Mathematical Analysis for Optically Thin Radiating ...Mathematical analysis for optically thin...
Transcript of Mathematical Analysis for Optically Thin Radiating ...Mathematical analysis for optically thin...
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Global Journal of Pure and Applied Mathematics.
ISSN 0973-1768 Volume 13, Number 6 (2017), pp. 1777-1798
Β© 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 π₯-axis is taken in the upward direction along the plate and π¦-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 π¦ and π‘ only. Now, under the usual Boussinesqβs approximation, the governing boundary layer equations are:
ππ£
ππ¦= 0 (1)
ππ’
ππ‘+ π£
ππ£
ππ¦= β
1
π
ππ
ππ₯+ π
π2π’
ππ¦2 + ππ½π(π β πβ) + ππ½πΆ(πΆ β πΆβ) β (
π
π +ππ΅0
2
π)π’ (2)
ππ
ππ‘+ π£
ππ
ππ¦=
π
πππ[π2π
ππ¦2 β
1
π
ππ
ππ¦] +
π
ππ(ππ’
ππ¦)
2
(3)
π2π
ππ¦2 β 3πΌ
2π β 16πβπΌπβ3 ππ
ππ¦= 0 (4)
ππΆ
ππ‘+ π£
ππΆ
ππ¦= π·
π2πΆ
ππ¦2 β πΆπ(πΆ β πΆβ) (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 π satisfies this non-linear differential equation.
The boundary conditions for the velocity, temperature and concentration fields are:
{π’ = π’π, π = ππ€ + π(ππ€ β πβ)π
ππ‘, πΆ = πΆπ€ + π(πΆπ€ β πΆβ)πππ‘ ππ‘ π¦ = 0
π’ = πβ = π0(1 + ππππ‘), π βΆ πβ , πΆ βΆ πΆβ ππ π¦ βΆ β
} (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:
π£ = βπ0(1 + ππ΄πππ‘) (7)
The negative sign indicates that the suction is towards the plate.
Outside the boundary layer, Eq. (2) gives:
β1
π
ππ
ππ₯= ππβ
ππ‘+π
π πβ +
π
π π΅0
2πβ (8)
Since the medium is optically thin with relatively low density and πΌ βͺ 1, the radiative heat flux given by Eq. (3), in the spirit of Cogley et al. [22] becomes:
ππ
ππ¦= 4πΌ2 (π β πβ) where πΌ
2 = β« πΏπππ΅
ππ
β
0
, (9)
where B is Planckβs 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
{
π’ =
π’
π0 , π£ =
π£
π0, π¦ =
π0π¦
π , πβ =
πβπ0
,
ππ =π’π
π0, π‘ =
π‘ π02
π ,
π =π β πβ
ππ€ β πβ , π =
πΆ β πΆβ
πΆπ€ β πΆβ , π =
π π
π02 , πΎ =
πΎ π02
π2 , πΆπ =
ππΆπ
π02
ππ = πππΆπ
π, ππ =
π
π· , π =
ππ΅02π
ππ02 , πΊπ =
ππ½ππ(ππ€ β πβ)
π0π02
,
πΊπ =ππ½πΆπ(πΆπ€ β πΆβ)
π0π02 , πΈπ =
π02
πΆπ(ππ€ β πβ), π 2 =
πΌ2(ππ€ β πβ)
ππΆπππ02
,
}
(10)
In view of Eqs. (4) and (7) β(10), Eqs. (2), (3) and (5) reduce to the following
dimensionless form:
ππ’
ππ‘β (1 + ππ΄πππ‘)
ππ’
ππ¦=ππβππ‘
+π2π’
ππ¦2+ πΊππ + πΊππ + π(πβ β π’) (11)
ππ
ππ‘β (1 + ππ΄πππ‘)
ππ
ππ¦=
1
ππ[π2π
ππ¦2β π 2π] + πΈπ (
ππ’
ππ¦)2
(12)
ππ
ππ‘β (1 + ππ΄πππ‘)
ππ
ππ¦=1
ππ
π2π
ππ¦2β πΆππ (13)
where π = π + πΎβ1
The corresponding dimensionless boundary conditions are:
{π’ = ππ , π = 1 + ππ
ππ‘, π = 1 + ππππ‘, ππ‘ π¦ = 0
π’ = πβ = 1 + ππππ‘, π βΆ 0, π βΆ 0 ππ π¦ βΆ β
} (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:
