effects of radiation on mhd boundary layer flow of combined heat and mass transfer … ·...
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Volume-7, Issue-1, January-February 2017
International Journal of Engineering and Management Research
Page Number: 137-150
Effects of Radiation on MHD Boundary Layer Flow of Combined Heat
and Mass Transfer over a Moving Inclined Plate in a Porous Medium
with Suction and Viscous Dissipation in Presence of Hall Current and
Chemical Reaction
V.Subhakanthi1, N. Bhaskar Reddy
2
1,2Department of Mathematics, SVU, Tirupati, INDIA
ABSTRACT This paper analyzes the chemical reaction and
radiation effects on heat and mass flow over a moving
inclined plate in porous medium with suction and viscous
dissipation in presence of Hall current. A suitable similarity
transformation is used to transform the nonlinear system of
partial differential equations into a system of ordinary
differential equations.to solve the resultant system a well
tested numerical technique Runge-Kutta fourth order is
used along with shooting technique. The behavior of
primary and secondary velocities, temperature and
concentration for variations in thermo physical parameters
are presented in graphs. transfer in magnetohydrodynamic
boundary layer The values of skin friction coefficient,
Nusselt number and Sherwood number are also computed
and are reported in tables.
Keywords-- heat and mass transfer-MHD- radiation-
viscous dissipation-chemical reaction
I. INTRODUCTION
The free convection processes involving the
combined mechanism of heat and mass transfer are
encountered in many natural processes, in many
industrial applications and in many chemical processing
systems. The study of free convective mass transfer flow
has become the object of extensive research as the
effects of heat transfer along with mass transfer effects
are dominant features in many engineering applications
such as rocket nozzles, cooling of nuclear reactors, high
sinks in turbine blades, high speed aircrafts, chemical
devices and process equipments.
The study of MHD flows have stimulated more
attention due its important applications in different
subject areas such as astrophysical, geophysical and
engineering problems. Free convection in electrically
conducting fluids through an external magnetic field has
been a subject of considerable research interest of a large
number of scholars for a long time due to its
miscellaneous applications in the fields of nuclear
reactors, geothermal engineering, liquid metals and
plasma flows etc. Fluid flow control under magnetic
forces is also applicable in MHD generators and a host
of magnetic devices used in industries. Jha[1] explained
the problem of MHD free convection and mass transfer
flow past an impulsively moving vertical plate through
porous medium when the vertical plate moves with
uniform acceleration and applied magnetic field is fixed
with the moving plate. Pioneer work on convective flow
in porous media are presented in the form of books and
monographs by Ingham and Pop[2], Ingham et al. [3],
Vafai[4] and Nield and Bejan [5].
A two dimensional steady MHD mixed
convection and mass transfer flow over a semi- infinite
porous inclined plate in the presence of thermal radiation
with variable suction and thermophoresis was studied by
Alam et al. [6]. Orthan Aydm and Ahmet Kaya[7]
considered MHD mixed convective heat transfer flow
about an inclined plate. Gnaneswara Reddy and Bhaskar
Reddy[8] presented mass transfer and heat generation
effects on MHD free convection flow over an inclined
vertical surface in a porous medium. Recently, Hitesh
Kumar[9] done his work on the heat transfer MHD
boundary layer flow through a porous medium.
The role of thermal radiation is of major
importance in some industrial applications such as glass
production and furnace design and in space technology
applications, cosmical flight aerodynamics, propulsion
systems, plasma physics and craft re-entry,
aerothermodynamics which operate at high
temperatures. Solving the governing equations become
quite complicated when radiation is taken into account
and hence many difficulties arise while solving such
equations. Viskanta and Grosh [10] were one of the
initial investigators to study the effects of thermal
radiation on temperature distribution and heat transfer in
an absorbing and emitting media flowing over a wedge.
They used Rosseland approximation for the radiative
flux vector to simplify the energy equation. Suneetha et
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al. [11] studied the effects of thermal radiation on
unsteady hydro magnetic free convection flow over an
impulsively started vertical plate with variable surface
temperature and concentration. Gnaneswara Reddy and
Bhaskar Reddy[12] reported the radiation and mass
transfer effects on unsteady MHD free convection flow
past a vertical porous plate with viscous dissipation by
using finite element method. Recently, Narahari and
Ishak[13] carried out an analysis to study the effects of
thermal radiation on unsteady free convection flow of an
optically thick fluid past a moving vertical plate with
Newtonian heating. Their interesting cases are impulsive
movement of the plate, uniformly accelerated movement
of the plate and exponentially accelerated movement of
the plate.
The viscous dissipation heat in the natural
convective flow is important, when the flow field is of
extreme size or at low temperature or in high
gravitational field.. such effects are also important in
geophysical flows and also in certain industrial
operations and are usually characterized by the Eckert
number. When the viscosity of the fluid is high, the
dissipation term becomes important. For many cases,
such as polymer processing which is operated at a very
high temperature, viscous dissipation cannot be
neglected. An extensive work on the viscous dissipative
heat effects on the study free convection and on
combined free and forced convection flows has been
done by Ostrach [14-18]. Numerical analysis of steady
non- Newtonian flows with heat transfer analysis, MHD
and non- linear slip effects was examined by Ellahi and
Hameed [19]. Ellahi et al. [20] explained the influence of
slip on steady flows in viscous fluid with heat and mass
transfer. Recently, Vajravelu[21] investigated unsteady
convective boundary layer flow of a viscous fluid at a
vertical surface with different fluid properties.
