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100
AMSE JOURNALS-AMSE IIETA publication-2017-Series: Modelling C; Vol. 78; N°1; pp 100-115
Submitted Jan. 2017; Revised March 15, 2017, Accepted April 15, 2017
A Theoretical Study of Heat and Mass Transfer in Forced
Convective Chemically Reacting Radiating MHD Flow through
saturated Porous Medium over Fixed Horizontal Channel
Pooja Sharma*, Ruchi Saboo**
*,**Department of Mathematics & Statistics, Manipal University Jaipur, Jaipur, India
(*pooja_2383@yahoo.co.in, **ruchipsaboo@gmail.com)
Abstract
In the present study; the effect of chemical reaction, viscous dissipation are considered in the
steady forced convective chemically reactive radiating MHD flow through the horizontal channel
with joule heating. The channel is saturated with uniform porous medium and having a insulated,
impermeable bottom wall. The governing equations for velocity, temperature and concentration
field are obtained and solved. The effect of various physical paramteres over these fluids are
considred and showed through graphs. The expression for mean velocity, mean temperature and
mean concentration are also presented. The numerical values of Nusselt number, skin-friction &
Sherwood number are discussed through the table. It was concluded that; for high depth porous
medium channel the flud mean temperature and species mean concentration both rise high. Fluid
flow can be made fast in this geometrical fluid flow by increasing the depth of the porous
medium
Keywords
Mass transfer, Thermal radiation, Chemical reaction, Heat source, MHD.
1. Introduction
The behaviour of synchronized heat and mass transfer in the presence of magnetic field has
fascinated numerous investigators due to its several kinds of practices in engineering and scinces
such as polymer solutions, biomedicine, plasma astronomy, geological physics, processing of
food, oceanlogy and in many fluid flow problems. In recent years, different fluid flows in porous
medium has been considered by many of the researchers in different geometries in the presence
of various physical condition. Kuznestov et al.[1] studied the mass exchange of the glass and the
101
effect of thermal insulation on that. Abdullah[2] proposed the study of composite materials in
place of thermal insulation. The mixed convection flow is discussed by Reddy et al.[3] in the
presene of thermal diffusion and chemical reaction. Yao et al.[4] studied the heat transfer in
porous copper foam heat pipes. Zeng et al.[5] considered the small cavity to examine the heat
transfer and characteristics of natural convection. Magnetic field effect with heat and mass
transfer on the partial slip flow through vertical insulated porous plate was investigated by
Baoku[6]. In view of that Thakur et al.[7] extended the study to examinethe MHD flow of
micropolar fluid with variable viscosity in the same geometry. MHD natural convection flow was
also discussed by Jha et al.[8] in the vertical parallel plate microchannel. Umavathi et al.[9]
inspected the oscillatory flow with heat transfer in a porous medium channel. Bakkas et al.[10]
also considered the same geometry with rectangular blocks relasing heat flux with natural
convection heat transfer. The same study in porous media square cavity covered with cold walls
was probed by Duan et al.[11]. Joule heating and magnetic field effects in combined convection
through a lid driven cavity in the presence of rolled bottom surface was investigated by Pravin et
al.[12]. Liuw et al.[13] studied the secondary flow and heat transfer in horizontal channels heated
from below. Makinde et al.[14] also considered the channel filled with soaked porous medium
for temporal stability analysis of hydromagnetic flow. Mounuddin et al.[15] studied the steady
fluid flow through channel saturated with porous medium with impermeable and thermally
insulated bottom. Raju [16] extended their study with viscous dissipation and joule heating in the
presence of transvese magnetic field. Recently Zeghbid et al.[17] deliberated the mixed
convection in lid-driven cavities. Heat and mass transfer study between hot and cold vertex cores
inside a vertex tube was considered by Rafiee et al.[18]. Niche et al.[19] investigated the doufour
and soret effects in unsteady double diffusive natural convection. Chemically reacting MHD
visco-elastic fluid flow with thermal diffusion was studied by popoola et al.[20]. MHD flow
through rotating channel with mass transfer was explored by Ahmed et al.[21].
