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Journal of Applied Science and Technology
Vol.7, No.2, December 2010, pp 9-16
ISSN 2218-841X
*Corresponding author:
E-mail: ibrahim25si@yahoo.com (M.I. Abdullah)
The Effects of Radiation and Viscous Dissipation on MHD Natural Convection
Flow along a Vertical Flat Plate in Presence of Joule Heating
M.M. Ali1, R. Akhter2, N.H.M.A. Azim3, M.I. Abdullah4*
1 Faculty of life science, Mawlana Bhashani Science and Technology University,
Tangail-1902, Bangladesh
2 Department of Mathematics, Bangladesh University of Engineering and Technology,
Dhaka-1000, Bangladesh
3 School of Business Studies, Southeast University, Dhaka-1000, Bangladesh
4 Department of Computer Science and Engineering, Islamic University, Kushtia-Jhenidah,
Bangladesh
(Received 14 April 2010; Revised 26 July 2010)
ABSTRACT
In this analysis, the effects of radiation, joule heating and viscous dissipation on Magneto-Hydrodynamic
(MHD) natural convection flow along a vertical flat plate in presence of heat conduction are investigated. The
governing equations associated with boundary conditions for this analysis are transformed into dimensionless
form using appropriate similarity transformations and then solved numerically adopting implicit finite
difference method. The resulting numerical solutions are presented graphically in terms of velocity profile,
temperature distribution, local skin friction coefficient in terms of shear stress and local heat transfer rate in
terms of Nusselt number and the effects of magnetic parameter (M), radiation parameter (R), Prandtl number
(Pr), viscous dissipation parameter (N) and joule heating parameter (J) on the flow have been studied.
Keywards: Radiation, Viscous dissipation, Joule heating, MHD, Finite difference method, Vertical flat plate.
INTRODUCTION
The concept of natural convection flow of an
incompressible, viscous and electrically conducting
fluid in presence of transverse magnetic field is
important from technical point of view. Such
problems have received much attention due to its
potential application in nuclear power plants, cooling
of transmission lines and electric transformer etc. In
addition, the significant applications of MHD natural
convection flow are observed in the field of stellar
and planetary magnetospheres, aeronautics, chemical
engineering and electronics. As the engineering
processes closely related with temperature,
accordingly radiation heat transfer has significant
influence on engineering. Due to its wide
applications in space technology such as cosmical
flight, aerodynamics rocket, propulsion systems,
plasma physics, spacecraft re-entry aerodynamics and
at high operator’s temperature, many researchers
have studied the effect of radiation on MHD free
convection flow. Takhar and Soundalgekar1 have
studied the effect of radiation on MHD free
convection flow of a gas past a sami-infinite vertical
plate using the Cogley-vincenti-Giles equilibrium
model (Cogley et al.). The problem of natural
convection-radiation interaction on boundary layer
flow with Rosscland diffusion approximation along a
vertical thin cylinder has been investigated by
Hossain and Alim2. Radiation effect on free
convection flow of fluid from a porous vertical plate
was studied by Hossain et al.3
Abdel-naby et al.4 have studied the radiation effects
on MHD unsteady free convection flow over a
vertical plate with variable surface temperature.
Furthermore, the viscous dissipation heat in the
natural convection flow is important when the flow
field is of extreme size or in high gravitational field.
Israel Cookey et al.5 have investigated the influence
of viscous dissipation and radiation on unsteady
magnetohydrodynamic free convection flow past a
finite vertical heated plate in an optically thin
environment with time dependent-suction. Effect of
viscous dissipation and radiation on natural
convection flow in a porous medium embedded with
vertical annulus was studied by Budruddin et al.6.
Hossain7 have analyzed the effect of viscous and
joule heating effects on MHD free convection flow
with variable plate temperature. Combined effect of
viscous dissipation and joule heating on coupling of
conduction and free convection along a vertical flat
plate have been discussed by Alim et. al.8. Analytical
solutions for momentum and energy equations have
been arrived and the effect of radiation on mean
velocity and temperature discussed.