It may be noted that when the density of an
electrically conducting fluid is low and /or applied
magnetic field is strong (Sutton and Sherman[22]), the
effects of Hall current become significant. It plays an
important role in determining flow features of the fluid
flow problems because induces secondary flow in the
fluid. Therefore it is of considerable interest to study the
effects of Hall current on MHD fluid flow problems.
Sato[23], Sherman and Sutton[22] have analyzed the
Hall effects on the steady hydromagnetic flow between
two parallel plates. These effects in the unsteady cases
were reported by Pop[24]. Seth et al. [25] present the
effects of Hall current on unsteady hydromagnetic
natural convection transient flow of a viscous,
incompressible, electrically conducting and heat
absorbing fluid past an impulsively moving vertical plate
fixed in a fluid saturated porous medium, under
boussinesq approximation, taking into the effects of
thermal diffusion when temperature of the plate has a
temporarily ramped profile. Flow through a porous
medium bounded by a vertical surface in presence of
Hallcurrent was considered by Sudhakar[26].
Gopichand[27] analyzed the unsteady stretching surface
in porous medium and explained the viscous dissipation
and radiation effects on MHD flow over it.
Another important aspect, which influences
heat transfer processes is the suction /injection. It is well
known that the effects of injection on the boundary layer
flow area of interest in reducing the drag force. Suction
and heat transfer characteristics were addressed by
Youn[28]. The effects of suction or injection on the free
convection boundary layers induced by a heated vertical
plate fixed in a saturated porous medium with an
exponential decaying heat generation were presented by
Ali[29]. Suction or blowing of a fluid through the
bounding surface can significantly change the flow field.
In general, suction tends to increase the skin friction,
whereas injection acts in the opposite manner. In many
engineering activities such as in the design of thrust
bearing and radial diffusers, and thermal oil recovery the
process of suction/ blowing plays a significant role
because of its importance.
Bhattacharya [30] explained the effects of
radiation and heat source/sink on unsteady MHD
boundary layer flow and heat transfer over a shrinking
sheet with suction /injection. The flow is permeated by
an externally applied magnetic field normal to the plane
of the flow in his work. The self similar equations
corresponding to the velocity, temperature and
concentration fields are obtained, and then solved
numerically by finite difference method using quasi
linearization technique. Lin et al. [31] investigates study
laminar boundary layer flow of power law fluids past a
flat surface with suction or injection and magnetic
effects. Recently, Cao et al. [32] analyzed the MHD
Maxwell fluid over a stretching plate with suction or
injection in the presence of nano particles.
Heat and mass transfer problems in the
presence of chemical reaction are of importance in many
processes, and have therefore received a considerable
amount of attention in recent times. Possible applications
can be found in processes such as drying, distribution of
temperature and moisture over agricultural fields and
groves of fruit trees, damage of crops due to freezing,
evaporation at the surface of a water body and energy
transfer in a wet cooling tower, and flow in a desert
cooler. In many chemical engineering processes,
chemical reactions take place between a foreign mass
and the working fluid which moves due to the stretching
of a surface. The order of the chemical reaction depends
on several factors. One of the simplest chemical reaction
is the first-order reaction in which the rate of reaction is
directly proportional to the species concentration .
Deka et al. [33] reported the effect of first order
homogeneous chemical reaction on the process of an
unsteady flow over an infinite vertical plate with a
constant heat and mass transfer. Muthucumaraswamy
and Ganesan [34] studied the flow characteristics in an
unsteady upward motion of an isothermal plate by taking
chemical reaction and injection into account. Reddy et
al. [35] analyzed the effects of radiation and chemical
reaction on an unsteady hydromagnetic natural
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convection flow over a moving vertical plate in a porous
medium.
Owing to the above mentioned studies, the
author made an attempt to investigate the combined
effects of chemical reaction, thermal radiation and Hall
current on the hydro magnetic free convective flow of
heat and mass transfer over a moving inclined plate in a
porous medium with suction and viscous dissipation.
The governing boundary layer equations (4.2.11) to
(4.2.14) subject to the boundary conditions (4.2.15) are
solved numerically by using Runge-Kutta fourth order
method along with shooting technique.
II. MATHEMATICAL ANALYSIS
A two dimensional steady laminar MHD
viscous incompressible electrically conducting and
chemically reacting fluid along a moving inclined plate
with an acute angle γ embedded in a porous medium, in
the presence of suction is considered. x- direction is
taken along the leading edge of the inclined plate and y
is normal to it and extends parallel to x -axis. Let wT
(> T ) be the uniform plate temperature, where T∞ is
the temperature of the fluid far away from the plate. Let
u, v and w be the velocity components along the x and y
axis and secondary velocity component along the z axis
respectively in the boundary layer region. Let wC be the
concentration of the fluid at the surface of the plate and
C be the free stream concentration. The flow is
subjected to the effect of thermal radiation and a
transverse magnetic field of strength B0, which is
assumed to be applied in the positive y direction, normal
to the surface. The induced magnetic field is also
assumed to be small compared to the applied magnetic
field so it is neglected. All the fluid properties are
assumed to be constant except for the density variations
in the buoyancy force term of the linear momentum
equation. The Hall effects and viscous dissipation are
taken into account .Joule heating term is neglected. The
sketch of the physical configuration and coordinate
system are shown in Fig 1.