{
π’(π¦, π‘) = π’0(π¦) + ππππ‘π’1(π¦) + 0(π
2) + β―
π(π¦, π‘) = π0(π¦) + ππππ‘π1(π¦) + 0(π
2) + β―
π(π¦, π‘) = π0(π¦) + ππππ‘π1(π¦) + 0(π
2) + β―
} (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(π2), we obtain:
π’0β³(π¦) + π’0
β² (π¦) β ππ’0(π¦) = βπ β πΊππ0(π¦) β πΊππ0(π¦) (16)
π’1β³(π¦) + π’1
β² (π¦) β (π + π)π’1(π¦)
= β(π + π) β π΄π’0β² (π¦) β πΊππ1(π¦) β πΊππ1(π¦) (17)
π0β³(π¦) + ππ π0
β²(π¦) β π 2 π0(π¦) = βπππΈπ [π’0β² (π¦)]2 (18)
π1β³(π¦) + ππ π1
β²(π¦) β (π 2 + πππ)π1(π¦) = βπππ΄ π0β²(π¦) β 2πππΈπ π’0
β² (π¦)π’1β²(π¦) (19)
π0β³(π¦) + ππ π0
β² (π¦) β ππ πΆππ0(π¦) = 0 (20)
π1β³(π¦) + ππ π1
β² (π¦) β ππ(π + πΆπ)π1(π¦) = βπ΄ππ π0β² (π¦) (21)
where prime denotes ordinary differentiation with respect to y.
The corresponding boundary conditions can be written as:
{π’0 = ππ, π’1 = 0, π0 = 1, π1 = 1, π0 = 1, π1 = 1 ππ‘ π¦ = 0
π’0 = 1, π’1 = 1, π0 βΆ 0, π1 βΆ 0, π0 βΆ 0, π1 βΆ 0 ππ π¦ βΆ β } (22)
The Eqs. (16) β (21) are still coupled and non-linear, whose exact solutions are not
possible. So we expand π’0 , π’1 , π0 , π1 , π0 , π1 in terms of πΈπ in the following form, as the Eckert number is very small for incompressible flows.
πΉ(π¦) = πΉ0(π¦) + πΈπ πΉ1(π¦) + 0(πΈπ2) (23)
where πΉ stands for any π’0 , π’1 , π0 , π1 , π0 , π1 .
Substituting Eq. (23) in Eqs. (16) β (21), equating the co-efficient of πΈπ to zero and neglecting the terms in πΈπ2 and higher order, we get the following equations:
The zeroth order equations are:
π’01β³ (π¦) + π’01
β² (π¦) β ππ’01(π¦) = βπ β πΊπ π01(π¦) β πΊπ π01(π¦) (24)
π’02β³ (π¦) + π’02
β² (π¦) β ππ’02(π¦) = βπΊπ π02(π¦) β πΊπ π02(π¦) (25)
π01β³ (π¦) + ππ π01
β² (π¦) β π 2 π01(π¦) = 0 (26)
π02β³ (π¦) + ππ π02
β² (π¦) β π 2 π02(π¦) = βππ[π’01β² (π¦)]2 (27)
π01β³ (π¦) + ππ π01
β² (π¦) β ππ πΆπ π01(π¦) = 0 (28)
π02β³ (π¦) + ππ π02
β² (π¦) β ππ πΆπ π02(π¦) = 0 (29)
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1784 Nava Jyoti Hazarika and Sahin Ahmed
and the respective boundary conditions are:
{π’01 = ππ, π’02 = 0, π01 = 1, π02 = 0, π01 = 1, π02 = 0 ππ‘ π¦ = 0
π’01 βΆ 1, π’02 βΆ 0, π01 βΆ 0, π02 βΆ 0,π01 βΆ 0, π02 βΆ 0 ππ‘ π¦ βΆ β} (30)
The first order equations are:
π’11β³ (π¦) + π’11
β² (π¦) β (π + π)π’11(π¦) = {β(π + π) β πΊπ π11(π¦)
βπΊπ π11(π¦) β π΄ π’01β² (π¦)
} (31)
π’12β³ (π¦) + π’12
β² (π¦) β (π + π)π’12(π¦) = βπΊπ π12(π¦) β πΊπ π12(π¦) β π΄ π’02β² (π¦) (32)
π11β³ (π¦) + ππ π11
β² (π¦) β π1π11(π¦) = βπππ΄ π01β² (π¦) (33)
π12β³ (π¦) + ππ π12
β² (π¦) β π1π12(π¦) = βπππ΄ π02β² (π¦) β 2ππ π’01
β² (π¦)π’11β² (π¦) (34)
π11β³ (π¦) + ππ π11
β² (π¦) β ππ(π + πΆπ)π11(π¦) = βπ΄ππ π01β² (π¦) (35)
π12β³ (π¦) + ππ π12
β² (π¦) β ππ(π + πΆπ)π12(π¦) = βπ΄ππ π02β² (π¦) (36)
where π1 = π 2 + πππ.