To the best of our knowledge, no one has done the
combined effect of radiation, joule heating and
viscous dissipation under the process of steady flow
but it has importance in the field of engineering heat
transfer problem. So, the objective of the present
study is to study the effect of radiation, joule heating
and viscous dissipation on MHD natural convection
flow of an incompressible, viscous and electrically
conducting fluid along a vertical plate under the
influence of transverse magnetic field in presence of
heat conduction. The governing partial differential
equations are reduced to locally non-similar partial
M.M. Ali et al. / J. Appl. Sci. Tech. Vol.7, No.2 (2010) 9-16
10
differential forms by using appropriate
transformations. The transformed boundary layer
equations are solved numerically adopting implicit
finite difference method together with Keller Box
Scheme technique 9, 10. Here, the assumption is
focused on the evaluation of the surface shear stress
in terms of local skin friction and the rate of heat
transfer in terms of local Nussetl number, velocity
profiles as well as temperature distribution for
selected values of parameters consisting magnetic
parameter M , radiation parameter R, Prandtl number
Pr, viscous dissipation parameter N and joule heating
parameter J.
MATHEMATICAL ANALYSIS
Let us consider a steady natural convection flow of
an incompressible, viscous and electrically
conducting fluid along a vertical flat plate of length l
and thickness b. A constant temperature, Tb is
maintained at the outer surface, where ∞
>TTb. ∞
T is
the ambient temperature of the fluid and uniform
magnetic field of strength H0 is applied along the y-
axis. The flow configuration and the coordinates
system are shown in Fig.-1.
The governing equation of such flow under the
Bousinesq approximation can be expressed within the
usual boundary layer as 4, 8:
0=
∂
∂
+
∂
∂
y
v
x
u (1)
ρ
σ
βν
uH
TTg
yu
y
u
v
x
u
uf
0
2
2
2
)( −−+
∂
∂
=
∂
∂
+
∂
∂
∞ (2)
()
p
bf
p
f
p
fff
CuH
TT
y
u
c
y
T
C
k
y
T
v
x
T
u
ρ
σ
ν
ρ
22
0
2
2
2
4+−Γ−
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
∂
∂
+
∂
∂
=
∂
∂
+
∂
∂
(3)
Where
λ
λ
λ
d
T
e
K
w
b
w⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
∂
∂
=Γ ∫∞
0
,
(
)
ww TKK
λλ
= is
the mean absorption coefficient,
λ
b
eis Plank’s
function and
T
is the temperature of the fluid in the
boundary layer. Where
ν
is the kinematic viscosity,
β is the Thermal expression co-efficient, σ is the
Electrical conductivity, Cp is the specific heat due to
constant pressure. The boundary conditions are:
()
()
⎪
⎪
⎭
⎪
⎪
⎬
⎫
>∞→→→
>=−=
∂
∂
===
∞0,,0
0,0
,0,,0
xyatTTu
xyatTT
kbk
y
T
xTTvu
f
bf
f
s
f
f
(4)
We observe that the equations (1) (2) and (3) together
with the boundary conditions (4) are nonlinear partial
differential equations. Equations (1) (2) and (3) may
now be non-dimensionalzed by using the following
dimensionless dependent and independent variables:
()
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎭
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎬
⎫
Γ
=
=
−
=
−
−
==
===
−
−
∞
∞
∞
−
−
2/1
2
2/1
2
0
2
3
4/1
2/1
4
,
,
,
,,,
Gr
l
R
Gr
lH
M
TTlg
Gr
TT TT
Gr
lv
v
Gr
lu
u
l
y
y
l
x
x
b
b
ν
μ
σ
ν
β
θ
ν
ν
(5)
Where ⎟
⎠
⎞
⎜
⎝
⎛
=
ρ
μ
v is the kinematics viscosity, Gr is
the Grashof number and θ is the non-dimensional
temperature. Using equation (5) in equations (1) (2)
and (3), we can obtain the governing equations in a
dimensionless form as:
0=
∂
∂
+
∂
∂
y
v
x
u (6)
θ
+
∂
∂
=+
∂
∂
+
∂
∂
2
2
yu
Mu
y
u
v
x
u
u (7)
2
2
2
2
)1(
Pr
1
uJ
y
u
N
R
y
y
v
x
u
+
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
∂
∂
+
−−
∂
∂
=
∂
∂
+
∂
∂
θ
θθθ
(8)
The corresponding boundary conditions are:
⎪
⎭
⎪
⎬
⎫
>∞→→→
>=
∂
∂
=−==
0,0,0
0,01,0
xyatu
xyat
y
pvu
θ
θ
θ
(9)
Here 2/1
2
2
0−
=Gr
lH
M
μ
σ
is the magnetic parameter,
2/1
2
4−
Γ
=Gr
l
R
ν
is the radiation parameter and the
In
su
lat
or
b
l
s
T
b
T
b
−
0
y
v
Interface
g
∞
T
Figure 1. Physical model and coordinate system.
