International Journal of Advancements in Research & Technology, Volume 2, Issue 7, July-2013
ISSN 2278-7763
170
Unsteady MHD thermal diffusive, radiative and free
convective flow past a vertical porous plate through nonhomogeneous porous medium
K.V.S.Raju1,
M.C.Raju2,
S.Venkataramana3 and
G.S.S.Raju4
1. Department of Mathematics, KORM college of Engineering, Kadapa, A.P, India, Email:
venky.sakku@gmail.com
2. Department of Humanities and Sciences, AITS, Rajampet, A.P, India, Email:
mcrmaths@yahoo.co.in
3. Department of Mathematics, Sri Venkateswara University, Tirupati, A.P, India.
4. Department of mathematics, JNTUACEP, Pulivendula, A.P, India.
ABSTRACT
The present paper deals with the analysis of unsteady free convection flow through a
porous medium of variable permeability bounded by an infinite porous vertical plate in slip
flow regime taking into account the radiation and temperature gradient dependent heat source.
The flow is considered under the influence of magnetic field applied normal to the flow. The
permeability of the porous medium and the suction velocity at the plate decreases
exponentially with time about a constant mean. Approximate solutions for velocity and
temperature fields are obtained using perturbation technique. The expressions for skinfriction and rate of heat transfer are also derived. The results obtained are discussed for
cooling case (G r > 0) of the plate. The effects of various physical parameters, encountered
into the problem, on the velocity field are numerically shown through graphs while the effects
on skin-friction and rate of heat transfer the numerically discussed through tables.
Keywords: MHD, thermal diffusion, radiation, free convection and non-homogeneous porous
medium.
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1. INTRADUCTION:
The study of flow through porous medium
finds
application
in
geophysics,
agricultural engineering and technology.
Further the free convection flow in
enclosures has become increasingly
important in engineering applications in
recent years due to fact growth of
technology, effecting cooling of electronic
equations
ranges
from
individual
transistors to mainframe computers and so
on [1]. Several authors including Gupta
and Sharma[2], Singh et. al., [3],
T.S.Reddy et. al. [4] has studied MHD
flow through porous medium in slip flow
regime. A Soret effect due to Natural
convection between heated inclined plates
with magnetic field was studied by Raju et
al. [5]. In their study Reddy et al. [6-8],
Copyright © 2013 SciResPub.
considered effect of thermal diffusion on
heat and mass transfer flow problems in
different geometries. Ravikumar et al. [9]
investigated, heat and mass transfer effects
on MHD flow of viscous fluid through
non-homogeneous porous medium in
presence of temperature dependent heat
source. MHD transient free convection and
chemically reactive flow past a porous
vertical plate with radiation and
temperature gradient dependent heat source
in slip flow regime was investigated by
Rao et al. [10]. Non-Darcy fully developed
mixed convection in a porous medium
channel with heat generation/absorption
and hydromagnetic effects were considered
by Chamkha [11]. Non-similar solutions
for heat and mass transfer by
hydromagnetic mixed convection flow
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International Journal of Advancements in Research & Technology, Volume 2, Issue 7, July-2013
ISSN 2278-7763
over a plate in porous media with surface
suction or injection and effects of heat
generation/absorption and thermophoresis
on hydromagnetic flow with heat and mass
transfer over a flat surface were considered
by
Chamkha
[12-13].
Magetohydrodynamics is attracting the
attention of the many authors due to its
applications in geophysics; it is applied to
study the stellar and solar structures,
interstellar matter, radio propagation
through the ionosphere etc. In engineering
in MHD pumps, MHD bearings etc. at high
temperatures attained in some engineering
devices, gas, for example, can be ionized
and so becomes an electrical conductor.
The ionized gas or plasma can be made to
interact with the magnetic and alter heat
transfer and friction characteristic. Since
some fluids can also emit and absorb
thermal radiation, it is of interest to study
the effect of magnetic field on the
temperature distribution and heat transfer
when the fluid is not only an electrical
conductor but also when it is capable of
emitting and absorbing thermal radiation.
This is of interest because heat transfer by
thermal radiation is becoming of greater
importance when we are concerned with
space applications and higher operating
temperatures. The effects of transversely
magnetic field, on the flow of an
electrically conducting fluid past an
impulsively started infinite isothermal
vertical plate was studied by Soundalgekar
et al. [14]. Again, Soundalgekar and
Takhar [15] studied the effect of radiation
on the natural convection flow of a gas
past a semi-infinite plate using the CoglyVincentine-Gilles equilibrium model. For
the same gas Takhar et al. [16] investigated
the effect of radiation on the MHD free
convection flow past a semi-infinite
vertical plate. Later, Hossain et al. [17]
studied the effect of radiation on free
171
convection from a porous vertical plate.
Muthucumarswamy and Kumar [18]
examined the thermal radiation effects on
moving infinite vertical plate in presence
of variable temperature and mass diffusion.
An analytical solution for unsteady free
convection in porous media has been