and respective boundary conditions are:
{ π’11 = 0, π’12 = 0, π11 = 1, π12 = 0, π11 = 1, π12 = 0 ππ‘ π¦ = 0
π’11 βΆ 1, π’12 βΆ 0, π11 βΆ 0, π12 βΆ 0, π11 βΆ 0, π12 βΆ 0 ππ‘ π¦ βΆ β } (37)
Solving Eqs. (24) β (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:
π’(π¦, π‘) =
{
π3πβπ3π¦ + π1π
βπ2π¦ + π2πβπ1π¦ + 1
+πΈπ { π½8π
βπ3π¦ + π½1πβπ2π¦ + π½2π
β2π3π¦ + π½3πβ2π2π¦ + π½4π
β2π1π¦
+π½5πβ(π2+π3)π¦ + π½6π
β(π1+π2)π¦ + π½7πβ(π1+π3)π¦
}
+ ππππ‘
[ {
πΊ6πβπ6π¦ + πΊ1π
βπ5π¦ + πΊ2πβπ2π¦ + πΊ3π
βπ4π¦
+πΊ4πβπ1π¦ + πΊ5π
βπ3π¦ + 1}
+πΈπ
{
πΏ19πβπ6π¦ + πΏ1π
βπ5π¦ + πΏ2πβπ2π¦ + πΏ3π
β2π3π¦
+πΏ4πβ2π2π¦ + πΏ5π
β2π1π¦ + πΏ6πβ(π2+π3)π¦ + πΏ7π
β(π1+π2)π¦
+πΏ8πβ(π1+π3)π¦ + πΏ9π
β(π3+π6)π¦ + πΏ10πβ(π3+π5)π¦
+πΏ11πβ(π3+π4)π¦ + πΏ12π
β(π2+π6)π¦ + πΏ13πβ(π2+π5)π¦
+πΏ14πβ(π2+π4)π¦ + πΏ15π
β(π1+π6)π¦ + πΏ16πβ(π1+π5)π¦
+πΏ17πβ(π1+π4)π¦ + πΏ18π
βπ3π¦ }
]
}
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Mathematical analysis for optically thin Radiating/ Chemically reacting fluid 1785
π(π¦, π‘) =
{
πβπ2π¦ + πΈπ {
π7πβπ2π¦ + π1π
β2π3π¦ + π2πβ2π2π¦ + π3π
β2π1π¦
+π4πβ(π2+π3)π¦ + π5π
β(π1+π2)π¦ + π6πβ(π1+π3)π¦
}
+ππππ‘
[
{π·2πβπ5π¦ + π·1π
βπ2π¦} +
πΈπ
{
π 17πβπ5π¦ + π 1π
βπ2π¦ + π 2πβ2π3π¦ + π 3π
β2π2π¦
+π 4πβ2π1π¦ + π 5π
β(π2+π3)π¦ + π 6πβ(π1+π2)π¦
+π 7πβ(π1+π3)π¦ + π 8π
β(π3+π6)π¦ + π 9πβ(π3+π5)π¦
+π 10πβ(π3+π4)π¦ + π 11π
β(π2+π6)π¦ +
π 12πβ(π2+π5)π¦ + π 13π
β(π2+π4)π¦ + π 14πβ(π1+π6)π¦
+π 15πβ(π1+π5)π¦ + π 16π
β(π1+π4)π¦ }
]
}
π(π¦, π‘) = πβπ1π¦ + ππππ‘{π2πβπ4π¦ + π1π
βπ1π¦}
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:
πΆπ =ππ€
ππ0π0= (
ππ’
ππ¦ )π¦=0
= ( ππ’0ππ¦
+ ππππ‘ππ’1ππ¦ )π¦=0
=
[ βπ3π3 βπ2π1 βπ1π2 + πΈπ {
βπ3π½8 βπ2π½1 β 2π3π½2 β 2π2π½3 β 2π1π½4 β(π2 +π3)π½5 β (π1 +π2)π½6 β (π1 +π3)π½7
}
+ππππ‘
[
(βπ6πΊ6 βπ5πΊ1 βπ2πΊ2 βπ4πΊ3 βπ1πΊ4 βπ3πΊ5)
+πΈπ
{
βπ6πΏ19 βπ5πΏ1 βπ2πΏ2 β 2π3πΏ3 β 2π2πΏ4 β 2π1πΏ5 β(π2 +π3)πΏ6 β (π1 +π2)πΏ7 β (π1 +π3)πΏ8