Motivated by above citied work we have made an attempt to extand the study of Raju et
al.[16] with mass transfer effects in the presence of heat source and with impact of thermal
radiation.
In the present study the heat and mass transfer in two dimensional chemically reactive steady
MHD forced convective flow of a viscous fluid through a horizontal channel soaked with porous
medium and fixed impermeable thermally insulated bottom surface in the presence of heat source
and thermal radiation.
102
2. Mathematical formulation
We consider a chemically reactive steady flow of a viscous incompressible, electrically
conducting, forced convective flow through a soaked porous medium of depth H with
impermeable and thermally insulated bottom in the presence of transverse magnetic field, heat
source and thermal radiation. The schematic diagram of the flow geometry is given in fig.1.
Fig.1. Flow configuration
According to figure 1, we consider a rectangular co-ordinate system, with the reference of
that the origin is considered on the bottom surface and
'
X
-axis is along the flow direction. The
height H of the porous medium is assumed in the
'
Y
-axis direction. Therefore the bottom is
considered as
'
Y
=0 and the free surface is considered as
'
Y
=H. The uniform magnetic field is
working in the transverse direction of the fluid flow. The upper side of the porous medium is
supposed as free surface which is open to environment of constant temperature T1.
The following assumptions are supposed in the considered physical problem:
(1) The induced magnetic field is negligible in comparison of the functional magnetic field due to
the small magnetic Reynolds number.
(2) The fluid is considered as electrically conducting fluid. The positive negative charged particles
are approximately equivalent in a small finite volume of the fluid. Therefore the total additional
charge mass and executed electric field strength is expected to be nil.
(3) The fluid flow is caused due to uniform horizontal pressure gradient at the left open side of the
channel along the bottom line. The fluid velocity is
'
U
in the
'
X
-direction.
(4) The generalized Darcy’s law suggested by Yamamoto & Iwamura[22] is anticipated in the
momentum equation, in which classical Darcy force is considered to be associated with
convective acceleration and Newtonian- viscous stresses.
Under the above assumptions, the governing equations of the fluid flow are specified as
below:
103
2 ' ' 2 ' ' '
( ) 0
10
0
' '2 '
P d U U B U g C C
X d Y K
(1)
2'
' 2 ' '
' 2 '2 ' '
00
10
0
' '2 ' '
Qdq
T d T dU r
CpU B U T T
Cp
X d Y d Y d Y
(2)
2' ''0
10
'2
dC k C C
l
dY
(3)
It is assumed that the medium is optically thin with relatively low density. Following
Cogely[23] equilibrium model, the radiative heat flux term is given by
'' ' ' '
4( ) ; ,
0
''
0
e
qb
rT T I I K d
YT
(4)
where
K
is the absorption coefficient at the wall and
b
e
is the Plank constant. Here we
assume that the temperature differences within the flow are sufficiently small.
The appropriate boundary conditions for the velocity, temperature and concentration fields
are given as follows:
''
''
0, 0, 0 0.
