) ,
-
Mathematical analysis for optically thin Radiating/ Chemically
reacting fluid 1795
𝐿1 =−𝐺𝑟𝑅17
𝑚52 −𝑚5 − (𝑁 + 𝑛)
, 𝐿2 =𝐴𝑚2𝐽1 − 𝐺𝑟𝑅1
𝑚22 −𝑚2 − (𝑁 + 𝑛)
, 𝐿3
=2𝐴𝑚3𝐽2 − 𝐺𝑟𝑅2
4𝑚32 − 2𝑚3 − (𝑁 + 𝑛)
,
𝐿4 =2𝐴𝑚2𝐽3 − 𝐺𝑟𝑅3
4𝑚22 − 2𝑚2 − (𝑁 + 𝑛)
, 𝐿5 =2𝐴𝑚1𝐽4 − 𝐺𝑟𝑅4
4𝑚12 − 2𝑚1 − (𝑁 + 𝑛)
,
𝐿6 =𝐴(𝑚2 +𝑚3)𝐽5 − 𝐺𝑟𝑅5
(𝑚2 +𝑚3)2 − (𝑚2 +𝑚3) − (𝑁 + 𝑛) , 𝐿7
=𝐴(𝑚1 +𝑚2)𝐽6 − 𝐺𝑟𝑅6
(𝑚1 +𝑚2)2 − (𝑚1 +𝑚2) − (𝑁 + 𝑛),
𝐿8 =𝐴(𝑚1 +𝑚3)𝐽7 − 𝐺𝑟𝑅7
(𝑚1 +𝑚3)2 − (𝑚1 +𝑚3) − (𝑁 + 𝑛), 𝐿9
=−𝐺𝑟𝑅8
(𝑚3 +𝑚6)2 − (𝑚3 +𝑚6) − (𝑁 + 𝑛) ,
𝐿10 =−𝐺𝑟𝑅9
(𝑚3 +𝑚5)2 − (𝑚3 +𝑚5) − (𝑁 + 𝑛) ,
𝐿11 =−𝐺𝑟𝑅10
(𝑚3 +𝑚4)2 − (𝑚3 +𝑚4) − (𝑁 + 𝑛) ,
𝐿12 =−𝐺𝑟𝑅11
(𝑚2 +𝑚6)2 − (𝑚2 +𝑚6) − (𝑁 + 𝑛) ,
𝐿13 =−𝐺𝑟𝑅12
(𝑚2 +𝑚5)2 − (𝑚2 +𝑚5) − (𝑁 + 𝑛) ,
𝐿14 =−𝐺𝑟𝑅13
(𝑚2 +𝑚4)2 − (𝑚2 +𝑚4) − (𝑁 + 𝑛) ,
𝐿15 =−𝐺𝑟𝑅14
(𝑚1 +𝑚6)2 − (𝑚1 +𝑚6) − (𝑁 + 𝑛) ,
𝐿16 =−𝐺𝑟𝑅15
(𝑚1 +𝑚5)2 − (𝑚1 +𝑚5) − (𝑁 + 𝑛) ,
𝐿18 =𝐴𝑚3𝐽8
𝑚32 −𝑚3 − (𝑁 + 𝑛)
, 𝐿17 =−𝐺𝑟𝑅16
(𝑚1 +𝑚4)2 − (𝑚1 +𝑚4) − (𝑁 + 𝑛) ,
𝐿19 = −(1 + 𝐿1 + 𝐿2 + 𝐿3 + 𝐿4 + 𝐿5 + 𝐿6 + 𝐿7 + 𝐿8 + 𝐿9 + 𝐿10+𝐿11
+ 𝐿12 + 𝐿13 + 𝐿14 + 𝐿15 + 𝐿16 + 𝐿17 + 𝐿18
) ,
𝑆1 =−𝑃𝑟𝑚3
2𝑃32
4𝑚32 − 2𝑃𝑟𝑚3 − 𝑅2
, 𝑆2 =−𝑃𝑟𝑚2
2𝑃12
4𝑚22 − 2𝑃𝑟𝑚2 − 𝑅2
, 𝑆3 =−𝑃𝑟𝑚1
2𝑃22
4𝑚12 − 2𝑃𝑟𝑚1 − 𝑅2
,
-
1796 Nava Jyoti Hazarika and Sahin Ahmed
𝑆4 =−2𝑃𝑟𝑚2𝑚3𝑃3𝑃1
(𝑚2 +𝑚3)2 − 𝑃𝑟(𝑚2 +𝑚3) − 𝑅2 , 𝑆5
=−2𝑃𝑟𝑚1𝑚2𝑃1𝑃2
(𝑚1 +𝑚2)2 − 𝑃𝑟(𝑚1 +𝑚2) − 𝑅2 ,