β(π3 +π6)πΏ9 β (π3 +π5)πΏ10 β (π3 +π4)πΏ11β(π2 +π6)πΏ12 β (π2 +π5)πΏ13 β (π2 +π4)πΏ14
β(π1 +π6)πΏ15 β (π1 +π5)πΏ16 β (π1 +π4)πΏ17 βπ3πΏ18}
]
]
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:
ππ’ = βπ₯
(ππππ¦β)π¦=0
ππ€ β πβ= ππ’π ππ₯
β1 = β( ππ
ππ¦ )π¦=0
= β( ππ0ππ¦
+ ππππ‘ππ1ππ¦ )π¦=0
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1786 Nava Jyoti Hazarika and Sahin Ahmed
= β
[ βπ2 + πΈπ {
βπ2π7 β 2π3π1 β 2π2π2 β 2π1π3 β (π2 +π3)π4β(π1 +π2)π5 β (π1 +π3)π6
}
+ππππ‘
[
(βπ5π·2 βπ2π·1)
+πΈπ
{
βπ5π 17 βπ2π 1 β 2π3π 2 β 2π2π 3 β 2π1π 4β(π2 +π3)π 5 β (π1 +π2)π 6 β (π1 +π3)π 7β(π3 +π6)π 8 β (π3 +π5)π 9 β (π3 +π4)π 10β(π2 +π6)π 11 β (π2 +π5)π 12 β (π2 +π4)π 13β(π1 +π6)π 14 β (π1 +π5)π 15 β (π1 +π4)π 16}
]
]
where π ππ₯ =π0π₯
π 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:
πβ = βπ₯
(ππΆππ¦β )
π¦=0
πΆπ€ β πΆβ ,
πβπ ππ₯β1 = β(
ππΆ
ππ¦ )π¦=0
= β( ππΆ0ππ¦
+ ππππ‘ππΆ1ππ¦ )π¦=0
= β[βπ1 + ππππ‘(βπ4π2 βπ1π1)]
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, Τ=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 Σ¨, 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, Τ=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 (Σ¨) 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, Τ=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
π’ , π£ Velocity components in π₯ , π¦ directions respectively,
π‘ Time,
π Pressure,
π Acceleration due to gravity,
π Permeability of porous medium,
π Temperature of the fluid in the boundary layer,
πβ Temperature of the fluid far away from the plate,
πΆ Species concentration in the boundary layer,
πΆβ Species concentration in the fluid far away from the plate,
π΅π Magnetic induction,
ππ Specific heat at constant pressure,
π Thermal conductivity,
π Radiative heat flux,
πβ Stefan-Boltzmann constant,
D Mass diffusivity and
πΆπ Chemical reaction.