0
H
0
H
(
)
0,xT
u
x
M.M. Ali et al. / J. Appl. Sci. Tech. Vol.7, No.2 (2010) 9-16
11
f
p
K
C
μ
=Pr is the Prandtl number,
()
∞
−
=TTlC
Gr
N
bp 2
2
ν
is
the viscous dissipation parameter,
)(
2/12
0
∞
−
=TTC GrH
Jbp
ρ
νσ
is
the joule heating parameter and 4/1
Gr
l
b
k
k
ps
f⎟
⎠
⎞
⎜
⎝
⎛
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
= is
a conjugate conduction parameter. The value of the
conjugate conduction parameter p depends on
s
f
k
k
l
b,
and Gr but each of which depends on the types of
considered fluid and the solid. Therefore in different
cases p is different but not always a small number. In
the present analysis we have taken p=1.The steam
function, similarity variable and the dimensionless
temperature are considered in the following form to
solve the equations (7) and (8) and for the boundary
conditions described in equation (9):
),()1(
,)1(
),,()1(
5/15/1
20/15/1
20/15/4
ηθ
η
ηψ
xhxx
xyx
xfxx
−
−−
−
+=
+=
+=
(10)
Where ψ is the dimensionless stream function which
is related to the velocity components such as y
u∂
∂
=
ψ
and
x
v∂
∂
−=
ψ
, ),(
η
xh is a dimensionless
temperature. By substituting (10) in equations (7)
and (8) and transforming these equations in the new
co-ordinate system, we obtain the transformed
equations:
)()1(
)1(10
56
)1(20
1516
10/1
5/22
x
f
f
x
f
fxhfx
Mxf
x
x
ff
x
x
f
∂
∂
′′
−
∂
′
∂
′
=+
′
+
−
′
+
+
−
′′
+
+
+
′′′
(11)
)(
)1()1(
)1(
)1(5
1
)1(20
1516
Pr
1
2
210/15/710/35/1
10/15/2
x
f
h
x
h
fxfxN
fxxJxxR
hxxRhf
x
hf
x
x
h
∂
∂
′
−
∂
∂
′
=
′′
+
′
++++
+−
′
+
−
′
+
+
+
′′
(12)
The boundary condition (9) become
⎪
⎭
⎪
⎬
⎫
∞→→∞→∞
′
=++
+−=
′
=
′
=
yatxhxf
yatxhxx
xxhxfxf
0),(,0),(
0)0,()1(
)1()0,(,0)0,()0,(
20/15/1
4/1
(13)
The equations (11) and (12) together with the
boundary condition (13) are solved numerically by
applying Implicit finite difference method with
Keller-Box (1978) Scheme. Its application to the
boundary layer flow problems have been analyzed
Cebeci, Brad and Shaw (1984) from the process of
numerical computation in practical point of view10. It
is important to calculate the values of the rate of heat
transfer and the skin friction co-efficient surface
shear stress. This can be written in the dimensionless
form as:
wf lGr
C
τ
μν
24/3−
=&
()
w
bf
uq
TTk Grl
N
∞
−
−
=
4/1
(14)
where
0=
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
∂
∂
=
y
wy
u
μτ
and w
q=
0=
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
∂
∂
−
y
f
fy
T
kare the
shearing stress and the heat flux
Thus the local skin friction co-efficient and rate of
heat transfer as
)0,()1( 20/35/2 xfxxCfx ′′
+= − (15)
)0,()1( 4/1 xhxNux ′
+−= − (16)
We have discussed the velocity profiles and
temperature distributions for various values of
Prandtl number, magnetic parameter, radiation
parameter, viscous dissipation parameter and joule
heating parameter in this paper.