𝑆6 =−2𝑃𝑟𝑚3𝑚1𝑃2𝑃3
(𝑚1 +𝑚3)2 − 𝑃𝑟(𝑚1 +𝑚3) − 𝑅2 , 𝑆7 = −(𝑆1 + 𝑆2 + 𝑆3 + 𝑆4 + 𝑆5 +
𝑆6),
𝐷1 =𝑃𝑟𝐴𝑚2
𝑚22 − 𝑃𝑟𝑚2 − 𝑁1
, 𝐷2 = 1 − 𝐷1, 𝑅1 =𝑃𝑟𝐴𝑚2𝑆7
𝑚22 − 𝑃𝑟𝑚2 − 𝑁1
,
𝑅2 =2𝑃𝑟𝐴𝑚3𝑆1 − 2𝑃𝑟𝑚3
2𝐺5𝑃3
4𝑚32 − 2𝑃𝑟𝑚3 − 𝑁1
, 𝑅3 =2𝑃𝑟𝐴𝑚2𝑆2 − 2𝑃𝑟𝑚2
2𝐺2𝑃1
4𝑚22 − 2𝑃𝑟𝑚2 − 𝑁1
,
𝑅4 =2𝑃𝑟𝐴𝑚1𝑆3 − 2𝑃𝑟𝑚1
2𝐺4𝑃2
4𝑚12 − 2𝑃𝑟𝑚1 − 𝑁1
, 𝑅5
=𝑃𝑟𝐴(𝑚2 +𝑚3)𝑆4 − 2𝑃𝑟𝑚2𝑚3(𝐺2𝑃3 + 𝐺5𝑃1)
(𝑚2 +𝑚3)2 − 𝑃𝑟(𝑚2 +𝑚3) − 𝑁1 ,
𝑅6 =𝑃𝑟𝐴(𝑚1 +𝑚2)𝑆5 − 2𝑃𝑟𝑚1𝑚2(𝐺4𝑃1 + 𝐺2𝑃2)
(𝑚1 +𝑚2)2 − 𝑃𝑟(𝑚1 +𝑚2) − 𝑁1 ,
𝑅7 =𝑃𝑟𝐴(𝑚1 +𝑚3)𝑆6 − 2𝑃𝑟𝑚3𝑚1(𝐺4𝑃3 + 𝐺5𝑃2)
(𝑚1 +𝑚3)2 − 𝑃𝑟(𝑚1 +𝑚3) − 𝑁1 ,
𝑅8 =2𝑃𝑟𝑚3𝑚6𝐺6𝑃3
(𝑚3 +𝑚6)2 − 𝑃𝑟(𝑚3 +𝑚6) − 𝑁1 , 𝑅9 =
2𝑃𝑟𝑚3𝑚5𝐺1𝑃3(𝑚3 +𝑚5)2 − 𝑃𝑟(𝑚3 +𝑚5) − 𝑁1
,
𝑅10 =2𝑃𝑟𝑚3𝑚4𝐺3𝑃3
(𝑚3 +𝑚4)2 − 𝑃𝑟(𝑚3 +𝑚4) − 𝑁1 , 𝑅11
=2𝑃𝑟𝑚2𝑚6𝐺6𝑃1
(𝑚2 +𝑚6)2 − 𝑃𝑟(𝑚2 +𝑚6) − 𝑁1 ,
𝑅12 =2𝑃𝑟𝑚2𝑚5𝐺1𝑃1
(𝑚2 +𝑚5)2 − 𝑃𝑟(𝑚2 +𝑚5) − 𝑁1 , 𝑅13
=2𝑃𝑟𝑚2𝑚4𝐺3𝑃1
(𝑚2 +𝑚4)2 − 𝑃𝑟(𝑚2 +𝑚4) − 𝑁1,
𝑅14 =2𝑃𝑟𝑚1𝑚6𝐺6𝑃2
(𝑚1 +𝑚6)2 − 𝑃𝑟(𝑚1 +𝑚6) − 𝑁1 , 𝑅15
=2𝑃𝑟𝑚1𝑚5𝐺1𝑃2
(𝑚1 +𝑚5)2 − 𝑃𝑟(𝑚1 +𝑚5) − 𝑁1
𝑅16 =2𝑃𝑟𝑚1𝑚4𝐺3𝑃2
(𝑚1 +𝑚4)2 − 𝑃𝑟(𝑚1 +𝑚4) − 𝑁1 ,
𝑅17 = −(𝑅1 + 𝑅2 + 𝑅3 + 𝑅4 + 𝑅5 + 𝑅6 + 𝑅7 + 𝑅8 + 𝑅9
+𝑅10 + 𝑅11 + 𝑅12 + 𝑅13 + 𝑅14 + 𝑅15) ,
-
Mathematical analysis for optically thin Radiating/ Chemically
reacting fluid 1797
𝑍1 =𝐴𝑚1𝑆𝑐
𝑚12 − 𝑆𝑐 𝑚1 − 𝑆𝑐(𝑛 + 𝐶𝑟)
, 𝑍2 = 1 − 𝑍1.
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