International Journal of Engineering Trends and Technology (IJETT) – Volume 10 Number 5 - Apr 2014
Thermophoresis and Chemical Reaction Effects on MHD DarcyForchheimer Mixed Convection in a Fluid Saturated Porous Media
N. Kishan1 and S. Jagadha2
1
2
Department of Mathematics, University College of Science, Osmania University, Hyderabad,A.P.,India
Department of Mathematics, Hyderabad Institute of Technology and Management, Medchal, R.R.District,A.P.India,
Abstract - The purpose of this work is to study the magnetic
hydrodynamic flow of mixed convection, heat and mass transfer
about an isothermal vertical flat plate embedded in a fluidsaturated porous medium and the effects of chemical reaction and
thermophoresis in both aiding and opposing flows. The governing
partial differential equations are converted into ordinary
differential equations by similarity transformations. The coupled
nonlinear ordinary differential equations are linearized by using
Quasi-linearization technique and implicit finite difference scheme
is used to solve these equations. The numerical computations are
carried out for the different physical parameters such as
thermophoretic, mixed convection, inertia parameter, buoyancy
ratio, Schmid number on the flow, and chemical reaction ,heat and
mass transfer characterized.
Keywords: MHD, chemical reaction, Porous media, Mixed
convection, Thermophoresis, Finite difference method
1.INTRODUCTION
Over the past two decades, studies in aerosol particle deposition
due to thermophoreis have gained importance for engineering
applications. The technological problem include particle
deposition onto wafers in the microelectronics industry, particle
surfaces produced by condensing vapor gas mixtures, particles
impacting the blade surface of gas turbines and others such as
filtration in gas cleaning and nuclear reactor safety.
Thermophoresis is the term describing the fact that small micron
sized particles suspended in a non-isothermal gas will acquire a
velocity in the direction of decreasing temperature. The gas
molecules coming from the hot side of the particles have a
greater velocity than those coming from the cold side. The
faster moving molecules collide with the particles more
forcefully. This difference in momentum leads to the particle
developing a velocity in the direction of the cooler temperature.
The velocity acquired by the particles is called the
thermophoretic velocity and the force experienced by the
suspended particles due to the temperature gradient is known as
the thermophoretic force. The magnitudes of the thermophoretic
force and velocity are proportional to the temperature gradient
and depend on many factors like thermal conductivity of
aerosol particles and carrier gas. Thermophoresis causes small
particles to deposit on cold surfaces. Thermophoresis principle
is utilized to manufacture graded index silicondioxide and
ISSN: 2231-5381
germanium dioxide optical fiber performs used in the field of
communications. In engineering particle, usually more than one
mechanism can act simultaneously and their interactions need to
be considered for accurate prediction of deposition rates. The
use of thermophoretic heaters has led to a reduction in chip
failures. In the same vein there is the potential application of
thermophoresis to remove radioactive aerosols from
containment domes in the event of a nuclear reactor accident.
In light of these various applications Goren [1] was one of the
first to study the role of thermophoresis in the laminar flow of a
viscous and incompressible fluid. He used the classical problem
of flow over a flat plate to calculate deposition rates and showed
that substantial changes in surfaces deposition can be obtained
by increasing the difference between the surface and free stream
temperatures. Chamkha and Pop [2] studied the effect of
thermophoretic particle deposition in free convection boundary
layer from a vertical flat plate. Coupled Heat and Mass Transfer
in Darcy-Forchheimer mixed convection from a vertical flat
plate embedded in a fluid-saturated porous medium under the
effects of radiation and viscous dissipation have also been
studied Salem.A.M. [3]. Seeddek[4], studied the effect of
viscous dissipation and thermophoresis on Darcy-Forchheimer
mixed convection flow, heat and mass transfer about an
isothermal vertical flat plate embedded in a fluid saturated
porous medium in both aiding and opposing flows.
