American Journal of Computational Mathematics, 2013, 3, 252-259
http://dx.doi.org/10.4236/ajcm.2013.33035 Published Online September 2013 (http://www.scirp.org/journal/ajcm)
Conjugate Effect of Radiation and Thermal Conductivity
Variation on MHD Free Convection Flow
for a Vertical Plate
Rowsanara Akhter1*, Mohammad Mokaddes Ali2, Babul Hossain2, M. Sharif Uddin1
1Department of Mathematics, Jahangirnagar University, Dhaka, Bangladesh
2Department of Mathematics, Mawlana Bhashani Science and Technology University, Tangail, Bangladesh
Received August 20, 2013; revised September 5, 2013; accepted September 11, 2013
Copyright © 2013 Rowsanara Akhter et al. This is an open access article distributed under the Creative Commons Attribution Li-
cense, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
ABSTRACT
A numerical investigation is performed to study the effect of thermal radiation on magnetohydrodynamic (MHD) free
convection flow along a vertical flat plate in presence of variable thermal conductivity in this paper. The governing
equations of the flow and the boundary conditions are transformed into dimensionless form using appropriate similar-
ity transformations and th en solved employin g the implicit fi nite d ifference metho d with Keller-box scheme. Results for
the details of the velocity profiles, temperature distributions as well as the skin friction, the rate of heat transfer and
surface temperature distributions are shown graphically. Results reveal that the thermal radiation is more significant in
MHD natural convection flow during thermal conductivity effect is considered. To illustrate the accuracy of the present
results, the results for the local skin fraction and surface temperature distribution excluding the extension effects are
compared with results of Merkin and Pop designed for the fixed value of Prandtl number and a good agreement were
found.
Keywords: Radiation; MHD; Thermal Conductivity; Finite Difference Method
1. Introduction
The physical phenomenon of free convection flow is
driven by temperature difference. Using these considera-
tions, the temperature variation generates a density gra-
dient which responsible for buoyancy forces. The buoy-
ancy effects are important in free convection flow of v is-
cous incompressible electrically conducting fluid. Many
practical applications of free convection flow exist, for
example in the heater and coolers of mechanical devices,
in chemical industries, in nuclear power plants, in the
formation of microstructures during the cooling of mol-
ten metal’s, in fluid flows around heat-dissipation fins,
and solar ponds etc. Moreover, MHD free convection
flow is used frequently in the field of stellar and plane-
tary magnetospheres, aeronautics, chemical engineering
and electronics. Furthermore, most of the engineering
processes are rela ted with a high temperature , accordingly ,
radiation heat transfer is significant to design the relevant
equipment of heat transfer process. In addition, radiation
effects on MHD free convection flow and heat transfer
are important in the context of space technology. Con-
sidering it’s important applications in engineering and
industrial fields, a number of theoretical and experimen-
tal work have been conducted extensively by many re-
searchers. Among them, Soundalgekar and Takhar [1]
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.
[2]). Hossain and Takhar [3] employed implicit finite dif-
ference methods to analyze the effect radiation on mixed
convection flow along a heated vertical flat plate with a
uniform free stream and a uniform surface temperature.
The effects of radiation and transverse magnetic field
near stretching sheet were investigated by Ghaly [4] in
the p r esence of a uniform free stream of constant v e lo c i t y,
temperature and concentration to show that radiation hav e
significant influences on the velocity and temperature
profiles. Abd El-Naby et al. [5] studied the radiation ef-
fects on MHD un steady free convection flow over a ver-
tical plate with variable surface temperature. Badruddin
et al. [6] explored the effect of radiation and viscous dis-
sipation on natural convection flow in a porous medium
*Corresponding a uthor.
C
opyright © 2013 SciRes. AJCM