studied by Magyari et al. [19].
Simultaneous radiative and convective heat
transfer in a variable porosity medium was
studied by Nagaraju et al. [20]. An
analytical solution for hydromagnetic
natural convection flow of a particulate
suspension through a channel with heat
generation or absorption effects was
studied by Subaie and Chamkha [21].
Recently Patil et al. [22] studied double
diffusive mixed convection flow over a
moving vertical plate in the presence of
internal heat generation and chemical
reaction. Soret effects due to natural
convection between heated inclined plates
were investigated by Raju et al. [23].
Motivated by above cited work, in
this chapter we have considered an
unsteady MHD free convection flow of a
viscous fluid past a vertical porous plate
embedded with porous medium in the
presence of thermal diffusion. In obtaining
the solution, the terms regarding radiation
effect and temperature gradient dependent
heat source are taken into account in
energy equation. The permeability of the
porous medium and the suction velocity
are considered to be exponentially
decreasing function of time.
2.
FORMULATION
OF
THE
PROBLEM:
We have considered a two
dimensional free convective, radiative and
thermal diffusive transient viscous,
incompressible and electrically conducting
fluid past an infinite vertical porous plate.
A rectangular Cartesian co-ordinate system
is considered. Let x*-axis is taken along
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ISSN 2278-7763
172
∂θ * * ∂θ *
k ∂ 2θ *
+
v
=
∂t*
∂y* ρ C p ∂y*2
the plate in the direction of the flow and
y*-axis is taken perpendicular to it. It is
assumed that a uniform magnetic field of
strength B o is applied in the perpendicular
direction of the flow. It is also considered
that the presence of non-homogeneous
porous medium and time dependent
suction velocity that adds near the porous
plate; that to in slip flow regime. Here the
flow is considered in the presence of
thermal radiation.
In addition with
temperature gradient dependent heat source
effect is taken into account. It is also
assumed that the induced magnetic field is
neglected due to the negligible effect of the
magnetic Reynolds number, which is taken
to be very small. Since, the length of the
plate is assumed to be infinite in length and
the flow is along the direction of x*-axis,
all the variables are function of y and t
only. Here the flow is due to the
buoyancy force caused by temperature
differences between the porous plate and
the fluid.
In this analysis Darcy’s
dissipation and viscous dissipation are
neglected in the case of small velocity [1],
under the above assumption and the usual
Bossiness’s approximation, the unsteady
flow is governed by the following
equations of continuity, momentum and
energy as follows
(3)
Q* ∂θ *
1 ∂qr
−
+
ρ C p ∂y* ρ C p ∂y*
*
∂C *
∂ 2θ *
∂ 2T *
* ∂C
+
=
+
v
D
D
1
2
2
∂t *
∂y *
∂y *
∂y *
(4)
The corresponding boundary conditions
relevant to the problem are:
 ∂u * 
u * = L1  *  , θ = θ w* , C = C w*
 ∂y 
at y* = 0
u * → 0 , θ * → θ ∞* , C * → C∞* as y* → ∞
(5)
Where u' and v' are the components of
velocity along X' and Y' directions, g is the
acceleration due to gravity ,β is the
coefficient of volume expansion is the
kinematic viscosity, k' is the permeability
of the porous medium, ρ is the density of
the fluid ,σ is the electrical conductivity of
the fluid , Bo is the uniform magnetic field,
T' is the temperature, Kt is the thermal
conductivity, Cp is the specific heat at
constant pressure ,qr is the radiative heat
flux, Q' is the heat source, Tw' is the
temperature of the wall as well as the
temperature of the fluid at the plate, T∞' is
the temperature of the fluid far away from
the plate. L being the mean free path and
m1 the Maxwell’s reflection coefficient.
The equation of continuity yields
that v is either a constant or some
function of time. Hence we assume from
equation (1), we assume the suction
velocity as
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∂V *
=0
∂y *
(1)
*
∂u*
* ∂u
u
g βT T * − Tα* + g β C C * − Cα*
+
=
∂t*
∂y*
(
)
(
σ Bo2 *
∂ 2u *
υ
u
+υ *2 − * u * −
k (t )
∂y
ρ
(
V * = −Vo* 1 + εe − n t
* 8
)
( 6)
*
(2)
Copyright © 2013 SciResPub.
)
Where V o >0 is the suction velocity at the
plate and n' is a positive constant. The
negative sign indicates that the suction
velocity acts towards the plate. The
radiative heat flux is taken in the following
form given by Takhar et al. [16],
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International Journal of Advancements in Research & Technology, Volume 2, Issue 7, July-2013
ISSN 2278-7763
∂q r
= 4 T * − Tα* I
∂y
(
)
(7)
α
∂ebλ
dλ , K λ n is the
∂T
Where I = ∫ K λ n
0
absorption coefficient at the plate and ebλ
is plank function. The permeability k'(t) of
the porous medium is considered in the
following form:
(
k * (t ) = K o* 1 + ε e − n t
* *
)
(8)
Now we introduce the following
dimensionless variables and parameters
υ
Sc =
y=
θ=
D
y *Vo*
υ
)
)
(
,n =
4υ n *
Vo*
2
,t =
Vo*t *
,
4υ
σ Bo2 υ
T * − Tα*
C * − Cα*
,
,
,
M
=
C
=
2
Tw* − Tα*
C w* − Cα*
ρVo*
Ko =
Gm =
R=
(
D1 Tw* − Tα*
u*
,
u= *,
Vo
υ C w* − Cα*
, So =
K o* Vo*
3
υ2
υgβ T (Tw* − Tα* )
υgβ C (C w* − Cα* )
4υI
ρC pVo*
2
3
,H =
Vo*
, Pr =
Q*
ρC pVo*
2
µC p
K
,
,
(9)
Substuting the dimensionless variables
given by equation (9). In the set of
equations (1) - (4) and in the corresponding
boundary condition of equation (5) we
obtained the following dimensionless
governing
equation
and
boundary
conditions respectively
1 ∂u
∂u
− (1 + ε e − nt ) = Grθ + GmC
u ∂t
∂y
(10)