Figure1 Physical configuration and coordinate system
Under the above assumptions the boundary layer
equations describing the flow field under consideration
are
The boundary conditions for the velocity,
temperature and concentration fields are
bx+C=C=C
,ax+T=T=T,0=w,V=v,ax=u
∞w
∞w at 0y
CCTTwu ,,0,0 as y
(6)
where u, v and w be the velocity components along the
x- axis and y- axis and secondary velocity component
along the z - axis respectively in the boundary layer
region. T and C are the temperature and concentration of
the fluid respectively. g- the gravitational acceleration,
T and c - the coefficients of thermal and
concentration expansions,γ - the acute angle or
inclination parameter, 0B - the magnetic field induction,
m- the hall parameter, - the kinematic viscosity, k -
permeability of the porous medium, - thermal
diffusivity, pc - the specific heat at constant pressure,
rq - the radiative heat flux, D- the mass diffusivity,
1k - chemical reaction rate, wT and wC - the temperature
and concentration of the fluid at the surface of the plate,
T - the temperature of the fluid far away from the plate
and C - the free stream concentration .
The second and third terms on the right hand
side of equation (4) are the viscous dissipative heat and
radiative heat flux respectively. The second term on right
hand side of the equation (5) is the species chemical
reaction.
0=y∂
v∂+
x∂
u∂ (1)
2
2
y∂
u∂υ=
y∂
u∂v+
x∂
u∂u +
( ) ( )( )
( ) uk′
υmw+u
m+1ρ
BσγcosCCβg+γcosTTβg 2
2
0
∞c∞T
(2)
( )( ) w
k′
υwmu
m+1ρ
Bσ+
y∂
w∂υ=
y∂
w∂v+
x∂
w∂u
2
2
0
2
2
(3)
+y∂
T∂α=
y∂
T∂v+
x∂
T∂u 2
2
y∂
q∂
cρ
1_
y∂
w∂+
y∂
u∂
cρ
υ r
p
22
p
(4)
CCk
y
CD
y
Cv
x
Cu 12
2
(5)
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Continuity equation (1) is identically satisfied by the
stream function yx, , defined as
xv
yu
, (7)
By using Rosseland approximation, the radiative heat
flux rq is given by
y
T
kqr
4
*
*
3
4 (8)
Where* is the Stefan – Boltzman constant and
*k is
the mean absorption coefficient. It should be noted that
by using Rosseland approximation, the present analysis
is limited to optically thick fluids.If the temperature
differences within the flow are sufficiently small then
equation (4.2.6)can be linearized by expanding 4T in a
Taylor series about the free stream temperature
T which after neglecting the higher order terms takes
the form 434 34 TTTT (9)
To transform equations (2) to (4) into a set of
ordinary differential equations, the following similarity
transformations and dimensionless variables are
introduced
Substituting the equations (7) to (10) into the
equations (2) to (5) we obtain
γcosφG+γcosθG+f′′f2
1+f′′′ cr
0=f′kgm+1
mMf′
m+1
M022
(11)
0112
1002200
kggm
Mf
m
Mmfgg
(12)
0Pr2
11
2
0
2
gfEcfR
(13)
02
1 ScKrScf (14)
The corresponding boundary conditions are
1,1,1,0, 0 fgFf w at 0
00 gff as (15) (15)
where prime ( ' ) denotes differentiation with respect to η.
η - the similarity parameter, f is the
dimensionless stream function,, - the
dimensionless temperature, - the dimensionless
concentration, ψ – the stream function, M- the magnetic
field parameter 0g - the secondary velocity parameter,
rG - the local thermal Grahsof number, cG - the
local solutal Grahsof number, K- the permeability
parameter, γ - the inclination parameter, m – Hall
current parameter, R- radiation parameter, Pr - the
Prandtl number, Ec - the Eckert number, Sc - the
Schmidt number, Kr - the chemical reaction parameter,
wF - the suction parameter.
III. SOLUTION OF THE PROBLEM
The governing boundary layer equations (11) to
(14) subject to the boundary conditions (15) are solved
numerically by using Runge-Kutta fourth order method
along with shooting technique. First of all higher order
non-linear differential equations (11) to (14) are
converted into simultaneous linear differential equations
of first order and they are further transformed into initial
value problem by applying the shooting technique (Jain
et al.[36]). The resultant initial value problem is solved
by employing Runge-kutta fourth order technique.
Numerical results are reported in figures for various
values of the physical parameters of interest.
From the process of numerical computation, the skin-
friction coefficient, the Nusselt number and Sherwood
number which are respectively proportional to 0f ,
0 and 0 are also sorted out and numerical
values are presented in a tabular form.
IV. RESULTS AND DISCUSSION
As a result of the numerical calculations, the
dimensionless velocity, temperature and concentration
are obtained and their behaviour have been discussed
for variations in governing parameters viz., M- the
magnetic field parameter 0g - the secondary velocity
parameter, rG - the local thermal Grahsof number,
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cG - the local solutal Grahsof number, K- the
permeability parameter, γ - the inclination parameter, m
– Hall current parameter, R- radiation parameter,, Pr -
the Prandtl number, Ec - the Eckert number, Sc - the
Schmidt number, Kr - the chemical reaction parameter,
wF - the suction parameter. The results are presented in
Figures from 4.2 – 1.38. Numerical results for the skin –
friction, Nusselt number and Sherwood number are
reported in Tables 1 and 2. A parametric study is carried
out to demonstrate the effects of governing parameters
on velocity, temperature and concentration profiles.