R. AKHTER ET AL. 253
by imposing finite element method (FEM). Furthermore,
it is also known that the physical property may change
significantly with temperature. To obtain better pred ictio n
of the flow behavior, it is necessary to take into account
this variation of thermal conductivity of fluid. Mishra et
al. [7] employed alternating direction implicit scheme
and collapsed dimension method to investigate the effect
of temperature dependent thermal conductivity and ra-
diation heat transfer on transient conduction for a 2-D
rectangular enclosure containing an absorbing, emitting
and scattering medium. Seddeek and Salama [8] applied
perturbation technique and shooting method to analyze
the effects of variable viscosity and thermal conductivity
on MHD unsteady two-dimensional laminar flow of a
viscous incompressible conducting fluid past a semi-
infinite vertical porous moving plate considering variable
suction. Sharma and Singh [9] obtained the analytical
and numerical solutions of the effects of thermal conduc-
tivity on MHD steady free convection flow of a viscous
incompressible electrically conducting liquid along an
inclined isothermal non-conducting porous plate in pres-
ence of viscous dissipation and Ohmic heating. The ef-
fects of thermal conductivity on unsteady MHD free
convective flow over an isothermal semi infinite vertical
plate were studied by Loganathan et al. [10] using im-
plicit finite-difference method of Crank-Nicholson type.
In this paper, the effects of radiation and variable ther-
mal conductivity on free convection flow for a vertical
flat plate in presence of transverse magnetic field are
studied. The detail derivation of the governing equations
for the flow and the parametric discussion depending on
the numerical results of the present simulations are pre-
sented in the following section.
2. Governing Equations of the Flow
We consider the conduction inside a vertical heated flat
plate and free convection flow of an incompressible, vis-
cous and electrically conducting fluid along that vertical
flat plate of length l and width b. The fluid properties are
assumed to be constant and the temperature b of the
outer surface of the plate is greater than ambient tem-
perature and a uniform magnetic field of strength H0
is imposed along the
T
T∞
y
-axis. Here the
x
-axis is taken
along the vertical flat plate in upward direction and also
the
y
-axis is normal to that plate. The effects of radia-
tion from the heated plate and thermal conductivity var ia-
tion within the two dimensional flow region are consid-
ered in this analysis. Moreover, thermal conductivity of
the fluid is assumed as
()
{
}
*
1
ff
kk T γ =+ T
∞∞
−. The
flow configuration and the coordinates system are shown
in following Figure 1.
The governing equations of the flow under the Bous-
sinesq approximations can be expressed within the usual
boundary layer as follows:
Figure 1. Physical model and coordinate system.
0
uv
xy
∂∂
+=
∂∂ (1)
()
2 20
2f
H
u uu u
uv gTT
xyy
σ
νβ
ρ
∞
∂∂∂
+= +−−
∂∂∂ (2)
(
14
ff f
)
f
fb
p
TT T
uvk TT
xyCyy ρ
∂∂ ∂

∂
+= −Γ−

∂∂ ∂∂
 (3)
where 0d
b
b
fb
e
KT
λ
λ
λ
∞ ∂
Γ= 

∂

,
()
bb
K
KT
λλ
= is the
mean absorption coefficient [5], b is the Plank’s func-
tion. The boundary conditions based on conduction are:
e
λ
()
()
0, ,0,
at 0, 0
0, at , 0
f
fs
fb
f
f
uvT Tx
TkTTy x
ybk
uTTy x
∞

== =

∂

=− =>

∂


→→ →∞>

(4)
Equations (1) to (3) are nonlinear dimensional partial
differential equations and these equations can be made
non-dimensional by using the following dimensionless
variables:
()
14 1/2
1/4
3
2
,, ,
, ,
f
b
b
xy ul
xyGruGr
ll
TT
vl
vGr TT
gl TT
Gr
ν
θ
ν
β
ν
−
∞
−
∞
∞

== =



−

== 
−


−

=


(5)

R. AKHTER ET AL.
254
Therefore, the dimensionless governing equations are:
0,
uv
xy
∂∂
+=
∂∂ (6)
2
2,
uu u
uvMu
xy y
θ
∂∂ ∂
++=+
∂∂ ∂ (7)
()
()
2
2
11
Pr
1
uv
x
y y
Ra
θθθ θ
γθ γ
θ