1
∂u 2 
u
+ 2 − M +
− nt
∂y 
1
K
e
+
ε

(
)
o

1 ∂θ
1 ∂ 2θ
∂θ
− nt ∂θ
− 1 + εe
=
− Rθ + H
2
∂y Pr ∂y
u ∂t
∂y
(11)
(
∂C
∂ 2θ
1 ∂C
1 ∂ 2C
− 1 + εe − nt
=
+
S
o
∂y S c ∂y 2
u ∂t
∂y 2
(12)
 ∂u 
u = h  , θ = 1 , C = 1 at y = 0
 ∂y 
(
)
u =0, θ = 0, C = 0 as y →∞
)
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(13)
3. SOLUTION OF THE PROBLEM
In order to solve the set of partial
differential equation (10) – (12), along
with the boundary conditions (13), we
reduce them into ordinary differential
equations .To obtain the solution, the
following perturbation method, which is
given by N.P. Singh et al. [1] is used for
‘ε’ << 1.
( )
⋅θ ( y ) + O (ε )
⋅ C ( y ) + O ( ε ) (14)
u ( y, t ) = uo ( y ) + ε e − nt ⋅ u1 ( y ) + O ε 2
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, Gr =
Vo*
173
θ ( y, t ) = θ o ( y ) + ε e − nt
1
C ( y, t ) = Co ( y ) + ε e − nt
1
2
2
Substituting equation (14) in set of
equation (10) – (12) and equating the coefficient of zero and first order terms of ‘ε’
and neglecting the higher order terms of
‘ε’, we obtain the following set of O.D.E in
terms of harmonic and non-harmonic
function (u o ,θ o , C o ) and (u1,θ1 ,C1 )
uo11 + uo1 − M 1uo =
−Grθ o − GmCo
(15)
θ o11 + (1 + H )Prθ o1 − RPrθ o = 0
(16)
C o11 + S c C o1 + S c S oθ o11 = 0
(17)
−Grθ1 − GmC1 − uo1 +
u111 + u11 − M 2u1 =
uo
ko
(18)
n