''
d T d C
U atY
d Y d Y
'' ' ' ' '
0,
11
'
dU T T C C atY H
dY
(5)
3. Method of solution
Introducing the following non-dimensional quantities:
22
' ' ' 2 2
2 2 '
2
3'1
0 1 0 1
2 4 ' 3
10 ' ' ' '3
4
22 '1 0 0
10
02 ' '
'
'0 1 0 '
, , , , , , Pr , ,
, , ,
16
, , , , ,
,
u p Cp a
X a x Y a y H a h U P M
a a K
L
Pp
Br C C C C C L x
a T T x a
a T T T
g C C a
Ba S
S N M Gc S N
U Cp U Cp
T
T T T T X
10 2
22 ,,
l
TTL L k a
ax
(6)
Using (6), the governing equations (1)-(4) are reduced in non-dimensional form as follows:
2
1
2,
du u L GcC
dy
(7)
2
22
2
2Pr ,
d d u
a L u Br Br M u S N
d y d y
(8)
104
2
20,
dC C
dy
(9)
The corresponding boundary conditions are reduced as:
0, 0, 0 0,
d d C
u at y
d y d y
0, 1, 1
du C at y h
dy
(10)
On solving the above differential equations (7)-(9) under the boundary conditions (10), the
exact solutions are obtained for the velocity, temperature and concentration distribution. These
expressions are given in eqution (11), (12) and (13). We have formed now forming the following
significant features of the flow:
1
6 5 1 1 ,
yy yy
L
u B e B e B e B e
(11)
22
28 29 1 2 4 4 19 20 21
22
22 23 ,
y y y y y y
yy
yy
R e R e R e R e R e R e R R e R e
R e R e
(12)
22
,
yy
C A e A e
(13)
The flow rate:
65 1 1 1 7
0
hhh hh
B e B e L B e B e
u dy h B
(14)
The mean velocity:
65 11
1
07
11
hh h
h
h
B e B e L B e
h
u u dy Be
hh B
(15)
The mean temperature:
22
28 29 20
1 2 4 4 21
19
2
2
23
22
030
11 22
22
h h h
h h h
hh
h
h
h
R e R e R e
R e R e R e R e R e
Rh
dy Re
Re
hh R
(16)
Mean Concentration:
12
0
11
hyy
ee
C C dy a a
h h y
(17)
Where A1 to A2, B1 to B7 and R1 to R30 are the constant and not given due to sake of
breavity.
105
4. Skin-friction coefficient
The non-dimensional shearing stress in terms of coefficient of skin-friction on the free
surface(Cf)y=h and insulated bottom walls(Cf)y=0 given by the folowing expressions respectively:
23
1.
w
fhyh
u
Cy
a
(18)
23
00
1.
w
fyy
u
Cy
a
(19)
5. Nusselt number
The rate of heat transfer in terms of Nusselt number on the free surface (Nu)y=h and insulated
bottom wall (Nu)y=0 is given by following expressions respectively
yh yh
d
Nu dy
(20)
00
yy
d
Nu dy
(21)
6. Sherwood number
The rate of mass transfer in terms of Sherwood number on the free surface (Sh)y=h and
insulated bottom wall (Sh)y=0 is given by following expressions respectively
yh yh
dC
Sh dy
(22)
00
yy
dC
Sh dy
(23)
106
Table:1 The numerical values of skin-friction coefficient, Nusselt number and Sherwood
number at the channel for different physical values when a=0.5
7. Result & Discussion
This study reflects the fluid flow configuration through a horizontal channel soaked with
porous medium having fixed impermeable thermally insulated bottom surface. The significant
effects of numerous constraints or parameters on the heat and mass transfer in forced convective
flow in the presence of heat source and thermal radiation are calculated. In addition of that mean
temperature, mean velocity and mean concentration of species are also considered as well. These
results are presented through graphs and tables.
The influence of different physical parameters are showed by the figures 2 and 3. Fig.2
represents that due to the high intensity of applied magnetic field and chemical reaction
coefficient, the fluid velocity is reduced throughout the channel. But the reverse effect can be
seen by increasing the mass buoyancy parameter Gc. It is observed from fig.3 that the fluid
velocity is upsurging from the bottom surface to the top free surface with the rise of permeability
parameter. When the permeability of any porous medium rises the fluid flow become easier
through it, due to the larger void space in porous medium and less resistance in it. This is the
basic reason that the fluid velocity is perceived to the zero at the insulated bottom surface and
progressively it rises as it extents to the free surface and achieves an extreme value nearby. We
can see the effect of depth of the channel on the velocity field by the fig.3. It is clear, as the depth
increases; the fluid velocity is upsurging at all the points of the channel in the same porous
medium.