∂∂∂ ∂ 
+= ++


∂∂ ∂ ∂



−−
y





(8)
The corresponding boundary conditions are:
0,1 at 0,0
0,0 at ,0
uvpy x
y
uyx
θ
θ
θ
∂
== −==>

∂

→→ →∞>
(9)
Here 2212
0
Hl
M
Gr
σ
μ
−
= is the magnetic parameter,
212
4l
Ra Gr
ν
−
Γ
= is the radiation parameter,
(
*
b
TT γγ
∞
=−
)
is the thermal conductivity parameter,
Pr
p
f
C
K
μ
= is the Prandtl number and
() ()
14
fs
pkkblGr= is a conjugate conduction pa-
rameter. The value of the conjugate conduction parame-
ter p depends on
()
bl ,
()
f
s
kk and but each of
which depends on the types of considered fluid and the
solid. The steam function and similarity variable and the
dimensionless temperature are considered in the follow-
ing form to solve the governing equations:
Gr
() ()
()
()()
120
45
120
15
15
15
1,
1,
1,
xxfx
yx x
xxhx
,
ψ
η
η
θη
−
−
−
−
=+
=+
=+
(10)
Using the above transformations, we obtain the fol-
lowing dim e nsionl ess g overning equatio ns:
() ()
()
2
110
25
16 1565
20 110 1
1
xx
ffff
xx
f
f
Mxxfhxff
x
x
++
′′′′′ ′
+−
++
′
∂∂

′′′′
−+ +=−

∂∂




)11(
()
() () ()
()
2
110
25
310
15
1
PrPr1Pr 1
16 1511
20 151
1
xx
hhh h
xx
x
f
hfhRaxx
xx
hf
Raxxx fh
xx
γγ
 
′′′′ ′
++
 
++
 
+′′
+−−+
++
∂∂

′′
++= −

∂∂

The boundary condition (9) become
() ()
() ()()()
() ()
14 120
15
,0,0 0,
,011,0 at 0
,0,,0 at .
fxfx
hxxxx hxy
fx hxy
′
== 


′=−+++= 

′∞→ ∞→→∞


)13(
In practical point of view, it is important to calculate
the values of the skin friction co-efficient in term of sur-
face shear stress and the rate of heat transfer in term of
the Nussetl number. These can be written in the dimen-
sionless form as:
()
34 214
and
f
ww
fb
GrlGr l
CNu
kT T
τ
μν
−−
∞
==
−q
(14)
where
0
w
y
u
y
τμ
=

∂
=
∂

is the shearing stress and
0
f
wf
y
T
qk
y=
∂

=− 
∂

is the heat flux. Thus the local skin
friction co-efficient and the local Nussetl number is ob-
tained using the new variable systems that is describes in
Equation (20) as follows:
() ()
()()
320
25
14
1,0
1,0
fx
x
Cxx fx
Nuxh x
−
−
′′
=+
=− +
and
(15)
The numerical value of the surface temperature distri-
bution are obtained from the following relation
() ()(
15
15
,0 1,0
)
x
xxhx θ
−
=+ (16)
We have discussed the velocity profiles and tempera-
ture distributions for various values of magnetic parame-
ter, radiation parameter, thermal conductivity variation
parameter and Prandtl number in the present investiga-
tion.
3. Method of Solution
The numerical solutions of this analysis are found by
using implicit finite difference method with Keller-box
[11] Scheme which is well documented by Cebeci and
Bradshaw [12].
4. Comparison of the Results
h
The comparison of the skin friction coefficients
()
f
x
C
and the surface temperature distribution be-
tween the present work and the work of Merin and Pop
[13] is presented in following Table 1. We observed in
this table, that the present analysis is an excellent agree-
ment with the published work.
() (
,0x θ
)
)12(
5. Results and Discussion
The main objective of the present work is to analyze