π’π Plate velocity,
ππ€ Temperature of the plate,
πΆπ€ Concentration of the plate,
πβ Free stream velocity,
π0 Constant,
π Constant
-
1794 Nava Jyoti Hazarika and Sahin Ahmed
A Real positive constant
π0 Non-zero positive constant
GREEK SYMBOL
π Density,
π½π Thermal expansion co-efficient,
π½πΆ Concentration expansion co-efficient,
π Kinematic viscosity,
π Electrical conductivity of the fluid,
πΌ Fluid thermal diffusivity,
Τ small such that Τ βͺ 1
APPENDIX
π1 =ππ + β1 + 4πππΆπ
2, π2 =
ππ + βππ2 + 4π 2
2, π3 =
1 + β1 + 4π
2 ,
π4 =ππ + βππ2 + 4ππ(π + πΆπ)
2 ,π5 =
ππ + βππ2 + 4π12
,π6
=1 + β1 + 4(π + π)
2 ,
π1 =βπΊπ
π22 βπ2 β π
, π2 =βπΊπ
π12 βπ1 β π
, π3 = ππ β 1 β π1 β π2 ,
π½1 =βπΊππ7
π22 βπ2 βπ
, π½2 =βπΊππ1
4π32 β 2π3 β π
, π½3 =βπΊππ2
4π22 β 2π2 βπ
,
π½4 =βπΊππ3
4π12 β 2π1 β π
, π½5 =βπΊππ4
(π2 +π3)2 β (π2 +π3) β π ,
π½6 =βπΊππ5
(π1 +π2)2 β (π1 +π2) β π , π½7 =
βπΊππ6(π1 +π3)2 β (π1 +π3) β π
,
π½8 = β(π½1 + π½2 + π½3 + π½4 + π½5 + π½6 + π½7), πΊ1 =βπΊππ·2
π52 βπ5 β (π + π)
,
πΊ2 =π΄π2π1 β πΊππ·1
π22 βπ2 β (π + π)
, πΊ3 =βπΊππ2
π42 βπ4 β (π + π)
, πΊ4 =π΄π1π2 β πΊππ1
π12 βπ1 β (π + π)
,
πΊ5 =π΄π3π3
π32 βπ3 β (π + π)
, πΊ6 = β(1 + πΊ1 + πΊ2 + πΊ3 + πΊ4 + πΊ5) ,
-
Mathematical analysis for optically thin Radiating/ Chemically reacting fluid 1795
πΏ1 =βπΊππ 17
π52 βπ5 β (π + π)
, πΏ2 =π΄π2π½1 β πΊππ 1
π22 βπ2 β (π + π)
, πΏ3
=2π΄π3π½2 β πΊππ 2
4π32 β 2π3 β (π + π)
,
πΏ4 =2π΄π2π½3 β πΊππ 3
4π22 β 2π2 β (π + π)
, πΏ5 =2π΄π1π½4 β πΊππ 4
4π12 β 2π1 β (π + π)
,
πΏ6 =π΄(π2 +π3)π½5 β πΊππ 5
(π2 +π3)2 β (π2 +π3) β (π + π) , πΏ7
=π΄(π1 +π2)π½6 β πΊππ 6
(π1 +π2)2 β (π1 +π2) β (π + π),
πΏ8 =π΄(π1 +π3)π½7 β πΊππ 7
(π1 +π3)2 β (π1 +π3) β (π + π), πΏ9
=βπΊππ 8
(π3 +π6)2 β (π3 +π6) β (π + π) ,
πΏ10 =βπΊππ 9
(π3 +π5)2 β (π3 +π5) β (π + π) ,
πΏ11 =βπΊππ 10
(π3 +π4)2 β (π3 +π4) β (π + π) ,
πΏ12 =βπΊππ 11
(π2 +π6)2 β (π2 +π6) β (π + π) ,
πΏ13 =βπΊππ 12
(π2 +π5)2 β (π2 +π5) β (π + π) ,