Fig. 2 and Fig. 3 show the effects of thermal
Grashof number Gr and solutal Grash of number Gc on
the velocity respectively. As shown the velocity
increases as Gr and Gc increases. Physically Gr > 0
means heating of the fluid or cooling of the boundary
surface, Gr < 0 means cooling of the fluid or heating of
the boundary surface and Gr = 0 corresponds to the
absence of free convection current. The effect of
inclination parameter on the velocity of the fluid is
shown in Fig. 4. It is noticed that increasing the
inclination parameter results a decrease in the velocity.
Fig. 5 displays the effect of magnetic field paramater on
the velocity of the fluid. The presence of a magnetic
field in an electrically conducting fluid induces a force
called Lorentz force, which opposes the flow. This
resistive force tends to slow down the flow, so the effect
of increase in M is to decrease the velocity. Fig. 6
illustrates the effect of Hall parameter m on the velocity.
It is observed that the velocity of fluid increases on
increasing the Hall parameter. Fig.7 shows the effect of
the thermal conductivity on the velocity of the fluid. It is
seen that velocity decreases on increasing the thermal
conductivity. Fig. 8 represents the effect of radiation
parameter on the velocity, and it is noticed that the effect
of radiation parameter on the velocity of the fluid is
slight.
The effect of the Prandtl number on the velocity
of the fluid is illustrated in Fig. 9. On increasing the
Prandtl number, the velocity of the fluid flow increases.
Fig. 10 depicts the effect of Eckert number on the
velocity of the boundary layer. A slight change in the
velocity is seen. The effect of Schmidt number on the
velocity of the fluid is shown in Fig. 11. A slight
decrease in the velocity of the fluid on increasing the
Schimdt number is noticed. The effect of chemical
reaction parameter on the velocity of the fluid flow is
illustrated in Fig. 12. It is found that on increasing the
chemical reaction parameter the velocity of the fluid is
decreasing. Fig. 13 shows the effect of suction parameter
on the velocity. It is observed that the velocity increases
on increasing the suction parameter. The effect of
magnetic parameter M on the secondary velocity of the
fluid is shown in Fig. 14. Increase in the secondary
velocity of the fluid is observed on increasing the
magnetic parameter. Fig. 15 shows the effect of radiation
parameter on secondary velocity of the fluid. On
increasing the radiation parameter secondary velocity of
the fluid is found to be decreased. The effect of
chemical reaction parameter on the secondary velocity of
the fluid is shown in Fig. 16. Decrease in the secondary
velocity is noticed from the figure on increasing the
chemical reaction parameter. Fig. 17 and Fig. 18
illustrates the effects of thermal and mass Grashofer
numbers Gr and Gc respectively on the temperature of
the fluid. Decrease in the temperature is noticed. The
effect of the Magnetic field parameter is shown in Fig.
19, increase in the temperature of the fluid is observed.
The effects of thermal conductivity on temperature of
the fluid is depicted in the Fig. 20. It is shown from the
figure that temperature increases on increasing the
thermal conductivity. The effects of radiation on
temperature of the fluid is illustrated in Fig. 21.
Decrease in the temperature of the fluid on increasing
the radiation is observed. Effects of Prandtl number on
temperature of the fluid is shown in Fig. 22. Increase in
the temperature is noticed. Hall parameter decreases the
temperature of the fluid as shown in Fig. 23. Effects of
Eckert number on thermal boundary layer is illustrated
in Fig. 24. Temperature of the fluid in the boundary
layer increases on increasing the Eckert number.
Chemical reaction effect on the thermal boundary layer
is depicted in Fig. 25. Thermal boundary layer thickness
increases on increasing the chemical reaction. Thermal
boundary layer thickness is increased on increasing the
inclination parameter γ as shown in Fig. 26. The effect
of suction parameter on the temperature is depicted in
Fig. 27. It is seen that temperature increases on
increasing the suction parameter.
Fig. 28 and Fig. 29 illustrate the effects of
thermal and mass Grashof numbers Gr and Gc on the
species concentration field. Concentration of the fluid in
the boundary layer decreases on increasing Gr and Gc.
Concentration of the fluid in the boundary layer
increases on increasing the magnetic parameter M as
shown in Fig. 30. The effect of hall parameter m on
concentration field was displayed in Fig. 31. A decrease
in the concentration is noticed on increasing the hall
parameter.
Fig. 32 depicts the effect of thermal
conductivity on the concentration field . Concentration
of the fluid increases on increasing the thermal
conductivity. The effect of Prandtl number on
concentration field is illustrated in Fig. 33. Effect of
Eckert number on concentration field was displayed in
Fig. 34. Decrease in the concentration on increasing the
Eckert number is noticed. Fig. 35 represents the effect of
Schmidt number on concentration field. It is interesting
to note that the chemical species concentration also
decreases within the boundary layer with an increase in
Schmidt number due to the combined effects of
buoyancy forces and species molecular diffusivity. Fig.
36 depicts the influence of chemical reaction rate on
concentration field. An increase in the value of chemical
reaction parameter decreases the concentration of
species in the boundary layer. This is due to the fact that
chemical reaction in this system results in consumption
of the chemical and hence results in decrease of
concentration. Fig.37.demonstrates the effect of
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inclination parameter on the concentration .There is a
slight change in the concentration of the fluid is
observed on increasing the inclination parameter γ. Fig.