R. AKHTER ET AL. 255
Table 1. Comparison of the present numerical results of the
skin friction (Cfx) and surface temperature (θ(x,0)) with
Prandtl number Pr = 1.00, and p = 1.00.
Merin and Pop [1 3] Present work (2013)
1
5
x
ξ
= fx
C
()
,0
x
θ
fx
C
()
,0
x
θ
0.7
0.8
0.9
1.0
1.1
1.2
0.430
0.530
0.635
0.745
0.859
0.972
0.651
0.686
0.715
0.741
0.762
0.781
0.424
0.529
0.635
0.744
0.860
0.975
0.651
0.687
0.716
0.741
0.763
0.781
MHD free convection flow in presence of thermal con-
ductivity variation with radiation effects. In this analysis,
the numerical solutions are calculated from the trans-
formed momentum and energy equations. The value of
conjugate conduction parameter is considered
for the simulation of the present problem and the values
of Prandtl number are considered 0.733, 0.930, 1.241 and
1.630 which corresponds to air, ammonia water and glyc-
erin, respectively. The detailed numerical solutions have
been obtained in terms of velocity, temperature, local s kin
friction, heat transfer rate and surface temperature for a
wide range of values of the parameters as M = 0.10, 0.50,
0.70 and 1.00, Ra = 0.01, 0.03, 0.06 and 0.08, γ = 0.01,
0.10, 0.15 and 0.20 and then presented graphically in
Figures 2-11, respectively.
1.0p=
The numerical values of velocity and temperature are
obtained from the solution of the Equations (11) and (12)
with the boundary condition (13) for different values of
magnetic parameter M when Pr = 0.733, Ra = 0.01 and γ
= 0.01 and are illustrated in Figures 2(a) and (b), respec-
tively. Here we observed that the velocity decreases for
the increasing values of M. This is to be expected be-
cause, the magnetic field acting along the horizontal di-
rection that introduces a retard force due to the interac-
tion between applied magnetic field and fluid flow which
acts against the fluid motion, as a result, the velocity of
the fluid decreases. But near the surface of the plate the
velocity increases and become maximum and then de-
crease and finally approaches to zero. Moreover, the ve-
locity profiles meet together after certain position of η
and cross the side. This is because, the gradient of de-
creasing of velocity decrease with the increasing value of
mag n e ti c pa r ame t e r . I n Figure 2(b), the temperature within
the boundary layer increases with the increasing values
of magnetic parameter M due to the interaction of applied
magnetic field and fluid motion that tends to heat the
fluid. Furthermore, the temperature decreases monotoni-
cally with increasing of η for each value of M. Thus the
magnetic field works to retard the fluid motion but in-
crease the temperature within the thermal boundary layer
region.
The variation of velocity and temperature for distinct
values of the radiation parameter Ra together with an
0.0 2.0 4.0 6.0 8.010.0
η
0.0
0.2
0.4
0.6
0.8
Velocity
M = 0.10
M = 0.50
M = 0.70
M = 1.00
(a)
0.0 2.0 4.0 6.0 8.010.0
η
0.0
0.4
0.8
1.2
1.6
2.0
Temperature
M = 0.10
M = 0.50
M = 0.70
M = 1.00
(b)
Figure 2. (a) Variation of velocity and (b) Variation of tem-
perature against η for varying of M wit h Ra = 0.01 γ = 0.01,
and Pr = 0.733.
individual certain value of Prandtl number, Pr, magnetic
parameter, M and thermal conductivity variation parame-
ter, γ are presented in Figures 3(a) and (b), respectively.
It can be seen that both the velocity and the temperature
of the fluid increase within the velocity boundary layer
and the temperature boundary layer, respectively for the
increasing Ra due to the absorbsion of emitted heat from
the heated plate that caused by the radiation effect. The
trend is observed to shift upward and the peak velocity
increases gradually with the increasing values of Ra. As
the velocity and temperature of the fluid increases with
the increasing value of radiation parameter as shown in
Figures 3(b) and (b), the thickness of the velocity and
thermal boundary layer increase.
Figures 4(a) and (b) illustrate the effects of thermal
conductivity variation parameter on velocity and tem-
perature profiles associated with the certain value of M,
Ra and Pr. Both figures reflect that the velocity and the
temperature of the fluid increases with the increasing
value of γ. The fact behind it’s that the increasing value of
thermal conductivity increases the energy transfer ability.
It is also seen that near the surface of the plate the veloc-
ity becomes maximum with increasing of γ then after the
peak position start to decrease and finally approaches to
zero. On the other hand, the maximum values of tem-
perature are occurred on the surface of the plate for each