πΏ14 =βπΊππ 13
(π2 +π4)2 β (π2 +π4) β (π + π) ,
πΏ15 =βπΊππ 14
(π1 +π6)2 β (π1 +π6) β (π + π) ,
πΏ16 =βπΊππ 15
(π1 +π5)2 β (π1 +π5) β (π + π) ,
πΏ18 =π΄π3π½8
π32 βπ3 β (π + π)
, πΏ17 =βπΊππ 16
(π1 +π4)2 β (π1 +π4) β (π + π) ,
πΏ19 = β(1 + πΏ1 + πΏ2 + πΏ3 + πΏ4 + πΏ5 + πΏ6 + πΏ7 + πΏ8 + πΏ9 + πΏ10+πΏ11 + πΏ12 + πΏ13 + πΏ14 + πΏ15 + πΏ16 + πΏ17 + πΏ18
) ,
π1 =βπππ3
2π32
4π32 β 2πππ3 β π 2
, π2 =βπππ2
2π12
4π22 β 2πππ2 β π 2
, π3 =βπππ1
2π22
4π12 β 2πππ1 β π 2
,
-
1796 Nava Jyoti Hazarika and Sahin Ahmed
π4 =β2πππ2π3π3π1
(π2 +π3)2 β ππ(π2 +π3) β π 2 , π5
=β2πππ1π2π1π2
(π1 +π2)2 β ππ(π1 +π2) β π 2 ,
π6 =β2πππ3π1π2π3
(π1 +π3)2 β ππ(π1 +π3) β π 2 , π7 = β(π1 + π2 + π3 + π4 + π5 + π6),
π·1 =πππ΄π2
π22 β πππ2 β π1
, π·2 = 1 β π·1, π 1 =πππ΄π2π7
π22 β πππ2 β π1
,
π 2 =2πππ΄π3π1 β 2πππ3
2πΊ5π3
4π32 β 2πππ3 β π1
, π 3 =2πππ΄π2π2 β 2πππ2
2πΊ2π1
4π22 β 2πππ2 β π1
,
π 4 =2πππ΄π1π3 β 2πππ1
2πΊ4π2
4π12 β 2πππ1 β π1
, π 5
=πππ΄(π2 +π3)π4 β 2πππ2π3(πΊ2π3 + πΊ5π1)
(π2 +π3)2 β ππ(π2 +π3) β π1 ,
π 6 =πππ΄(π1 +π2)π5 β 2πππ1π2(πΊ4π1 + πΊ2π2)
(π1 +π2)2 β ππ(π1 +π2) β π1 ,
π 7 =πππ΄(π1 +π3)π6 β 2πππ3π1(πΊ4π3 + πΊ5π2)
(π1 +π3)2 β ππ(π1 +π3) β π1 ,
π 8 =2πππ3π6πΊ6π3
(π3 +π6)2 β ππ(π3 +π6) β π1 , π 9 =
2πππ3π5πΊ1π3(π3 +π5)2 β ππ(π3 +π5) β π1
,
π 10 =2πππ3π4πΊ3π3
(π3 +π4)2 β ππ(π3 +π4) β π1 , π 11
=2πππ2π6πΊ6π1
(π2 +π6)2 β ππ(π2 +π6) β π1 ,
π 12 =2πππ2π5πΊ1π1
(π2 +π5)2 β ππ(π2 +π5) β π1 , π 13
=2πππ2π4πΊ3π1
(π2 +π4)2 β ππ(π2 +π4) β π1,
π 14 =2πππ1π6πΊ6π2
(π1 +π6)2 β ππ(π1 +π6) β π1 , π 15
=2πππ1π5πΊ1π2
(π1 +π5)2 β ππ(π1 +π5) β π1
π 16 =2πππ1π4πΊ3π2
(π1 +π4)2 β ππ(π1 +π4) β π1 ,
π 17 = β(π 1 + π 2 + π 3 + π 4 + π 5 + π 6 + π 7 + π 8 + π 9
+π 10 + π 11 + π 12 + π 13 + π 14 + π 15) ,
-
Mathematical analysis for optically thin Radiating/ Chemically reacting fluid 1797
π1 =π΄π1ππ
π12 β ππ π1 β ππ(π + πΆπ)
, π2 = 1 β π1.
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