In recent years, a great deal of interest has been generated in the
area of convective heat transfer from a vertical flat plate
embedded in a porous medium because of its wide-range of
applications in various fields such as thermal insulation, the
enhanced recovery of petroleum resource and geophysical
flows. The disposal of nuclear waste into the earth’s crust or the
seabed is an area of particular interest in the study of convection
in a porous medium. Extensive studies of mixed convection in
porous media have been performed over the last five decades
covering a broad range of fields including several different
physical effects since such studies have important aplications in
various fields such as geothermal resources, thermal insulation
for structures, modeling of oil reservoir, oil extraction,
underground disposal of unclear waste, packed bed catalyst
reactors, storage of heat generating materials, food processing
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International Journal of Engineering Trends and Technology (IJETT) – Volume 10 Number 5 - Apr 2014
and casting and welding in manufacturing processes (Aly et
al.[5]. Mixed convection flows arise when, besides buoyancy,
there acts an external force, because of which there will be a
free stream or a moving boundary in such problems. In mixed
convection studies, there can arise two types of flows assisting
flow and opposing flow, depending on whether the plate is hot
or cold as compared to the ambient temperature and based on
the direction of the free stream or the moving boundary.
Aim of the present work is to investigate the effects of MHD
thermophoresis and chemical reaction on mixed flow of heat
and mass transfer in a fluid saturated porous medium. A
similarity transformation has been utilized to convert the partial
differential equations into ordinary differential equations and
then the solution of numerical problem is solved using implicit
finite difference method along with Gauss-seidel iterative
scheme wih the help of C program.
The combined heat and mass transfer problems with chemical
reactions are of importance in many processes, and therefore
have received a considerable amount of attention in recent years.
In processes, such as drying, evaporation at the surface of a
water body, energy transfer in a wet cooling tower and the flow
in a desert cooler, the heat and mass transfer occurs
simultaneously. Chemical reactions can be codified as either
homogeneous or heterogeneous processes. A homogeneous
reaction is one that occurs uniformly through a given phase. In
contrast, a heterogeneous reaction takes place in a restricted
region or within the boundary of a phase. A reaction is said to be
the first order if the rate of reaction is directly proportional to
the concentration itself. In many chemical engineering
processes, a chemical reaction between a foreign mass and the
fluid does occur. These processes take place in numerous
industrial applications, such as the polymer production, the
manufacturing of ceramics or glassware, the food processing [6]
and so on. Das et al.[7] considered the effects of a first order
chemical reaction on the flow past an impulsively started
infinite vertical plate with constant heat flux and mass transfer.
2.MATHEMATICALFORMULATION
The effects of thermophoresis with chemical reaction on forced
convection flow play an important role in the context of space
technology and processes involving high temperatures. In light
of these various applications, England and Emery [8] studied the
thermal radiation effect of an optically thin gray gas bounded by
a stationary vertical plate. N.Kishan et al. studied about
thermophoresis and viscous dissipation effects on Darcy
Forchheimer MHD mixed convection in a fluid saturated
porous media[9,10].
In view of possible applications, effect of magnetic field on
different convective problems was studied by several
researchers. Sobha and Ramakrishna [11] studied free
convective heat transfer in a porous medium subjected to a
magnetic field when the plate temperature varies as a power
function of the vertical coordinate. Acharya et al [12] studied
the effect of magnetic field on free convection in a porous
medium with constant heat flux. Rao, Lakshmi Prasannam and
Raja Rani [13] studied mixed convection in a porous medium
with magnetic field, variable viscosity and varying wall
temperature.
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Consider the steady laminar hydromagnetic coupled heat and
mass transfer by mixed convection boundary layer flow over a
vertical flat plate. A uniform transverse magnetic field of
strength βo is applied parallel to the y axis. The plate is
maintained constant temperature Tw and concentration Cw
which is embedded in a fluid-saturated porous medium of
ambient temperature T∞ and concentration C∞ respectively.