RESULTS AND DISCUSSION
The resulting numerical solutions for the velocity and
the temperature field are shown in Fig. 2 to fig. 11.
The values of Prandtl number are taken to be 0.733,
0.900, 1.000 and 1.446 which correspond to air,
ammonia, steam and water at temperatures KT,
1300 ,
KT,
273 and Ct,
120 , respectively. The detailed
numerical solutions have been obtained a wide range
of values of the magnetic parameter M= (0.10, 0.50,
0.80, 1.00), radiation parameter R= (0.001, 0.010,
0.30, 0.050), viscous dissipation parameter N= (0.01,
0.20, 0.50, 1.00) and joule heating parameter J=
(0.05, 0.20, 0.50, 0.80).
0.0 2.0 4.0 6.0 8.0 10.0
η
0.0
0.1
0.2
0.3
0.4
0.5
Velocity
M=0.10
M=0.50
M=0.80
M=1.00
(a)
0.0 2.0 4.0 6.0 8.0 10.0
η
0.0
0.2
0.4
0.6
0.8
Temperature
M=0.10
M=0.50
M=0.80
M=1.00
(b)
M.M. Ali et al. / J. Appl. Sci. Tech. Vol.7, No.2 (2010) 9-16
12
Figure 2. (a) Variation of velocity and (b) variation
of temperature against
η
for varying of M with Pr =
0.733, R = 0.001, N=0.01and J = 0.05.
Fig. 2 (a) shows that the velocity profiles of fluid
decreases with the increase of magnetic parameter
due to the interaction of the applied magnetic field
and flowing fluid particles. But for each value of M,
the velocity is zero at the boundary wall and then
increase to the peak value as
η
increases and then
turn to decrease, finally approach to zero. Moreover,
we have seen that the velocity profiles meet together
after certain position of
η
and cross the side. This is
because, the gradient of decreasing velocity increases
with the increase of magnetic parameter. Reverse
situation is observed from Fig. 2(b) in the case of
temperature field. It shows that the magnetic field
works to increase the values of temperature within
the flow region and also increase the thickness of the
thermal boundary layer.
0.0 2.0 4.0 6.0 8.0 10.0
η
0.0
0.1
0.2
0.3
0.4
0.5
Velocity
R=0.001
R=0.010
R=0.030
R=0.050
(a)
0.0 2.0 4.0 6.0 8.0 10.0
η
0.0
0.2
0.4
0.6
0.8
Temperature
R= 0.001
R= 0.010
R= 0.030
R= 0.050
(b)
Figure 3. (a) Variation of velocity and (b) variation
of temperature against
η
for varying of R with Pr =
0.733, M= 0.50, N=0.01and J = 0.05.
The effect of radiation parameter R on velocity and
the temperature distributions for selected value of
Prandtl number Pr, magnetic parameter M,
dissipation parameter N and joule heating parameter J
are presented in Fig. 3 (a) and Fig. 3 (b), respectively.
The velocity and temperature of the fluid increase
with the increase of R due to the absorption of heat
while radiation imitates from the heated plate. It can
be noted that, near the surface of the flat plate,
velocity increases significantly and becomes
maximum. It then decreases along horizontal
direction. The maximum values of the velocity are
0.3909, 0.4070, 0.4380 and 0.4650 for R = 0.001,
0.010, 0.030 and 0.050 at η = 1.3827, 1.4279 1.4741
and 1.5214. From Fig. 3 (a) and also numerical
values we have seen that the velocity profiles shift
upward and the position of the velocity peak moves
toward the boundary layer for the increasing R. As
the temperature of the fluid increases with increasing
radiation as shown in Fig. 3(b), accordingly the
thickness of the boundary layer and also boundary
layer velocity increases observed in Fig. 3.
0.02.04.06.08.010.0
η
0.0
0.1
0.2
0.3
0.4
0.5
Velocity
Pr=0.733
Pr=0.900
Pr=1.000
Pr=1.446
(a)
0.0 2.0 4.0 6.0 8.0 10.0
η
0.0
0.2
0.4
0.6
0.8
Temperature
Pr=0.733
Pr=0.900
Pr=1.000
Pr=1.446
(b)
Figure 4. (a) Variation of velocity and (b) variation
of temperature against
η
for varying of Pr with M =
0.50, R = 0.001, N=0.01and J = 0.05.