R. AKHTER ET AL.
256
0.0 2.0 4.0 6.0 8.010.0
η
0.0
0.2
0.4
0.6
0.8
Velocity
Ra = 0.01
Ra = 0.03
Ra = 0.06
Ra = 0.08
(a)
0.0 2.0 4.0 6.0 8.010.0
η
0.0
0.3
0.6
0.9
1.2
1.5
1.8
Temperature
Ra = 0.01
Ra = 0.03
Ra = 0.06
Ra = 0.08
(b)
Figure 3. (a) Variation of velocity and (b) Variation of tem-
perature against η for varying of Ra with M = 0.50 γ = 0.01,
and Pr = 0.733.
0.02.0 4.06.08.010.0
η
0.0
0.2
0.4
0.6
0.8
1.0
1.2
Velocity
γ = 0.01
γ = 0.10
γ = 0.15
γ = 0.20
(a)
0.0 2.0 4.0 6.0 8.010.0
η
0.0
0.6
1.2
1.8
2.4
3.0
Temperature
γ = 0.01
γ = 0.10
γ = 0.15
γ = 0.20
(b)
Figure 4. (a) Variation of velocity and (b) Variation of tem-
perature against η for varying of γ with M = 0.50, Ra = 0.01
value of γ and
and Pr = 0.733.
then turn to decrease asymptotically and
ature dis-
tri
ively reveal that the skin
fr
finally approach to zero. These phenomenons are dem-
onstrated in Figures 4(a) and (b), respectively.
The effect of Pr on the velocity and temper
butions is displayed in Figures 5(a) and (b), respec-
tively. From Figure 5(a) observed that the velocity of the
fluid decrease as Pr increases. It is due to the fact that for
increasing Pr, density of the fluid increases which creates
a negative force to flow and then fluid does not move
freely. Furthermore, Figure 5(b) shows that the tem-
perature profiles for change in Pr from 0.733 to 1.630
and seen that the thermal boundary layer thickness de-
crease for increasing Pr, because of the increased Pr de-
crease the thermal d iffusivity, which leads to the decre ase
of the energy transfer ability.
Figures 6(a) and (b), respect
iction coefficient and the heat transfer rate for some
selected values of M with Pr = 0.733, Ra = 0.01, and
0.01
γ
=. The increased value of M leads to decrease the
tion along the plate due to the fact the effect of
magnetic field parameter opposes the fluid flow. Increas-
ing fluid temperature for increasing M decrease the rate
of heat transfer from the plate to fluid. This is because,
the increased temperature reduces the temperature dif-
ference between the heated plate and fluid within the
boundary layer.
skin fric
0.02.04.06.08.010.0
η
0.0
0.2
0.4
0.6
0.8
Velocity
Pr =0.733
Pr =0.930
Pr =1.241
Pr =1.630
(a)
0.02.0 4.0 6.0 8.010.0
η
0.0
0.4
0.8
1.2
1.6
2.0
Temperature
Pr =0.733
Pr =0.930
Pr =1.241
Pr =1.630
(b)
Figure 5. (a) Variation of velocity and (b) Variation ofm- te
perature against η for varying of Pr with, M = 0.50, Ra =
0.01 and γ = 0.01.