The x -coordinate is measured along the plate from its leading
edge and the y -coordinate normal to it. Allowing for both
Brownian motion of particles and thermophoretic transport, the
governing boundary layer equations are:
+
1+
=0
(1)
+
+
=
+
=
= ±
(
−
−
+ )
(
(2)
(3)
) − ( − )
(4)
The boundary conditions are given by
→ 0: = 0,
→ ∞: =
=
, = ,
=
,
, = (5)
where u,v are velocity components along x,y coordinates,
respectively, T and C are, respectively, the temperature and
concentration,, Cf Forchheimer coefficient, K1 the Darcy
permeability, g is the acceleration due to gravity, ν is the
kinematic viscosity, βT the coefficient of tthermal expansion, βc
the coefficient concentration expansion, cp is the specific heat
of the fluid at constant pressure, qr the radiative heat flux, and
D is the mass diffusivity. In Eq.(2) the plus sign corresponds to
the case where the buoyancy force has a component “aiding”
the forced flow and the minus signs refer to the “opposing” case
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International Journal of Engineering Trends and Technology (IJETT) – Volume 10 Number 5 - Apr 2014
In Eqs. (3) and (4) the thermophoretic velocity V T was given
by Talbot et al [14]
∇
=−
=−
(6)
Where k is the thermophoretic coefficient, which is given by
Batchelor [15 ]
⁄
=(
)
⁄
(7)
Where Cm , Cs are C1 constants and λg and λp are the thermal
conductivities of the fluid and diffused particles, respectively,
C1 is the Cunningham correction factor and Kn is the Knudsen
number.
Now we define the following dimensionless variables for mixed
convection
= Pe
⁄
,
( )=( −
= Pe
)/(
( ) = ( −
( ),
−
"
"
+
)
"
+2
"
(8)
"
−
(
+
)
==
⁄
=−
(0)
(13)
⃒ = 0
ℎ =(
−
)
=−
/
(0)
(14)
⃒ = 0
The governing boundary layer and thermal layer (9)-(11) with
boundary conditions (12) are coupled non-linear ordinary
differential equations.
(1 +
)
=±
=(
Applying the Quasi-linearization technique to the non-linear
equation (9) we obtain as
),
−
)/(
⁄
Where ψ is the stream function that satisfies the continuity
equation and η is the dimensionless similarity variable. With
these changes of variables, Eq. (1) is identically satisfied and
Eqs. (2) – (4) are transformed to
(1 +
The important physical quantities of our interest are the Nusselt
number Nu and Sherwood Sh . These can be defined as
follows:
+
)
(9)
=0
(10)
+ 2 λ ) " + 2 λF " =±
(
+ +
−
+
−
=0
set ⃒ − f⃒ < 10
Using an implicit finite difference scheme for the equation
(15),(10) and (11), we obtain
a [i] f[i-1]+b [i] f[i] + c[i] f[i+1]=d[i]
a1[i]θ[i-1] +b1[i]θ[i] + c1[i] θ[i+1] = d1[i]
(11)
a[i] = 1 +
The corresponding boundary conditions take the form
(0) = 0, (0) = 1,
(0) = 1,
(∞) = 1, (∞) = 0, (∞) = 0.
(12)
Where the primes denote differentiation with respect to η,
⁄ is the inertia parameter, Rax = ( g ) ×
Λ =
( − ) /
is the thermal Rayleigh number, Pex =
/
is the local Peclet number, N = ( − )/ ( − ) is
the buoyancy ratio, = − ( − )/ is the thermophoretic
parameter,
chemical reaction parameter δ = ⁄ and
Ha =
/ is the magnetic parameter.
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"
(15)
Where F is the nth iterative value of f which is a known
function and f is unknown function at (n+1)th iteration. Here we
a2[i] [i-1] +b2[i] [i] + c2[i] [i+1] = 0
"
)+2λ
+ 2 λ F1[i] – 0.5*h*2 λF2[i]
b[i] = -2* ( 1 +
+ 2 λ F1[i])
c[i] = = 1 +
+ 2 λ F1[i] + 0.5*h*2 λF2[i]
d[i] = h*h* ±
(θ +
)+2 λF2[i]F1[i]
a1[i] = 1 – τ Pr θ – 0.5*h*(0.5*f[i] – 2 τ Pr
b1[i] = -2* 1 – τ Pr θ[i] - τ Pr
[i]
c1[i] = 1- τ Prθ[i]+ 0.5*h*(0.5*f[i] – 2 τ Pr
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)
[i])
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International Journal of Engineering Trends and Technology (IJETT) – Volume 10 Number 5 - Apr 2014
d1[i] = h*h* ( –τ P rθ2 θ-τ Pr θ12)
field and therefore as magnetic parameter Ha increase, so does
the retarding force and hence the velocity profile
decreases
whereas the effective magnetic parameter Ha is to increase
temperature as well as concentration profiles with the increase
of magnetic parameter Ha.