S.No
S
Pr
r
H
c
γ
N
M
(Cf)y=0
(Cf)y=h
(Nu)y=0
(Nu)y=h
(Sh)y=
0
(Sh)y=h
I
0.5
0.71
1
.5
3
1
1
1
-
-
-31.5
-35.7
-
-
II
1
0.71
1
.5
3
1
1
1
-
-
-33.2
-47.5
-
-
III
0.5
7.0
1
.5
3
1
1
1
-
-
-23.2
-52.39
-
-
IV
0.5
0.71
.5
.5
3
1
1
1
1.64
-0.86
-47.7
-53.62
-
-
V
0.5
0.71
1
3
1
1
1
1.64
-0.86
-8.80
-14.50
0.33
0.64
VI
0.5
0.71
1
.5
5
1
1
1
-1.04
-4.00
-59.9
-78.81
0.46
0.886
VII
0.5
0.71
1
.5
3
3
1
1
2.388
-0.22
1.35
3.2995
2.71
0.425
VIII
0.5
0.71
1
.5
3
1
2
1
1.15
-2.85
-33.9
-68.22
-
-
IX
0.5
0.71
1
.5
3
1
1
2
-
-
-26.4
-51.84
-
-
107
Fig.2. Velocity distribution versus y when h=1.
Fig.3. Velocity distribution verus y when M=1.5, Gc=3 and γ=1.
Fig.4 and 5 illustrate that the fluid temperature is rising with the increases of magnetic
intensity, Brinkman number, heat source and chemical reaction parameter while adverse
behaviour is observed due to the upsurge in Prandtl number, radiation parameter and Grashof
number for mass transfer. From fig.6 it is seen that fluid temperature is growing with the increase
of porosity parameter while reverse effect is seen in the case of increment of depth size of
channel.
Fig.4. Temperature distribution versus y when h=1, Gc=3, γ=2, N=1.
108
Fig.5. Temperature distribution versus y when h=1, S=1, Br=0.5, M=1.5, Pr=0.71.
Fig.6. Temperature distribution versus y when Gc=3, S=1, Br=0.5, M=1.5, Pr=0.7, γ=1,
N=1.
It is provided from fig.7 that species concentration is declining throughout the fluid flow
due to the increase of chemical reaction parameter and depth size.
Fig.7. Temperature distribution versus y when Gc=3, S=1, Br=0.5, M=1.5, Pr=0.7, γ=1,
N=1.
The effect of various parameters on mean velocity is reflected by fig.8 and 9. It is
observed that the mean velocity of the fluid is growing with the intensity of magnetic field,
chemical reaction coefficient, permeability parameter, mass buoyancy and depth of channel.
109
Fig.8. Concentration distribution versus y.
Fig.9. Mean velocity versus y when Gc=3, γ=1, α=0.5
From fig.10 and 11, it is clear that the mean temperature rises with the increase of heat
source, Brinkman number, radiation parameter, chemical reaction coefficient, Prandtl number
and mass buoyancy; while the reverse effect is observed due to increase in intensity of magnetic
field. Fig.12 shows that when we increase the depth of open channel the value of the mean
temperature of the fluid flow become high at insulated bottom wall and then reduced gradually
when approaches to open surface. Adverse behaviour is observed in the case of permeability
parameter.
110
Fig.10 Mean velocity versus y when h=1, M=0.5.
Fig.11. Mean temperature versus y when Gc=3, γ=1, α=0.5, Pr=0.71, M=0.5, h=1.
Fig.12. Mean temperature versus y when S=0.5, Br=1, N=1, γ=1, α=0.5, h=1.
Fig.13 shows that the mean concentration is upsurging with the rising values of chemical
reaction coefficient and depth of the open channel.
Fig.13. Mean concentration versus y.
111
Table1 depicts the values of skin-friction coefficient, Nusselt number and Sherwood number
at insulated bottom surface as well as free top surface. At insulated bottom surface the value of
skin-friction coefficient is increasing with the upsurging values of mass buoyancy. While it is
decreasing with the increase of Grashof number of mass transfer. The same behaviour is observed
for skin-friction coefficient at free surface.