The variation of velocity profile and temperature
distribution for different values of Prandtl number Pr
with magnetic parameter M = 0.50, radiation
parameter R = 0.001, viscous dissipation parameter N
= 0.01 and joule heating parameter J=0.05 are shown
in Fig. 4(a) and Fig. 4 (b) respectively. The viscosity
of the fluid is measured with the increase of Pr. As a
results fluid does not move freely. Then it can be
concluded that, velocity of fluid decrease when the
value of Pr increases. Fig. 4 (b) shows temperature
distribution of boundary layer that decreases due to
the increase of Pr. The temperature at the solid/fluid
interface is reduced because the temperature at the
plate considered is constant. As a result, thermal
boundary layer thickness decreases with the increase
of Pr.
Fig. 5 (a) and Fig. 5(b) illustrates the effects of N on
velocity and temperature for Pr = 0.733, M = 0.50, R
=0.001 and J =0.05. From Fig. 5(a), it is seen that the
velocity profiles increases slightly due to the increase
of viscous dissipation parameter N along the η
direction and also we have seen that near the surface
of the plate the velocity is increased to a maximum
value with the increase of viscous dissipation
parameter N and then after the peak position start to
decrease and finally approaches to zero. The
maximum values of the velocities are 0.3909, 0.3954,
0.4026 and 0.4151 which occurs for N = 0.01, 0.20,
M.M. Ali et al. / J. Appl. Sci. Tech. Vol.7, No.2 (2010) 9-16
13
0.50 and 1.00 respectively. On the other hand, from
Fig. 5(b), we have seen that the same result is holds
for temperature distributions within the boundary
layer due to increasing of viscous dissipation
parameter N. The maximum value of temperature are
0.8800, 0.8941, 0.9178 and 0.9615 for viscous
dissipation parameter N =0.01, 0.20, 0.50 and 1.00,
respectively. Each of which occurs at the surface of
the plate. This is because; the increased value of
dissipation parameter N produces heat and also
dissipates viscosity of fluid.
0.0 2.0 4.0 6.0 8.0 10.0
x
0.0
0.1
0.2
0.3
0.4
0.5
Vlocity
N= 0.01
N= 0.20
N= 0.50
N= 1.00
(a)
0.0 2.0 4.0 6.0 8.0 10.0
η
0.0
0.2
0.4
0.6
0.8
1.0
Temperature
N=0.01
N=0.20
N=0.50
N=1.00
(b)
Figure 5. (a) Variation of velocity and (b) variation
of temperature against
η
for varying of N with Pr =
0.733, M=0.50, R = 0.001 and J = 0.05.
0.02.04.06.08.010.0
η
0.0
0.1
0.2
0.3
0.4
0.5
Velocity
J=0.05
J=0.20
J=0.50
J=0.80
(a)
0.0 2.0 4.0 6.0 8.0 10.0
η
0.0
0.2
0.4
0.6
0.8
1.0
Temperature
J= 0.05
J= 0.20
J= 0.50
J= 0.80
(b)
Figure 6. (a) Variation of velocity and (b) variation
of temperature against
η
for varying of J with Pr =
0.733, M=0.50, R = 0.001 and N=0.01.
The figures (6.a-b) shows that the effect of joule
heating parameter on velocity and temperature field
while the controlling parameters are Pr=0.733,
M=0.50, R=0.001 and N=0.01. As the effect of joule
heating parameter produce temperature in the
conductor, therefore the velocity and temperature of
the fluid increase associated with the increasing
values of joule heating parameter J. Moreover, the
maximum values of the velocity are 0.3909, 0.3987,
0.4146 and 0.4209 for J =0.05, 0.20, 0.50 and 0.80
and each of which occurs at
η
=1.3827. Furthermore,
the maximum value of temperature are 0.8800,
0.8889, 0.9078 and 0.9282 for joule heating
parameter J= 0.05, 0.20, 0.50 and 0.80, respectively
and each of which occurs at the surface of the plate.
Fig. 7(a) and Fig. 7(b), reveal that the skin friction
coefficient and the rate of heat transfer for some
values of M with Pr = 0.733, R = 0.001, N =0.01 and
J=0.05. The velocity decreases as shown in the Fig.