a2[i] =( 1/Sc) – 0.5*h* ( 1/2Pr)f[i]]) –τθ1[i]
b2[i] = -2( 1/Sc) - h*h* (τ θ2[i] –δ)
c2[i] =( 1/Sc) + 0.5*h* ( 1/2Pr)f[i] –τθ1[i] ) and
Figures 2(a)–(c) demonstrated the velocity, temperature and
concentration profiles for various values of mixed convection
parameterRa ⁄Pe . Infact, whenRa ⁄Pe ≫ 1
flow is
dominated by nature of convection, while whenRa ⁄Pe ≪ 1
forced convection takes the leading role.
Thus, when
Ra ⁄Pe = 1 the effects of nature and forced convection
achieve equal importance and flow is truly under mixed
convection condition. From the figure 2(a) it is observed that
the external velocity asRa ⁄Pe > 0, due to the effect of the
buoyancy force. When the free stream and buoyancy force are
in opposite direction, the buoyancy force retard the fluid on the
boundary layer, acting somewhat like an adverse pressure
gradient. The effect of Ra ⁄Pe
leads to decrease the
temperature as well as concentration profiles is shown in figs.
2(b) and 2(c ) for Ra ⁄Pe ≫ 0.
A1[i] = 1 – τ Pr θ[i]
B1[i] =0.5*f[i] – 2 τ Pr [i]
C1[i] = =τ Pr
[i]
D1[i] = –τ P rθ2 θ-τ Pr θ12
A2[i] = 1/Sc
B2[i] = ( 1/2Pr)f[i] –τθ1[i]
C2[i] = – τ θ2[i] -δ
Here the step size taken as h = 0.05 is obtained the numerical
solution with ηmax= 3 and solved the algebraic system of
equations by using Gauss-Seidel iterative method and five
decimal accuracy is the criterion for convergency.
3. RESULTS AND DISCUSSION
The set of differential equations (9) (10) and (11) are coupled
non-linear ordinary differential equations. These equations are
linearized first, by using Quasi-linearization technique (Bellman
Kalaba)[16]. The obtained linearized set of ordinary differential
equations are solved using implicit finite difference method
through the Gauss Seidel iterative scheme with the help of C
programming.
In order to get an insight into the physical situation of the
problem we have computed the numerical values of the velocity,
temperature and concentration for the different flow parameters
such as magnetic parameter Ha, Prandtl number Pr,
Thermophoretic parameter τ, mixed convection parameter
⁄
, buoyancy ratio N, Inertia parameter Λ, thermal
Rayleigh numberRa , Peclet number
and chemical reaction
δ.
The effective of magnetic parameter on velocity profile,
temperature profile and concentration profile are shown in
figures 1(a)–(c). The presence of transverse magnetic field sets
in Lorentz force which results in retarding force on the velocity
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Figs.3(a)-(c) show the effect of inertia parameter Λ on the
velocity, temperature and concentration profiles. It is observed
from these figures that an increase in the value of the inertia
parameter Λ leads to decrease in the velocity profile, which
occurs at the surface. This happens because the form drag of the
porous medium is increased when the inertia effect is included.
Further we observed that the temperature and concentration
profiles increase owing to increase in the value of inertia
parameter Λ.
The effect of buoyancy ratio parameter N on the velocity,
temperature and concentration is shown in figs.4(a)–(c). It is
seen from figure 4(a) that velocity profile
increases with the
increase of buoyancy ratio parameter N, but the concentration
and temperature profiles are decreasing with the increasing N
was observed from figures 4(b) and 4(c ).
Figures 5(a)–(c) show the effect of thermophoretic parameter τ
on velocity, temperature and concentration profiles.
It is
observed that an increase in the thermophoretic parameter τ
leads to decrease in the velocity as well as temperature and
concentration profiles.This is accompanied by a decrease in the
concentration and slight increase in the fluid temperature. This
means that effective of increase τ is limited to increase in the
walls slope of the convection profile without any significant
effect on the concentration boundary layer.
The effect of chemical reaction δ on the velocity, temperature
and concentration is shown in figs. 6(a)-(c). The velocity profile
decreases with the increase of chemical reaction parameter
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International Journal of Engineering Trends and Technology (IJETT) – Volume 10 Number 5 - Apr 2014
whereas temperature profile increase with the increase of
chemical reaction parameter δ. The concentration profile
decrease with the increase of chemical parameter δ.