R. AKHTER ET AL. 257
0.04.08.012.0 16.020.0
x
0.0
1.0
2.0
3.0
4.0
5.0
Skin
f
ricti o n
M = 0.10
M = 0.50
M = 0.70
M = 1.00
(a)
0.04.08.012.0 16.0 20.0
x
0.0
1.0
2.0
3.0
4.0
Heat transfer
M=0.10
M=0.50
M=0.70
M=1.00
(b)
Figure 6. (a) Variation of skin friction and (b) Variation of
Figures 7(a) and (b) plotted the numerical values of
th
r
th
on the skin friction (Cfx)
an
heat transfer against x for varying of M with Ra = 0.01, γ =
0.01 and Pr = 0.733.
e local sk in fr iction coefficie nt (Cfx) and heat transfer rate
(Nux) for different values of Ra associated with the dis-
tinct values of controlling parameter. As the radiation ef-
fect increases the fluid motion as well as temperature with-
in the boundary layer wh ich are mentioned earlier in Fig-
ures 3(a) and (b), respectively. Accordingly, th e correspon-
ding skin friction increases and heat transfer rate decreas-
es with the increasing value of Ra along the x direction.
The variation of skin friction and heat transfer rate fo
e effect of conductivity variation parameter are depi cted
in Figures 8(a) and (b), respectively. The incr easing value
of γ generates greater buoyancy force which therefore
increases the friction between the inner surface of the
vertical plate and moving fluid particles. Thus the skin
friction increases for the greater value of γ that is demon-
strate in Figure 8(a). Moreover, an increase in the value
of γ leads to increase the energy transfer ability within
the flow region, as a result, the heat transfers rate in-
creases with the increasing of γ.
The effects of Prandtl number
d heat transfer rate (Nux) against x for the fixed values
of M, Ra and γ are shown respectively in Figures 9(a)
and (b). The increased values of Pr decrease both the
velocity and temperature of the fluid within the boundary
layer, consequently, the related skin friction on the plate
decreases but the heat transfer rate from heated plate to
fluid increases that has been exposed in Figures 9(a) and
048 121620
x
0.8
1.6
2.4
3.2
4.0
Skin
f
r
icti o n
Ra = 0.01
Ra = 0.03
Ra = 0.06
Ra = 0.08
(a)
048 121620
x
0.0
0.8
1.6
2.4
3.2
4.0
Heat transfer
Ra =0.01
Ra =0.03
Ra =0.06
Ra =0.08
(b)
Figure 7. (a) Variation of skin friction and (b) Variation of
heat transfer against x for varying of Ra with M = 0.50, γ =
0.01 and Pr = 0.733.
0.04.08.012.0 16.0 20.0
x
0.0
1.0
2.0
3.0
4.0
5.0
Skin
f
rictio n
γ = 0.01
γ = 0.10
γ = 0.15
γ = 0.20
(a)
0.04.08.012.0 16.0 20.0
x
0.0
1.6
3.2
4.8
6.4
8.0
Heat trans
f
er
γ=0.01
γ=0.10
γ=0.15
γ=0.20
(b)
Figure 8. (a) Variation of skin friction and (b) Variation of
0.01 and Pr = 0.733.
heat transfer against x for varying of γ with M = 0.50, Ra =