38 shows the effect of suction parameter on the
concentration. It is found that concentration increases on
increasing the suction.
Table. I shows the comparison of results of
present work with that of Ali et al.[37], and it is found
that there is a good agreement. Numerical computations
of skin friction coefficient, Nusselt number and
Sherwood number for different values of Gr, Gc, M, m,
K, Ec, Sc, γ, Fw, R and Kr =0 are reported in Table II
and Table 3.
V. CONCLUSIONS
A two dimensional steady laminar MHD
viscous incompressible electrically conducting and
chemically reacting fluid along a moving inclined plate
with an acute angle γ embedded in a porous medium, in
the presence of suction has been studied. In addition to
this, thermal radition and external magnetic field
strength are also considered. The governing boundary
layer equations are solved numerically using well tested,
highly efficient Runge-Kutta fourth order method along
with shooting technique. From the present study we
arrive at the following significant observations.
On comparing the present results with previous work, it
is found that there is a good agreement.
Increasing buoyancy ratio parameters Gr and
Gc increases the velocity, but decreases the
temperature and concentration.
Increasing the magnetic field parameter
increases the temperature and the concentration,
but reduces the velocity.
Increasing the hall parameter enhances the
velocity, but reduces the temperature and
concentration.
Increasing the thermal conductivity parameter
rises the temperature and concentration, but
decreases the velocity.
Increasing the radiation parameter results a
slight change in the velocity, but reduces the
temperature.
Increasing the prandtl number increases the
velocity, temperature as well as the
concentration.
Increasing the Eckert number results a slight
change in the velocity, increases the
temperature, but decreases concentration.
Increasing the Schmidt number decreases the
velocity and the concentration
Increasing the suction parameter results an
increase in the velocity, temperature and the
concentration.
Increasing the chemical reaction parameter
increases the temperature, but reduces the
velocity and concentration.
Increasing inclination parameter decreases the
velocity and increases the temperature and there
is a slight change in the concentration.
Secondary velocity increases on increasing the
magnetic parameter, but reduces o increasing
the radiation and chemical reaction parameters.
Skin fraction coefficient and Nusselt number
decreases where as sherewood number
increases on increasing chemical reaction
parameter.
Skin friction coefficient and Sherwood number
decreases, whereas Nusselt number increases
with an increase in the radiation parameter.
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
0.0
0.2
0.4
0.6
0.8
1.0
Gr = 0.2,0.4,0.6,0.8
Gc=0.1, M = 1,m = 0.1,k = 1,
R= 1, Pr = 0.7, Ec = 0.01,
Sc = 0.22, Kr = 1,
Figure 2: Velocity profiles for different Gr.
0 1 2 3 40.0
0.2
0.4
0.6
0.8
1.0
Gc = 0.2,0.4,0.6,0.8
Gr= 0.1, M = 1,m = 0.1,k = 1,
R= 1, Pr = 0.7, Ec = 0.01,
Sc = 0.22, Kr = 1, = 30°
Figure 3: Velocity profiles for different Gc.
0 1 2 3 40.0
0.2
0.4
0.6
0.8
1.0
Gr= 0.1, Gc=0.1, M = 1,m = 0.1,
k = 1, R= 1, Pr = 0.7, Ec = 0.01,
Sc = 0.22, Kr = 1
Figure 4: Velocity profiles for different γ.
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0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.00.0
0.2
0.4
0.6
0.8
1.0
M = 1, 2, 3, 4
Gr= 0.1, Gc=0.1, m = 0.1,k = 1,
R= 1, Pr = 0.7, Ec = 0.01,
Sc = 0.22, Kr = 1, = 30°
Figure 5 :Velocity profiles for different M.
0 1 2 3 40.0
0.2
0.4
0.6
0.8
1.0
m = 0.2,0.4,0.6,0.8
Gr= 0.1, Gc=0.1, M = 1,k = 1,
R= 1, Pr = 0.7, Ec = 0.01,
Sc = 0.22, Kr = 1, = 30°
Figure 6 : Velocity profiles for different m.
0 1 2 3 40.0
0.2
0.4
0.6
0.8
1.0
k= 1,2,3,4
Gr= 0.1, Gc=0.1, M = 1,m = 0.1,
R= 1, Pr = 0.7, Ec = 0.01,
Sc = 0.22, Kr = 1, = 30°
Figure 7: Velocity profiles for different k.
0 1 2 3 40.0
0.2
0.4
0.6
0.8
1.0
Gr= 0.1, Gc=0.1, M = 1,m = 0.1,
k = 1,Pr = 0.7, Ec = 0.01,
Sc = 0.22, Kr = 1, = 30°
R = 1,2,3,4
Figure 8: Velocity profiles for different R.
0 1 2 3 40.0
0.2
0.4
0.6
0.8
1.0
Pr = 0.7,1.7,2.7,7.7
Gr= 0.1, Gc=0.1, M = 1,
m = 0.1,k = 1, R= 1, Ec = 0.01,
Sc = 0.22, Kr = 1, = 30°
Figure 9: Velocity profiles for different Pr.
0 1 2 3 40.0
0.2
0.4
0.6
0.8
1.0
Ec= 0.01,0.03,0.05,0.07
Gr= 0.1, Gc=0.1, M = 1,m = 0.1,
k = 1, R= 1, Pr = 0.7,Sc = 0.22,
Kr = 1, = 30°
Figure 10: Velocity profiles for different Ec.