2
The effect of Schmidt number Sc on the concentration
distribution is shown in figure 7. It is noticed that with the
effect of Schmidt number Sc is to decrease the concentration
distribution in the boundary layer. This is due to the fact that
increase of Sc causes a decrease of concentration boundary
layer, which results decreases of the concentration profile.
4
Ha = 0
Ha = 1
f
Ha = 2
2
f
'
Rax/Pex=0.1
Rax/Pex=0.2
Rax/Pex=0
1.5
1
0.5
Rax/Pex= -0.1
Rax/Pex=-0.2
0
0
1
2
η
3
Fig. 2(a).Effect of mixed convection parameter Rax/Pex on velocity profile.
Pr= 0.73, Sc = 1, τ=0.5, δ=1, Ha = 0, Λ = 0, N=2.
1
0
0
η
1
2
3
Fig. 1(a). Effect of Magnetic Parameter Ha on velocity profile
Pr=0.73, Sc=1, δ=1, Λ= 0, N=2, τ=0.5 ,Rax/Pex=1
Rax/Pex=0
Rax/Pex=1
Rax/Pex=2
Rax/Pex=3
Rax/Pex=4
θ
1
0.5
0.8
Ha = 0
Ha = 1
Ha = 2
Ha = 3
0.6
θ
0.4
0
0
1
η 2
3
Fig. 2(b). Effect of mixed convection parameter Rax/Pex on temperarure
profiles. Pr= 0.73, Sc = 1, τ=0.5, δ=1, Ha = 0, Λ = 0, N=2
0.2
0
0
2
η
4
1
Fig. 1(b). Effect of Magnetic Parameter Ha on temperature profile
Pr=0.73, Sc=1, δ=1, Λ= 0, N=2, τ=0.5, Rax/Pex=1
1
Rax/Pex=0
Rax/Pex=1
Rax/Pex=2
Rax/Pex=3
Rax/Pex=4
0.8
0.6
Ha = 0
Ha = 1
Ha = 2
Ha = 3
0.5
0.4
0.2
0
0
1
η
2
3
Fig. 2(c). Effect of mixed convection parameter Rax/Pex on
concentration profiles. Pr= 0.73, Sc = 1, τ=0.5, δ=1,
Ha = 0, Λ = 0, N=2
0
0
1
η
2
3
Fig. 1(c ). Effect of Magnetic parameter Ha on concentration profile
Pr=0.73, Sc=1 δ=1, Λ= 0, N=2, τ=0.5, Rax/Pex=1
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International Journal of Engineering Trends and Technology (IJETT) – Volume 10 Number 5 - Apr 2014
4
4
Λ=0
f'
f'
Λ = 0.5
Λ = 0.8
N=0
N=1
2
2
N=2
0
0
0
0
2
1
η
4
η
2
3
Fig. 4(a). Effect of buoyancy ratio parameter N on velocity profiles.
Pr= 0.73, Sc = 1, τ=0.5, δ=1, Ha = 0, Λ = 0, Rax/Pex= 1
Fig. 3(a)Effect of inertia parameter Λ on velocity profiles.
Pr= 0.73, Sc = 1, N = 2, δ=1, Ha = 0, τ= 0.5, Rax/Pex= 1
1
1
Λ=0
θ
0.8
Λ = 0.5
0.8
N=0
Λ = 0.8
0.6
θ
N=1
0.6
N=2
0.4
0.4
0.2
0.2
0
0
1
2
η
0
3
0
Fig.3(b)Effect of inertia parameter Λ on temperature profiles.
Pr= 0.73, Sc = 1, N = 2, δ=1, Ha = 0, τ= 0.5, Rax/Pex= 1
1 η
2
3
Fig. 4(b). Effect of buoyancy ratio parameter N on temperature profiles.
Pr= 0.73, Sc = 1, τ=0.5, δ=1, Ha = 0, Λ = 0, Rax/Pex= 1
1
0.8
1
N=0
Λ=0
Λ = 0.5
0.8
N=1
0.6
N=2
Λ = 0.8
0.6
0.4
0.4
0.2
0.2
0
0
0
0
1
η
2
η
2
3
3
Fig. 3(c )Effect of inertia parameter Λ on concentration profiles.