θ111 + (1 + H )Prθ11 −  R −  Prθ1 = −θ o1 Pr
4

(19)
1 11
n
C1 + C1 − C1 + C o1 + S oθ o11 = 0 (20)
Sc
4
Where M 1 =M+
1
1
1
, M 2 =M+
−
Ko
Ko 4
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International Journal of Advancements in Research & Technology, Volume 2, Issue 7, July-2013
ISSN 2278-7763
The corresponding boundary conditions
are
u o = hu o1 , u1 = hu1 , θ o = 1 , θ o = 0 , C o = 1 ,
C1 = 0 at y =0
174
The expression for the skin-friction (τ) at
the plate is given by
 du 
 duo 
 du1 
− nt
=
τ =


 +ε   e
dy
dy
dy
  y 0=

y 0 =
 y 0
=
u o → 0 , u1 → 0 , θ o → 0 , θ1 → 0 , C o = 1 ,
= ( − K 6C10 + K 2l12 + SC l10 ) + e(− K8C12 − K 2l25 (31)
(21)
C1 = 0 at y → ∞
The solutions of these coupled linear
differential equations (15) – (20) under the
corresponding boundary conditions are
given by
− K 4l26 − M 5 l15 + SC l27 − K 6l28 )e
− nt
The expression for the rate of heat transfer
in terms of Nusselt number (N u) is given
by
 dθ 
 dθ o 
 dθ1  − nt
=
Nu =


 +ε   e
1
(32)
 dy  y 0=
 dy  y 0 =
 dy  y 0
=
l12 + l10 + h ( k2l12 + Sc l10 )  e − k6 y
uo ( y )
=
(1 + hk6 )
=
(− K 2 ) + el1 ( K 4 − K 2 )e− nt
−l12 e − k2 y − l10 e − Sc y
The expression for the rate of heat transfer
(22)
in terms of Sherwood number (Sh ) is given
− M y
by
u ( y ) = C e − k8 y + l e − k2 y + l e − k4 y + l e 5
1
−l27 e
12
− Sc y
25
+ l28e
26
15
− k6 y
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(23)
(24)
θo ( y) = e− k y
2
(
=
θ1 ( y ) l1 e − k2 y − e − k4 y
)
(25)
Co ( y ) =
(1 − l2 ) e− Sc y + l2e− k2 y
(26)
− M5 y
+ l3e − Sc y + l7 e − k2 y + l6 e − k4 y
(27)
The constants are given in the
appendix. Finally the expressions for
velocity, temperature and concentration
given by
C1 ( y ) =
−l8e
−k y 
 1 
l12 + l10 + h ( k2l12 + Sc l10 ) e 6
 1 + hk 


6)
u (
+


− k2 y
−S y
− l10 e c
−l12e

 C e − k8 y + l e − k2 y + l e − k4 y 
25
26
 12
 e− nt
ε

− M5 y
−k y 
−S y
− l27 e c + l28e 6 
+l15e
(28)
θ =+
e − k y ε l1 ( e − k y − e − k y )  e − nt
2
2
4
(29)
C=
(1 − l2 ) e− Sc y + l2e− k2 y
+ε  −l8e

− M5 y
 dC 
 dC0 
 dC1  − nt
=

  + e  e
 dy  y 0=
 dy  y 0 =
 dy  y 0
=
+ l3e− Sc y + l7 e− k2 y + l6 e− k4 y  e− nt