0 1 2 3 40.0
0.2
0.4
0.6
0.8
1.0
Sc= 0.2,0.6,1.0,1.4
G r= 0.1, Gc =0.1, M = 1,m = 0.1,
k = 1, R= 1, Pr = 0.7, Ec = 0.01,
Kr = 1, = 30°
Figure 11: Velocity profiles for different Sc.
0 1 2 3 40.0
0.2
0.4
0.6
0.8
1.0
Kr = 1,2,3,4
Gr= 0.1, Gc=0.1, M = 1,m = 0.1,
k = 1, R= 1, Pr = 0.7, Ec = 0.01,
Sc = 0.22, = 30°
Figure 12: Velocity profiles for different Kr.
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0 1 2 3 40.0
0.2
0.4
0.6
0.8
1.0
Fw = - 0.5, 0.5, 1
Gr= 0.1, Gc=0.1, M = 1,m = 0.1,
k = 1, R= 1, Pr = 0.7, Ec = 0.01,
Sc = 0.22, Kr = 1, = 30°
Figure 13: Velocity profiles for different Fw.
0 1 2 3 40.000
0.002
0.004
0.006
0.008
0.010
0.012
g0
M= 0.5, 1,1.5, 2
Figure 14: Secondaryvelocity profiles
for different M.
0 1 2 3 40.000
0.002
0.004
0.006
0.008
0.010
0.012
R= 0.2, 4,8,10g
0
Gr= 0.1, M = 1,m = 0.1,
k = 1,Pr = 0.7, Ec = 0.01,
Sc = 0.22, Kr = 1, = 30°
Figure 15: Secondaryvelocity profiles
for different R.
0 1 2 3 40.000
0.002
0.004
0.006
0.008
0.010
0.012
g0
Kr= 1,2,3,10
Gr= 0.1, M = 1,m = 0.1,
k = 1, R= 1, Pr = 0.7, Ec = 0.01,
Sc = 0.22, = 30°
Figure 16: Secondary velocity profiles for different Kr.
0 1 2 3 40.0
0.2
0.4
0.6
0.8
1.0
Gr = 0.2,0.4,0.6,0.8
Gc=0.1, M = 1,m = 0.1,
k = 1, R= 1, Pr = 0.7, Ec = 0.01,
Sc = 0.22, Kr = 1, = 30°
Figure 17: Temperature profiles for different Gr.
0 1 2 3 40.0
0.2
0.4
0.6
0.8
1.0
Gc = 0.2,0.4,0.6,0.8
Gr= 0.1, M = 1,m = 0.1,
k = 1, R= 1, Pr = 0.7, Ec = 0.01,
Sc = 0.22, Kr = 1, = 30°
Figure 18: Temperature profiles for different Gc
0 1 2 3 40.0
0.2
0.4
0.6
0.8
1.0
M = 1, 2, 3, 4
Gr= 0.1, Gc=0.1,m = 0.1,
k = 1, R= 1, Pr = 0.7, Ec = 0.01,
Sc = 0.22, Kr = 1, = 30°
Figure19: Temperature profiles for different M.
0 1 2 3 40.0
0.2
0.4
0.6
0.8
1.0
Gr= 0.1, Gc=0.1, M = 1,m = 0.1,
R= 1, Pr = 0.7, Ec = 0.01,
Sc = 0.22, Kr = 1, = 30°
k= 1, 2, 3, 4
Figure 20: Temperature profiles for different k.
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0 1 2 3 40.0
0.2
0.4
0.6
0.8
1.0
R = 1, 5, 10, 15
Gr= 0.1, Gc=0.1, M = 1,m = 0.1,
k = 1, Pr = 0.7, Ec = 0.01,
Sc = 0.22, Kr = 1, = 30°
Figure 21: Temperature profiles for different R.
0 1 2 3 40.0
0.2
0.4
0.6
0.8
1.0
Pr = 0.7,1.7,2.7,7.7
Gr= 0.1, Gc=0.1, M = 1,m = 0.1,
k = 1, R= 1, Ec = 0.01,
Sc = 0.22, Kr = 1, = 30°
Figure 22: Temperature profiles for different Pr.
0 1 2 3 40.0
0.2
0.4
0.6
0.8
1.0
m= 0.1, 10, 20, 30
Gr= 0.1, Gc=0.1, M = 1,k = 1,
R= 1, Pr = 0.7, Ec = 0.01,
Sc = 0.22, Kr = 1, = 30°
Figure 23: Temperature profiles for different m.
0 1 2 3 40.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
Ec= 2,4,6,8
Gr= 0.1, Gc=0.1, M = 1,m = 0.1,
k = 1, R= 1, Pr = 0.7, Sc = 0.22,
Kr = 1, = 30°
Figure 24: Temperature profiles for different Ec.
0 1 2 3 40.0
0.2
0.4
0.6
0.8
1.0
Kr= 0.0001,0.01,1,100
Gr= 0.1, Gc=0.1, M = 1,m = 0.1,
k = 1, R= 1, Pr = 0.7, Ec = 0.01,
Sc = 0.22, = 30°
Figure 25: Temperature profiles for different Kr.
0 1 2 3 40.0
0.2
0.4
0.6
0.8
1.0
Gr= 0.1, Gc=0.1, M = 1,m = 0.1,
k = 1, R= 1, Pr = 0.7, Ec = 0.01,
Sc = 0.22, Kr = 1,
Figure 26: Temperature profiles for different γ.