Pr= 0.73, Sc = 1, N = 2, δ=1, Ha = 0, τ= 0.5, Rax/Pex= 1
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1
Fig. 4(c ). Effect of buoyancy ratio parameter N on concentration profiles.
Pr= 0.73, Sc = 1, τ=0.5, δ=1, Ha = 0, Λ = 0, Rax/Pex= 1
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International Journal of Engineering Trends and Technology (IJETT) – Volume 10 Number 5 - Apr 2014
4
4
τ = 0.0
f'
τ = 1.5
f' 3
δ=1
2
δ=3
δ=5
τ = 2.5
2
1
0
0
2
0
4
η
0
Fig. 5(a) Effect of thermophoresis parameter τ on velocity profiles.
Pr= 0.73, Sc = 1, N = 2, δ=1, Ha = 0, Λ = 0, Rax/Pex= 1
1
η
2
3
4
Fig.6(a )Effect of chemical reaction δ on velocity profiles.
Pr= 0.73, Sc = 1, N = 2, τ=0.5, Ha = 0, Λ = 0, Rax/Pex= 1
1
2
θ
τ = 0.0
τ = 1.5
τ = 2.5
0.8
0.6
δ=1
δ=7
θ
0.4
1
0.2
0
0
1
η
2
3
0
0
Fig. 5(b) Effect of thermophoresis parameter τ on temperature profiles.
Pr= 0.73, Sc = 1, N = 2, δ=1, Ha = 0, Λ = 0, Rax/Pex= 1
η
θ2
1
3
4
Fig.6(b )Effect of chemical reaction δ on temperature profiles.
Pr= 0.73, Sc = 1, N = 2, τ=0.5, Ha = 0, Λ = 0, Rax/Pex= 1
1
0.8
1
τ = 0.0
τ = 1.5
τ = 2.5
0.8
0.6
δ=1
δ =3
δ=5
0.6
0.4
0.4
0.2
0.2
0
0
0
0
η 1
2
1
2
η 3
4
Fig.6(c )Effect of chemical reaction δ on concentration profiles.
Pr= 0.73, Sc = 1, N = 2, τ=0.5, Ha = 0, Λ = 0, Rax/Pex= 1
3
Fig. 5(c )Effect of thermophoresis parameter τ on concentration profiles.
Pr= 0.73, Sc = 1, N = 2, δ=1, Ha = 0, Λ = 0, Rax/Pex= 1
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Shx/Pex1/2
Nux/Pex1/2
International Journal of Engineering Trends and Technology (IJETT) – Volume 10 Number 5 - Apr 2014
1
Sc = 0.3
The heat and mass transfer results in terms of local Nusselt
⁄
⁄
Sc = 0.5
0.8
number Nu ⁄Pe
and Sh ⁄Pe
as functions of the
Sc=0.8
mixed convection parameter is displayed in figs.( 8) and (9)
0.6
respectively. The effect of the thermophoretic parameter τ is to
⁄
increasing Nu ⁄Pe
value .The local Sherwood number in
0.4
⁄
terms of Sh ⁄Pe
verses Ra ⁄Pe is drawn for the different
⁄
values
of
τ
is
shown
in fig.(9). The Sh ⁄Pe
of increasing
0.2
⁄
⁄
with the increase of τ. The Sh Pe
value increases along
η
0
with the Ra ⁄Pe
. This is because increasingRa ⁄Pe
increase the momentum transport in the boundary layer and
0
1
2
3
more heat and mass species are carried out of the surface, thus
Fig..7. Effect of Schmid tnumber Sc on concentration profiles.
Pr= 0.73, δ=1, N = 2, Λ =0, Ha = 0, τ= 0.5, Rax/Pex= 1
decreasing the thickness of the thermal and concentration
boundary layer and hence increasing the heat and mass transfer
rates.
CONCLUSION
The
numerical
calculation
are carried out for velocity,
1.2
temperature and concentration profiles and analyzed.