(30)
Copyright © 2013 SciResPub.
=(− SC + l2 ( SC − K 2 ) + e(−l8 M 5 − SC l3 − K 2l7 − K 4l6 )e− nt
(33)
4. RESULTS AND DISCUSSION:
In order to guide the physical
insight of the problem, numerical
computations have been carried out for
velocity, temperature and concentration
distributions and as well as the coefficient
of skin friction, rate of heat transfer in
threw form of Nusselt number and the rate
of mass transfer in the form of Sherwhood
number and the effects of various physical
parameters on these flow quantities are
studied through graphs and tables. The
values of Prandtl number are chosen as
0.71 and 7.0 which correspond to air and
water respectively. The values of Schmidt
number are chosen as 0.16, 0.22, 0.64, 0.78
and 0.98 which correspond to water vapor,
NH 3 , CO 2 etc. The values all other
parameters are arbitrarily chosen.
Velocity profiles are displayed through
figures 1- 9. Effects of Modified Grashof
number Gm on velocity distribution are
shown in figure 1, from this figure it is
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International Journal of Advancements in Research & Technology, Volume 2, Issue 7, July-2013
ISSN 2278-7763
noticed that velocity increases with an
increasing in modified Grashof number. A
similar effect is noticed in the presence of
thermal Grashof number which is shown in
fig. 2. Fig. 3 depicts the effects of magnetic
parameter M on velocity. From this figure
it is noticed that velocity decreases as M
increases. It is due the fact that magnetic
force which is applied in the normal
direction to the flow produces a drag force
which is known as Lorentz force. The
presence of slip on the boundary layer is
presented in figure 4, as h increase velocity
u also increases. Effects of Radiation
parameter R, Permeability parameter Ko
Schmitt number Sc and Soret number So
are presented through figures 5-8. From
these figures it is observed that velocity
decreases as R and Ko increase, where as it
shows reverse effect in the case of Sc and
So.
Temperature profiles with the variations in
heat source parameter H, radiation
parameter R are displayed through figures
9 – 10. It is seen that temperature decreases
with an increase in both radiation
parameter and heat source parameter,
which shrink the thermal boundary layer.
Concentration profiles are displayed with
the variations in Schmidt number Sc and
soret number So in figures 11-12
respectively.From these figures it is
noticed that concentration decreases with
an increase in Sc and So.
The effects of all these parameters
on Skin friction, Nusselt number and
Sherwood number are presented in tables
1-3. From table 1 we observe that skin
friction increases with an increase in
magnetic parameter M, Schmidt number
Sc, soret number So, time t, Prandtl
number Pr, Permeability parameter Ko ,
modified grashof number Gm, and heat
source parameter H .Where as in the case
of other parameters
such as Grashof
175
number Gr, slip parameter h and radiation
parameter R increase
skin friction
decreases.Similarly the effects of radiation
parameter R, heat source parameter H and
Prandtl number Pr are presented in table 2.
Nusselt number decreases with an increase
in R,H and Pr respectively. In table 3
variations in Sherwood number with
respect to Schmidt number Sc and Soret
number So are shown. Sherwood number
is observed to increase with an increase in
Schmidt number Sc and Soret number So.
4
3
2
1
Pr=0.71
R=1
n=0.2
Gm=5,10,15,20
So=0.5
k0=10
Gr=15
H=1
t=1
Sc=0.60
M=2
Gr=15
h=0.4
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Copyright © 2013 SciResPub.
u
0
-1
-2
-3
-4
-5
0
0.5