R. AKHTER ET AL.
258
0.04.08.012.016.020.0
x
0.0
0.8
1.6
2.4
3.2
4.0
Skin
f
rictio n
Pr =0.733
Pr =0.930
Pr =1.241
Pr =1.630
(a)
0.04.08.012.0 16.0 20.0
x
0.0
0.8
1.6
2.4
3.2
4.0
Heat transfer
Pr =0.733
Pr =0.930
Pr =1.241
Pr =1.630
(b)
Figure 9. (a) Variation of skin friction and (b) Variation of
heat transfer against x for varying of Pr with M = 0.50, a =
r a particular value of Pr, the local skin
friction coefficient and heatfer rate increase mono-
n parameter and Prandtl
nu
sis, we have studied numerically the
on MHD free convection flow under
R
0.01 and γ = 0.01.
(b). Moreover, fo trans
tonically along the x direction.
The influence of magnetic field parameter, radiation
parameter, conductivity variatio
mber on the interfacial temperature are depicted in Fig-
ures 10 and 11, respectively. The overall temperature
profiles shift upward as well as the thermal boundary
layer thickness increases with the increasing values of M,
Ra, and γ observed in Figures 2(b), 3( b) and 4(b), re-
spectively. Consequently, the surface temperature inc r ea se
for enlarging valu es of M, Ra and γ, respectively. On the
other hand, increasing Pr decreases the fluid temperature
which results interfacial temperature decreases that is
shown in Figure 11(b).
6. Conclusions
In the present analy
effects of radiation
the influence of thermal conductivity variation for a ver-
tical flat plate and the numerical solutions of the trans-
formed governing equations associated with the specified
boundary are obtained for different values of related
physical parameters including magnetic parameter, radia-
tion parameter, thermal conductivity variation parameter
and Prandtl number. The particular conclusions in this
0.04.08.012.0 16.0 20.0
x
0.0
0.4
0.8
1.2
1.6
2.0
Sur
f
ace temperature
M=0.10
M=0.50
M=0.70
M=1.00
(a)
048 121620
x
0.8
1.0
1.2
1.4
1.6
1.8
Su
r
f
acetempe
r
atu
r
e
Ra = 0.01
Ra = 0.03
Ra = 0.06
Ra = 0.08
(b)
Figure 10. (a) Variation of surface temperature against x for
varying M and (b) Variation of surface temperature against
x for varying of Ra.
0.0 4.0 8.012.016.020.0
x
0.0
0.4
0.8
1.2
1.6
2.0
2.4
Sur
f
ace temperature
γ = 0.01
γ = 0.10
γ = 0.15
γ = 0.20
(b)
0.04.08.012.0 16.0 20.0
x
0.8
1.0
1.2
1.4
1.6
1.8
Sur
f
acetemperature
Pr =0.733
Pr =0.930
Pr =1.241
Pr =1.630
(b)
Figure 11. (a) Variation of surface temperature against x for
varying γ and (b) Variation of surface t emperature against x
for varying of Pr.

R. AKHTER ET AL.
259
e for decreasing
[1] V. M. Soundalgekar and H. S. Takhar, “Radiative Con-
vective Flow Pical Plate,” Model-
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Dimensional Transient Conduction and Radiation
study can be listed as follows:
The velocity of the fluid within the boundary layer and
the skin friction at the interface increas
values of magnetic parameter, M, Prandtl number, Pr and
increasing values of the radiation parameter, Ra and
thermal conductivity variation parameter, γ.
The increasing value of M, Ra and γ leads to increase
in the value of temperature within the thermal boundary
layer as well as the surface temperature on the plate a
wo-
Heat
Transfer with Temperature Dependent Thermal Conduc-
tivity,” International Communications in Heat and Mass
Transfer, Vol. 32, No. 3-4, 2005, pp. 305-314.
http://dx.doi.org/10.1016/j.icheat ma sst ran sfer.200 4.05.01 5
[8] M. A. Seddeek and F. A. Salama, “The Effects of Tem-
peratute Dependent Viscosity and Thermal Conductivity
nd
the effect of M and Ra decrease heat transfer rate from
plate to fluid within the boundary layer but opposite re-
sults hold for increasing of γ and Pr.
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on Na-
Nomenclature
f
T: Temperature of the fluid
f
x
C: Local skin friction coefficient
p
C: Specific heat at constant pressure
imensionless stream function f: D
g: Acceleration due to gravity
Gr : Grashof number
h: Derature imensionless temp
f
k,
s
k: Fluid and solid thermal conductivities
M: Magnetic parameter
Nux: Local Nusselt number
p: Conjugate conduction parameter
Pr: Prandtl number
u, v: Velocity components
v: Dimensionless velocity cou, mponents
x
y
, : Cartesian co-ordinates
x, y: Dimensionless Cartesian co-ordinate
β:oeion
γ: Thermal conductivity variation paramete
Cfficient of thermal expansr
es
η: Dimensionless similarity variable
θ: Dimensionless temperature
μ, v: Dynamic and kinematic viscositi
ρ: Density of the fluid
σ: Electrical conductivity