0 1 2 3 40.0
0.2
0.4
0.6
0.8
1.0
Fw=-1,- 0.5, 0.5, 1
Gr= 0.1, Gc=0.1, M = 1,m = 0.1,
k = 1, R= 1, Pr = 0.7, Ec = 0.01,
Sc = 0.22, Kr = 1, = 30°
Figure 27: Temperature profiles for different Fw.
0 1 2 3 40.0
0.2
0.4
0.6
0.8
1.0
Gr= 1,3,5,7
Gc=0.1, M = 1,m = 0.1,k = 1,
R= 1, Pr = 0.7, Ec = 0.01,
Sc = 0.22, Kr = 1, = 30°
Figure 28: Concentration profiles for different Gr.
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0 1 2 3 40.0
0.2
0.4
0.6
0.8
1.0
Gc = 1,3,5,7
Gr= 0.1, M = 1,m = 0.1,k = 1,
R= 1, Pr = 0.7, Ec = 0.01,
Sc = 0.22, Kr = 1, = 30°
Figure 29: Concentration profiles for different Gc.
0 1 2 3 40.0
0.2
0.4
0.6
0.8
1.0
M= 0.1, 1, 5, 10
Gr= 0.1, Gc=0.1, m = 0.1,k = 1,
R= 1, Pr = 0.7, Ec = 0.01,
Sc = 0.22, Kr = 1, = 30°
Figure 30: Concentration profiles for different M.
0 1 2 3 40.0
0.2
0.4
0.6
0.8
1.0
m= 10,90
Gr= 0.1, Gc=0.1, M = 1,k = 1,
R= 1, Pr = 0.7, Ec = 0.01,
Sc = 0.22, Kr = 1, = 30°
Figure 31: Concentration profiles for different m.
0 1 2 3 40.0
0.2
0.4
0.6
0.8
1.0
k= 0.1, 1, 5, 10
Gr= 0.1, Gc=0.1, M = 1,m = 0.1,
R= 1, Pr = 0.7, Ec = 0.01,
Sc = 0.22, Kr = 1, = 30°
Figure 32: Concentration profiles for different k.
0.0 0.5 1.0 1.5 2.0 2.5 3.0
0.0
0.2
0.4
0.6
0.8
1.0
1.2
Pr = 0.7 ,1,1.5,2
Gr=0.1,Gc=0.1,M=1, = 0.1,
Pr=0.7,R=0.5,Ec=0.01,Sc=0.22,
N = 0.1, = 45 ,Kr = 0.5.
Figure 33: Concentration profiles for different Pr.
0 1 2 3 40.0
0.2
0.4
0.6
0.8
1.0
Ec=1,50
Gr= 0.1, Gc=0.1, M = 1,m = 0.1,
k = 1, R= 1, Pr = 0.7, Sc = 0.22,
Kr = 1, = 30°
Figure 34: Concentration profiles for different Ec.
0 1 2 3 40.0
0.2
0.4
0.6
0.8
1.0
Sc= 0.22,0.6 1.0,1.4
Gr= 0.1, Gc=0.1, M = 1,m = 0.1,
k = 1, R= 1, Pr = 0.7, Ec = 0.01,
Kr = 1, = 30°
Figure 35: Concentration profiles for different Sc.
0 1 2 3 40.0
0.2
0.4
0.6
0.8
1.0
Kr= 1,5,10,15
Gr= 0.1, Gc=0.1, M = 1,m = 0.1,
k = 1, R= 1, Pr = 0.7, Ec = 0.01,
Sc = 0.22, = 30°
Figure 36: Concentration profiles for different Kr.
0 1 2 3 40.0
0.2
0.4
0.6
0.8
1.0
Gr= 0.1, Gc=0.1, M = 1,m = 0.1,
k = 1, R= 1, Pr = 0.7, Ec = 0.01,
Sc = 0.22, Kr = 1,
Figure 37: Concentration profiles for different γ.
0 1 2 3 40.0
0.2
0.4
0.6
0.8
1.0
Fw= -1, -0.5,0.5,1
Gr= 0.1, Gc=0.1, M = 1,m = 0.1,
k = 1, R= 1, Pr = 0.7, Ec = 0.01,
Sc = 0.22, Kr = 1, = 30°
Figure 38: Concentration profiles for different Fw.
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TABLE I
Computations showing comparision of present results for 0f and - ( )0θ′ at the plate with Gr,Gc, M, m, K, Ec, Sc, γ
and Fw for, R=0, Kr =0.with that of Ali[37].
TABLE II
Variation of 0f , 0 , 0 for different Gr, Gc, M, m, k, R with
P r = 0.7,Ec = 0.01, Sc = 0.22, Kr = 1, γ = 30°and Fw = 1.