1
1.5
2
2.5
3
y
Figure.1. Effect of Gm on Velocity
14
Pr=0.71
R=1
n=0.2
So=0.5
k0=10
Gr=15
12
10
, H=1
, t=1
,Sc=0.60
,M=2
,Gr=15
,h=0.4
u
8
6
4
2
Gr=10,15,20,25
0
0
0.2
0.4
0.6
0.8
1
y
1.2
1.4
1.6
1.8
2
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160
Figure.2 Effect of Gr on Velocity
Pr=0.71
R=1
n=0.2
So=0.5
k0=10
Gr=15
140
120
5
M=2,3,5
4
176
H=1
t=1
Sc=0.60
M=2
Gr=15
h=0.4
100
u
3
80
R=1,2,3,4
u
2
60
1
Pr=0.71
R=1
t=1
Sc=0.64
M=2
Gr=15
0
-1
-2
-3
0
0.5
1
H=1
e=0.1
n=0.2
So=0.5
KO=10
Gm=15
1.5
2
2.5
40
20
0
0
1
2
3
4
5
6
7
8
9
10
y
Figure 5. Effect of R on Velocity
3
y
6
Figure 3 Effect of M on Velocity
K0=10,20,50
4
140
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2
120
u
100
Pr=0.71
R=1
t=1
Sc=0.64
KO=10
Gm=15
u
80
60
H=1
e=0.1
n=0.2
So=0.5
Gr=15
h=0.4
0
Pr=0.71
R=1
t=2
So=0.5
KO=15
Gm=10
-2
-4
H=2
e=0.1
n=0.2
M=1
Gr=10
h=0.1
40
-6
0
0.5
1
1.5
2.5
2
3
y
20
Figure 6. Effect of K0 on Velocity
0
h=0.1,0.2,0.3,0.4
0
0.5
1
1.5
2
2.5
3
y
10
Figure4. Effect of h on Velocity
Sc = 0.16, 0.22,0.60,0.90,1.20
5
0
u
-5
-10
Pr=0.71 H=1
R=1
e=0.1
t=1
n=0.2
Sc=0.64 So=0.5
Gr=15
KO=10
Gm=15 h=0.4
-15
-20
-25
-30
0
0.5
1
1.5
2
2.5
3
y
Figure.7 Effect of Sc on Velocity
Copyright © 2013 SciResPub.
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177
6
1
4
Sc=0.16,0.22,0.6,0.78
0.9
0.8
Sc=0.5,1.0,1.5,2.0
2
Pr=0.71
R=1
t=2
So=2
KO=50
Gm=10
0
-2
H=2
e=0
n=0.2
M=2
Gr=10
h=0.1
0.6
C
u
0.7
0.5
0.4
0.3
-4
0.2
-6
0
0.5
1
1.5
2
2.5
3
0.1
y
0
0
0.5
1
2
1.5
2.5
3
y
Figure 8. Effect of So on Velocity
Figure 11. Effect of Sc on Concentration
1
1
0.9
So=1,2,3
IJOART
Pr=0.71
e=0.1
n=0.2
So=0.5
KO=10
Gm=15
0.8
0.7
0.5
Pr=0.71
R=1.0
e=0.1
M=2
H=0.1
So=3
0.9
0.8
0.7
H=0.1
y=0.01
t=1.0
KO=20
n=0.5
C
θ
0.6
R=1
t=1
Sc=0.64
M=2
Gr=15
h=0.4
0.4
0.6
0.3
0.5
H=0.1, 0.5,1,1.5
0.2
0.4
0.1
0
0
1
0.5
2
1.5
3
2.5
y
0
0.5
1
1.5
2
2.5
3
y
Figure.9 Effect of H on Temperature
Figure 12. Effect of So on Concentration
1
Table 2. Variations in
Nusselt number
0.9
Pr=0.71 H=1
t=1
n=0.2
Sc=0.64 So=0.5
M=2
KO=10
Gr=15 Gm=15
h=0.4
0.8
0.7
0.6
θ
R=1, 1.5, 2
0.5
0.4
0.3
0.2
0.1
0
0
0.5
1
1.5
2
2.5
3
y
Figure 10. Effect of radiation parameter R
on Temperature
Copyright © 2013 SciResPub.
R
H
Pr
Nu
1
2
3
1
1
1
1
2
2
2
1
2
3
2
0.71
0.71
0.71
0.71
0.71
0.71
0.71
-2.4227
-2.6622
-2.8679
-1.8119
-2.4227
-3.0771
-1.8119
IJOART
International Journal of Advancements in Research & Technology, Volume 2, Issue 7, July-2013
ISSN 2278-7763
1
2
7.0
-14.4836
Table.3. Variations in
Sherwood number:
Sc
So
Sh
0.16
0.22
0.38
0.62
0.78
0.16
0.16
0.16
2
2
2
2
2
3
4
5
4.4748
6.1528
10.6275
17.3394
21.8140
9.3391
12.5254
15.7117
5. CONCLUSIONS:
In this chapter we studied
the analysis of unsteady free convection
flow through a porous medium of variable
permeability bounded by an infinite porous
vertical plate in slip flow regime in the
presence of radiation and temperature
gradient dependent heat source. The nondimensional governing equations are
solved by perturbation technique. In the
analysis of the flow the following
conclusions are made.
178
vi. Sherwood number increases with an
increase of Sc and So.
6. REFRENCES
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Atul K Singh, “ MHD free convection
flow of viscous fluid past a porous
vertical plate through non homogeneous
porous medium with radiation and
temperature gradient dependent heat
source in slip flow regime”, Ultra
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2. M.Gupta. and S.Sharma., MHD flow of
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3. N.P.Singh,R.V.Singh, and Singh,Atul
Kumar, Flow of a dusty viscoelastic