1
 The velocity profile is decreased with the increase of
0.8
magnetic parameter Ha , thermophoretic parameter τ ,
0.6
chemical reaction δ and inertia parameter ⋀ whereas
τ= 0.5
reverse phenomena is observed in the case of
0.4
τ= 1.5
buoyancy ratio parameter N and mixed convection
0.2
parameter Rax/Pex .
τ= 2.5

The temperature profile increases with the increase of
0
magnetic parameter Ha, inertia parameter ⋀ and
0
2 Rax/Pex
4
chemical reaction δ while, with the increase of
Fig. 8 Effect of Nusselt number on temperature profiles.
thermophoretic parameter τ, buoyancy ratio parameter
Pr= 0.73, Sc = 1,δ=1, N = 2, Λ =0, Ha = 0, τ= 0.5, Rax/Pex= 1
N and mixed convection parameter Rax/Pex the
temperature profile decreases.

The concentration profile is increased with the
increase of magnetic parameter Ha, inertia parameter
⋀ whereas the reverse phenomena is seen in
thermophoretic parameter τ, buoyancy ratio parameter
N and mixed convection parameter Rax/Pex , chemical
4
reaction δ and Sc.
 The effect of thermophoretic parameter τ is to
increasing Nusselt number and Sherwood number.
2
NOMENCLATURE
τ= 0.5
τ= 1.5
τ= 2.5
0
0
Rax/Pex
2
4
Fig. 9. Effect of Eckert number on concentration profiles.
Pr= 0.73, Sc = 1, δ=1,N = 2, Λ =0, Ha = 0, τ= 0.5, Rax/Pex= 1
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β0
βT
βC
Cp
T
Tw
T∞
C
Cf
g
magnetic field of strength
coefficient of thermal expansion
coeffient of concentration expansion
the specific heat at constant pressure
the temperature of the fluid
the surface temperature of the plate
the free stream temperature
the concentration of the species
Forchheimer coefficient
acceleration due to gravity
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International Journal of Engineering Trends and Technology (IJETT) – Volume 10 Number 5 - Apr 2014
Ha
Sc
f
VT
Pex
Rax
N
qr
k
K1
D
δ
Nux
Shx
θ
η
ψ
Λ
w
∞
[10\. N.Kishan and Deepa, Viscous Dissipation Effects on Stagnation Point
Flow and Heat Transfer of a. MicropolarFluid with Uniform Suction or
Blowing., Advances in applied science research, 2012-3-1-430-439.
[11]. Sobha.V.V. and Ramakrishna.K., Convective Heat Transfer Past a Vertical
Plate embedded in a Porous Medium with an Applied Magnetic Field, IE(I)
Journal – MC, 84(2003) 130 – 134.
[12]. Acharya.M., Dash.G.C. and Singh.L.P.,Magnetic Field Effects on the Free
Convection and Mass Transfer Flow through Porous Medium with Constant
Suction and Constant Heat Flux, Indian J. Pure Appl. Math., 31(2000) 1
[13]. C.N.B.Rao, V.Laksmi Prasannam and T.Rajarani, Mixed Convection in a
Porous Medium with Magnetic Field, Variable Viscosity and Varying wall
Temperature. International Journal of Computational Mathematical Ideas,
2(1)(2010)13-21.
[14]. L.Talbot. R.K. Cheng, R.W.Scheffer,andD.P.Wills, Thermophoresis of
particles in a heated boundary layer, J. Fluid Mech. (1980),101 ,737-758.
[15]. G.K Batchelor and C. Shen, Thermophoresis deposition of particles in gas
flowing over cold surfaces, J. Colloid interface Sci. (1985), 107 ,21-37
[16]. R.B.Bellman, and R.E.Kalaba,Quasi-linearization and Non-Linear
boundary value problem, Elsevier, Newyork. (1965)
magnetic field parameter
Schmidt number
dimensionless stream function
the thermophoretic velocity
Thermal diffusity
kinematic viscosity
thermophoretic parameter
Local Peclet number
the thermal Rayleigh number
buoyancy ratio
Radioactive heat flux
thermophoretic coefficient
Darcy-permeability
mass diffusity
chemical reaction
Nusselt number
Sherwood number
density of the fluid
dimensionless temperature
dimensionless concentration
similarity parameter
stream function
inertia parameter
condition at wall
condition at infinity
conductivity of the fluid
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ISSN: 2231-5381
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