At insulated bottom wall the value of Nusselt number is growing up with the increase in
depth of the channel, chemical reaction coefficient and Prandtl number, while it is reducing with
the increment in the values of heat source parameter, Brinkman number, mass buoyancy and
radiation parameter. Nusselt number at free surface is rising with the upsurged values of chemical
reaction parameter and depth of the channel, whereas declining with the increase values of heat
source, Prandtl number, Brinkman number, mass buoyancy and radiation parameter. It is seen
through the table-1 at insulated bottom surface, the Sherwood number is escalating with the
raised values of chemical reaction coefficient but it is decreasing with the increase value of open
channel. At free surface, Sherwood number is declining with the upsurge values of chemical
reaction coefficient and channel depth.
8. Conclusion
In the present study we considered a horizontal channel in which, bottom surface is made
with insulated material while top surface is free surface. We deliberated the steady flow of
viscous fluid of fixed depth through a porous medium under forced convection and transverse
magnetic field. The flow is created by a constant horizontal pressure gradient parallel to the
insulated fixed bottom surface. A theoretical study has been covered by the effects of various
parameters considered in the flow problem. The following conclusions are made by the obtained
results:
(1) Velocity and mean velocity of the fluid is increasing from the insulated bottom surface to
the free surface with the increment in porosity parameter α. Since, when the permeability
of the medium increases then the resistance of the medium becomes stumpy; therefore the
fluid may flow easily through it. This is the reason that the velocity at insulated surface at
bottom level is seen to be zero and then it takes its maximum value at top free surface.
(2) Velocity and mean velocity of the fluid show there positive behaviour with the increase in
mass buoyancy and depth of the porous medium. Both are in increasing order with these
parameters. In the case of large values of depth of porous medium the velocity and mean
velocity attain its high values throughout the fluid flow. This is concluded that by
increasing the depth of porous medium; the fluid flow can be made fast in this
112
geometrical fluid flow. This study has been done in the presence of transverse magnetic
field and chemical reaction coefficient.
(3) Mean velocity shows its good approach in the presence of chemical reaction effect. Due to
its presence mean velocity attains its high values at insulated bottom wall, but it is getting
reduce rapidly when it approaches to free surface.
(4) By using the effects of chemical reaction and external heat source; the temperature and
mean temperature both can be extended for their high values. In the case of mean
temperature; it shows its positive behaviour with respect to the radiation coefficient and
intensity of the transfer magnetic field.
(5) For high depth porous medium channel, the fluid mean temperature and species mean
concentration both rise high.
(6) By increasing in the values of porosity parameter; the mean temperature of the fluid
decreases. This is because of the existence of joule heating which diminishes the
temperature due to the free expansion.
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Nomenclature
u
Non-dimensional velocity along the x-
axis
Kl
Chemical reaction parameter
K’
Dimensional porous parameter
Pr
Prandtl number
'
U
Velocity of the fluid along X’-axis
Nu
Nusselt number
g
Acceleration due to gravity
Gc
Grashof number for mass transfer
Sh
Sherwood number
H
Depth of the porous medium along Y’-axix
B0
Strength of uniform magnetic field
S
Heat source parmeter
C
Non-dimensional of concentration of
species
Br
Brinkman number
N
Radiation parameter
Η
Sum of magnetic parameter and permeability
a
Some standard length
Mean Temperature
T0
Temperature of the upper open surface
Κ
Thermal conductivity
115
T’
Temperature of the fluid
Α
Permeability parameter in the dimensionless
form
T1
Temperature at the bottom surface
Μ
Kinematic viscosity
Cp
Specific heat at constant pressure
Volumetric coefficient of expansion with species
concentration
C0
Species concentration at the upper open
surface
σ
Electrical conductivity
C1
Species concentration at the bottom
surface
Ρ
Density of the fluid
u
Mean velocity
Chemical reaction coefficient
C
Mean Concentration
θ
Dimensionless temperature of the fluid
P
Pressure of the fluid
M
Hartmann number
Cf
Skin-friction coefficient