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“ The effect of slip condition, Radiation
and chemical reaction on unsteady
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5. M.C.Raju, S.V.K Varma, P.V.Reddy
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Natural convection between Heated
Inclined Plates with Magnetic Field”,
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0379-4318.
6. N. Ananda Reddy, S. V. K. Varma
and M. C. Raju, “Thermo diffusion
and
chemical
effects
with
simultaneous thermal and mass
diffusion in MHD mixed convection
flow with ohmicheating”., Journal of
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Engineering 6(2009) 84-93.
IJOART
i. Velocity increases with increase in Gm,
Gr, h, Ko and So but it shows the reverse
effects in case of M, R Ko and Sc.
ii. Temperature decreases with an increase
of H and R.
iii. Concentration decreases with increase of
Sc and So.
iv. Skin friction increases with increase in
M, Sc, So, t, Pr, Ko, Gm and H but it
shows the reverse effects in case of Gr, h
and R.
v. Nusslet number decreases with an
increase of R, H and Pr
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International Journal of Advancements in Research & Technology, Volume 2, Issue 7, July-2013
ISSN 2278-7763
7.
N. Ananda Reddy, M.C.Raju and
S.V.K. Varma., “Soret effects on
MHD
three
dimensional
free
convection Couette flow with heat and
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mechanics, 2(1), 2010,pp.51-60.
8. M.C.Raju,
S.V.K.Varma,
N.
AnandaReddy.,
“MHD
Thermal
diffusion Natural convection flow
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Future Engineering and Technology.Vol.6,
No.2, pp.45-48
9. Ravikumar V, Raju M.C, Raju
&G.S.S., Heat and Mass Transfer
Effects on MHD Flow of Viscous Fluid
through Non-Homogeneous Porous
Medium in Presence of Temperature
Dependent Heat Source, International
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1597-1604.
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G.V.Reddy,
M.C.Raju,
S.V.K.Varma, MHD transient free
convection and chemically reactive
flow past a porous vertical plate with
radiation and temperature gradient
dependent heat source in slip flow
regime, IOSR Journal of Applied
Physics, Vol. 3(6), 2013, 22-32.
11. A.J. Chamkha, "Non-Darcy Fully
Developed Mixed Convection in a
Porous Medium Channel with Heat
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653 -675, 1997.
12. A.J. Chamkha, “Non-Similar Solutions
for Heat and Mass Transfer by
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Over a Plate in Porous Media with
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or
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13. A.J. Chamkha and C. Issa, “Effects of
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Generation/Absorption
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Thermophoresis on Hydromagnetic
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Over a Flat Surface ”. International
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& Fluid Flow, Volume 10, pp. 432449, 2000
14. Soundalgekhar V. M, Gupta S.K. and
Birajdar N.S., Effects of mass transfer
and free convection flow past currents
on MHD stokes problem for a vertical
plate, Nuclear Eng. Des., 53, pp.339346.
15. Soundalgekar V.M. and Takhar H.S.,
Radiative convective flow past a semiinfinite vertical plate, Modelling
Measure and Cont., 51, 31-40, 1992.
16. Tahkar H.S, Gorla S.R. and
Soundalgekar V.M., Radiation effects
on MHD free convection flow of a
radiating gas past a semi-infinite
vertical plate, Int. J. Numerical
Methods heat fluid flow, 6, 77-83,