2(a), due to the increasing M .Accordingly, the skin
friction on the plate decreases as observed in Fig.
7(a). Moreover, the temperature within the boundary
layer increases (Fig. 2(b)) for the increasing M. As a
result, the heat transfer rate from the plate to fluid
decreases as shown in figure 7(b).
0.01.02.03.04.05.0
x
0.0
0.3
0.6
0.9
1.2
Skin friction
M=0.10
M=0.50
M=0.80
M=1.00
(a)
0.0 1.0 2.0 3.0 4.0 5.0
x
0.0
0.1
0.2
0.3
0.4
0.5
Heat transfer
M=0.10
M=0.50
M=0.80
M=1.00
(b)
Figure 7. (a) Variation of skin friction and (b)
variation of heat transfer against x for varying of M
with Pr = 0.733, R = 0.001, N = 0.01 and J=0.05.
The variation of the local skin friction coefficient Cfx
and local rate of heat transfer Nux for different values
of R associated with Pr = 0.733, M = 0.50, N = 0.01
and J=0.05 are illustrated in Fig. 8 (a) and Fig. 8 (b).
Radiation increases the fluid motion as mentioned in
Fig. 3(a) and increases the shear stress at the wall, for
which local skin friction increase with the increasing
M.M. Ali et al. / J. Appl. Sci. Tech. Vol.7, No.2 (2010) 9-16
14
R. This phenomenon demonstrated in Fig. 8(a).The
increased value of the radiation parameter increases
temperature (Fig. 3(b)) which after decrease the rate
of heat transfers along the x-direction. Moreover,
heat transfers depend on temperature variation. As
the temperature of fluid increase within the boundary
layer, the heat transfer rate also decreases.
0.0 1.0 2.0 3 .0 4.0 5.0
x
0.0
0.3
0.6
0.9
1.2
Skin friction
R=0.001
R=0.010
R=0.030
R=0.050
(a)
0.0 1.0 2.0 3.0 4.0 5.0
x
0.0
0.1
0.2
0.3
0.4
0.5
Heat transfer
R=0.001
R=0.010
R=0.030
R=0.050
(b)
Figure 8. (a) Variation of skin friction and (b)
variation of heat transfer against x for varying of R
with Pr = 0.733, M=0.50, N = 0.01 and J=0.05.
The effects of Prandtl number on the skin friction Cfx
and heat transfer rate Nux with the increasing of axial
distance x for the fixed values of magnetic parameter,
radiation parameter, viscous dissipation parameter
and joule heating parameter are shown in Fig. 9(a)
and Fig. 9(b), respectively. The values of Pr are
proportional to the viscosity of the fluid. So the
increased values of Pr decrease the velocity of the
fluid within the boundary layer, as a result,
corresponding skin friction coefficient also decreases
(Fig. 9(a)). For a particular value of Pr the local skin
friction coefficient increases monotonically due to
the increase of x. From Fig. 9(b), it is observed that
heat transfer rate increases due to increase of Pr.
Furthermore for a particular value of Pr the local heat
transfer rate decreases monotonically due to increase
of x.
Fig. 10(a) and Fig. 10(b) illustrates the local skin
friction coefficients and rate of heat transfer for
different values of viscous dissipation parameter N
with increasing of x and controlling parameter Pr =
0.733, M =0.50, R = 0.001 and J = 0.05. It can be
seen that an increase in the dissipation parameter N is
associate with the increase of skin friction
coefficients against x. Moreover, for a particular
value of N the shear stress coefficient increases
vertically. Furthermore, temperature of the fluid
increase with the increasing of N shown in Fig.5 (b),
in that case temperature difference between the
surface of the plate and the fluid reduces for which
the rate of heat transfer decreases. So, the heat
transfer rate decrease with the increase of viscous
dissipation parameter N (viewed in Fig. 10(b)).
0.0 1.0 2.0 3.0 4.0 5.0
x
0.0
0.3
0.6
0.9
1.2
Skin friction
R=0.001
R=0.010
R=0.030
R=0.050
(a)
0.0 1.0 2.0 3.0 4 .0 5.0
x
0.0
0.1
0.2
0.3
0.4
0.5
Heat transfer
Pr=0.733
Pr=0.900
Pr=1.000
Pr=1.446
(b)
Figure 9. (a) Variation of skin friction and (b)
variation of heat transfer against x for varying of Pr
with M=0.50, R = 0.001, N = 0.01 and J=0.05.