Gr Gc M m K Pr Ec Sc
γ
de
g
Fw
Present work Ali [37]
0f
- ( )0θ′ 0f
- ( )0θ′
1
2
3
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
1
2
3
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
1
1
1
1
1
1
1
2
3
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.2
0.3
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
1
1
1
1
1
1
1
1
1
1
1
1
1
2
3
1
1
1
1
1
1
1
1
1
1
1
1
1
1
0.7
0.7
0.7
0.7
0.7
0.7
0.7
0.7
0.7
0.7
0.7
0.7
0.7
0.7
0.7
0.7
1.7
7.7
0.7
0.7
0.7
0.7
0.7
0.7
0.7
0.7
0.7
0.7
0.7
0.01
0.01
0.01
0.01
0.01
0.01
0.01
0.01
0.01
0.01
0.01
0.01
0.01
0.01
0.01
0.01
0.01
0.01
0.01
0.05
0.1
0.01
0.01
0.01
0.01
0.01
0.01
0.01
0.01
0.22
0.22
0.22
0.22
0.22
0.22
0.22
0.22
0.22
0.22
0.22
0.22
0.22
0.22
0.22
0.22
0.22
0.22
0.22
0.22
0.22
0.22
0.6
2.6
0.22
0.22
0.22
0.22
0.22
30
30
30
30
30
30
30
30
30
30
30
30
30
30
30
30
30
30
30
30
30
30
30
30
30
45
60
30
30
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
0.2
0.4
-0.898306
-0.398999
0.074336
-0.789894
-0.392932
0.001118
-1.14962
-1.46438
-1.73085
-1.14962
-1.14077
-1.1269
-1.14962
-1.46652
-1.73456
-1.14962
-1.1488
-1.14548
-1.14962
-1.14951
-1.14937
-1.14962
-1.15355
-1.16052
-1.14962
-1.16522
-1.18555
-1.32787
-1.28076
0.292389
0.291575
0.289413
0.224745
0.237695
0.249848
0.212403
0.205491
0.200752
0.212403
0.212579
0.212861
0.212403
0.205471
0.200725
0.212403
0.169949
0.0401351
0.212403
0.202479
0.19008
0.212403
0.212102
0.211644
0.212403
0.211814
0.211045
0.275895
0.258952
-0.743451
-0.313333
0.103076
-0.745896
-0.307984
0.12476
-1.14453
-1.46032
-1.72746
-1.14453
-1.13565
-1.12174
-1.14453
-1.46247
-1.73118
-1.14453
-1.14311
-1.13925
-1.14453
-1.14431
-1.14404
-1.14453
-1.14389
-1.14131
-1.14453
-1.16103
-1.18257
-1.32363
-1.27621
0.208039
0.235192
0.258818
0.207936
0.236809
0.26331
0.180087
0.167525
0.158989
0.180087
0.180409
0.180922
0.180087
0.16749
0.158941
0.180087
0.113843
0.003882
3
0.180087
0.16123
0.137683
0.180087
0.180181
0.180592
0.180087
0.178894
0.177333
0.303219
0.268367
Gr Gc M m k R 0f 0 0
0.2
0.4
0.6
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.2
0.4
0.6
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
1
1
1
1
1
1
1
2
3
1
1
1
1
1
1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.2
0.4
0.6
0.1
0.1
0.1
1
1
1
1
1
1
1
1
1
1
1
1
1
2
3
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1.10482
-1.01549
-0.926519
-1.10954
-1.02946
-0.949488
-1.14962
-1.46438
-1.73085
-1.14077
-1.10914
-1.06687
-1.14962
-1.46652
-1.73456
0.214188
0.217706
0.221155
0.213806
0.216589
0.21934
0.212403
0.205491
0.200752
0.212579
0.213232
0.214162
0.212403
0.205471
0.200725
0.463457
0.464835
0.466196
0.463316
0.46442
0.465518
0.462762
0.459924
0.457976
0.462837
0.463114
0.463507
0.462762
0.459914
0.457962
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TABLE III
Variation of 0f , 0 and 0 for different Pr, Ec, Sc, kr, γ, fw with Gr = 0.1,
Gc =0.1,M =1,m = 0.1,k =1 and R=1 .
0.1
0.1
0.1
0.1
0.1
0.1
1
1
1
0.1
0.1
0.1
1
1
1
1
2
3
-1.14962
-1.15006
-1.14995
0.212403
0.236925
0.230594
0.462762
0.462749
0.462752
Pr Ec Sc Kr γ fw 0f 0 0
0.7
1.7
7.7
0.7
0.7
0.7
0.7
0.7
0.7
0.7
0.7
0.7
0.7
0.7
0.7
0.7
0.7
0.7
0.7
0.01
0.01
0.01
0.01
0.03
0.05
0.01
0.01
0.01
0.01
0.01
0.01
0.01
0.01
0.01
0.01
0.01
0.01
0.01
0.22
0.22
0.22
0.22
0.22
0.22
0.22
0.6
1.0
0.22
0.22
0.22
0.22
0.22
0.22
0.22
0.22
0.22
0.22
1
1
1
1
1
1
1
1
1
1
2
3
1
1
1
1
1
1
1
300
300
300
300
300
300
300
300
300
300
300
300
300
450
600
600
600
600
600
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
-1
-0.5
0.5
1
-1.14962
-1.1488
-1.14548
-1.14962
-1.14957
-1.14951
-1.14962
-1.15355
-1.15597
-1.14962
-1.15257
-1.15471
-1.14962
-1.16522
-1.18555
-1.64629
-1.5062
-1.25785
-1.14962
0.212403
0.169949
0.0401351
0.212403
0.20744
0.202479
0.212403
0.212102
0.21193
0.212403
0.212179
0.212026
0.212403
0.211814
0.211045
0.392528
0.340839
0.250749
0.212403
0.462762
0.462787
0.462900
0.462762
0.462763
0.462764
0.462762
0.688359
0.84358
0.462762
0.637593
0.780226
0.462762
0.462532
0.462231
0.572475
0.543122
0.488276
0.462762
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