1996.
17. Hossain A.M, Alim M.A. and Rees,
D.A.S., Effect of radiation on free
convection from a porous a vertical
plate, Int.J.Heat Mass Transfer, 42,
181-191, 1999.
18. Muthucumarswamy R. and Kumar
G.S., Heat and Mass Transfer effects
on moving vertical plate in the
presence of thermal radiation, Theoret.
Appl. Mach., 31(1), 35-46, 2004.
19. Magyari E, Pop I. and Keller B.,
Analytical solutions for unsteady free
convection flow through a porous
media, J.Eng. math., 48, 93-104, 2004.
20. P. Nagaraju, A.J. Chamkha, H.S.
Takhar, and B.C. Chandrasekhara,
“Simultaneous
Radiative
and
Convective Heat Transfer in a Variable
Porosity Medium” Heat and Mass
IJOART
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IJOART
International Journal of Advancements in Research & Technology, Volume 2, Issue 7, July-2013
ISSN 2278-7763
Transfer, Volume 37, pp. 243-250,
2001.
21. M. Al-Subaie and A. J. Chamkha,
“Analytical Solutions for Hydromagnetic
Natural Convection Flow of a Particulate
Suspension Through a Channel with
Heat Generation or Absorption Effects.”
Heat and Mass Transfer, Volume 39, pp.
701-707, 2003.
22. P.M. Patil, S. Roy and A.J. Chamkha,
“Double Diffusive Mixed Convection
Flow over a Moving Vertical Plate in the
Presence of Internal Heat Generation and
Chemical Reaction.” Turkish Journal of
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Volume 33, pp. 193-206, 2009.
23. M.C.Raju, S.V.K.Varma, P.V.Reddy
and Sumon Saha, Soret effects due to
natural convection between heated
inclined plated with magnetic field,
Journal of Mechanical Engineering,
Vol. ME39, No.Dec 2008, 43-48.
7. Appendix:
180
),
,
,
,
,
,
,
,
,
,
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,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
Copyright © 2013 SciResPub.
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International Journal of Advancements in Research & Technology, Volume 2, Issue 7, July-2013
ISSN 2278-7763
181
Table 1: Variations in Skin friction
M
Sc
So
t
Pr
K
1
2
3
4
1
1
1
1
1
1
0.16
0.16
0.16
0.16
0.22
0.60
0.78
0.22
0.22
0.22
0.16
0.16
0.16
0.16
0.16
0.16
0.16
0.16
0.16
0.16
0.16
0.16
0.16
0.16
0.16
0.16
0.16
0.16
0.16
0.16
0.16
0.16
0.16
0.16
0.16
0.16
0.16
0.16
0.5
0.5
0.5
0.5
0.5
0.5
0.5
10
15
20
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
1
1
1
1
1
1
1
1
1
1
0.5
1.0
1.5
2.0
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
0.71
0.71
0.71
0.71
0.71
0.71
0.71
0.71
0.71
0.71
0.71
0.71
0.71
0.71
0.71
7.0
0.71
0.71
0.71
0.71
0.71
0.71
0.71
0.71
0.71
0.71
0.71
0.71
0.71
0.71
0.71
0.71
0.71
0.71
0.71
0.71
0.71
0.71
15
15
15
15
15
15
15
15
15
15
15
15
15
15
15
15
10
20
30
40
15
15
15
15
15
15
15
15
15
15
15
15
15
15
15
15
15
15
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
o
G
r
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
05
10
15
20
10
10
10
10
10
10
10
10
10
10
10
10
10
10
G
m
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
05
10
15
20
10
10
10
10
10
10
10
10
10
10
H
h
R
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
1
0.5
01
1.5
2.0
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
0.1
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
0.2
0.3
0.4
0.1
0.1
0.1
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
1
1
1
1
1
1
1
1
1
2
2.5
3.0
IJOART
Copyright © 2013 SciResPub.
13.5862
-49.7483
31.0249
27.6132
20.3152
59.3448
66.7688
26.1758
31.2704
35.5990
19.2401
20.3152
21.2880
22.1682
19.2401
28.5473
16.3763
20.5014
21.6662
22.2158
22.8607
16.3763
9.8919
3.4075
1.2817
22.8607
43.4628
60.0880
-59.5579
1.2817
11.6073
13.5172
78.5320
-95.3412
-124.8214
15.6280
14.6049
14.4756
IJOART