0.01.02.03.04.05.0
x
0.0
0.3
0.6
0.9
1.2
1.5
Skin friction
N=0.01
N=0.20
N=0.50
N=1.00
(a)
0.0 1.0 2.0 3.0 4.0 5.0
x
-0.2
0.0
0.2
0.4
Heat transfer
N=0.01
N=0.20
N=0.50
N=1.00
(b)
Figure 10. (a) Variation of skin friction and (b)
variation of heat transfer against x for varying of N
with Pr = 0.733, M=0.50, R = 0.001 and J=0.05.
Fig. 11(a) and Fig. 11(b) illustrates the local skin
friction coefficients and heat transfer rate for
M.M. Ali et al. / J. Appl. Sci. Tech. Vol.7, No.2 (2010) 9-16
15
different values of J with increasing x and controlling
parameter Pr = 0.733, R = 0.001, M = 0.50 and
N=0.01. The velocity of the fluid increase with the
increase of J that has been shows in Fig.6 (a),
accordingly the corresponding skin friction
coefficient increase associated with the increase of J.
But reverse situation is observed in Fig.11 (b) for the
case of heat transfer rate within the boundary layer
for increasing of joule heating parameter J.
0.0 1.0 2.0 3.0 4.0 5.0
x
0.0
0.4
0.8
1.2
1.6
2.0
Skin friction
J=0.0 5
J=0.2 0
J=0.5 0
J=0.8 0
(a)
0.0 1.0 2.0 3.0 4.0 5.0
x
-0.4
-0.2
0.0
0.2
0.4
Heat transfer
J=0.0 5
J=0.2 0
J=0.5 0
J=0.8 0
(b)
Figure 11. (a) Variation of skin friction and (b)
variation of heat transfer against x for varying of J
with Pr = 0.733, M=0.50, R = 0.001 and N = 0.01.
CONCLUSION
In this analysis the effect of radiation and joule
heating parameter on Magneto Hydrodynamic
(MHD) natural convection flow along a vertical flat
plate in presence of heat conduction and viscous
dissipation have been investigated for some selected
values of pertinent parameters including magnetic
parameter, radiation parameter, Prandtl number,
viscous dissipation parameter and joule heating
parameter. From the present investigation, it may be
concluded that the velocity of the fluid and the skin
friction at the interface decrease with the increase of
magnetic field and Prandtl number while they
increase with the increase of radiation parameter,
viscous dissipation parameter and joule heating
parameter. The temperature of the fluid increases
with the increase of magnetic field, radiation
parameter, viscous dissipation parameter and joule
heating parameter but decrease for Prandtl number.
Moreover, the rate of heat transfer decreases with
increasing magnetic field, radiation parameter,
viscous dissipation parameter and joule heating
parameter but it increases for increasing Prandtl
number.
M.M. Ali et al. / J. Appl. Sci. Tech. Vol.7, No.2 (2010) 9-16
16
Nomenclature:
b Plate thickness
fx
C Local skin friction coefficient
p
C Specific heat at constant pressure
f Dimensionless stream function
g Acceleration due to gravity
r
G Grashof number
h Dimensionless temperature
0
H Strength of magnetic field
J Joule heating parameter
sf kk , Fluid and solid thermal conductivities
l Length of the plate
M Magnetic parameter
N Viscous dissipation parameter
ux
N Local Nusselt number
p Conjugate conduction parameter
P
r
Prandtl number
w
q Heat flux
R Radiation parameter
b
T Temperature at outside surface of the
plate
f
T Temperature of the fluid
w
T Average temperature of porous plate
∞
T Temperature of the ambient fluid
u,v Velocity components
u,v Dimensionless velocity components
x
,
y
Cartesian co-ordinates
x
,y Dimensionless Cartesian co-ordinate
β
Coefficient of thermal expansion
η
Dimensionless similarity variable
θ
Dimensionless temperature
μ
Viscosity of the fluid
ν
Kinematic viscosity
ρ
Density of the fluid
σ
Electrical conductivity
w
τ
Shearing stress
ψ
Shearing stress
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