Open Journal of Fl ui d Dyn a mi cs, 20 1 1, 1, 1-11
doi:10.4236/ojfd.2011.11001 Published Online December 2011 (http://www.SciRP.org/journal/ojfd)
1
Radiation Ef f ect on Natu ral Convecti o n near a Vertical
Plate Embedded in Porous Medium
with Ramped Wall Temperature
Sanatan Das, Mrinal Jana, Rabindra Nath Jana
Department of Ap pl i e d M athematics, Vidyasagar University, Midnapore, India
Received November 16, 2011; revised December 7, 2011; accepted De cember 20, 2011
Abstract
Radiation effect on the natural convection flow of an optically thin viscous incompressible fluid near a ver-
tical plate with ramped wall temperature in a porous medium has been studied. The exact solution of mo-
mentum and energy equations is obtained by the use of Laplace transform technique. The variations in fluid
velocity and temperature are shown graphically whereas the numerical values of shear stress and the rate of
heat transfer at the wall are presented in tabular form for various values of flow parameters. The results show
that the fluid velocity increases with increase in Grashof number, Darcy number and time parameters
whereas the fluid velocity decreases with increase in the radiation parameter and Prandtl number for ramped
temperature as well as isothermal wall temperature. It is found that an increase in radiation parameter leads
to rise the temperature for both ramped wall temperature as well as isothermal wall temperature. Further, it is
found that an increase in Prandtl number leads to fall the temperature for both ramped wall temperature as
well as isothermal wall temperature. The shear stress at the wall decreases with increases in either Prandtl
number or porosity parameter while the result shows reverse in the case of radiation parameter. Finally, the
rate of heat transfer is increased with increase in the radiation parameter for both ramped wall temperature as
well as isothermal wall temperature.
Keywords: Natural Convection, Darcy Number, Radiation Parameter, Prandtl Number, Porous Medium,
Ramped Wall Temperature and Isothermal Wall Temperature
1. Introduction
The phenomenon of natural convection arises in fluids
when temperature changes cause density variations lead-
ing to buoyancy forces acting on the fluid particles. Such
flows which are driven by temperature differences abound
in nature and have been studied extensively because of
its applications in engineering, geophysical and astro-
physical environments. Comprehensive literature on va-
rious aspects of free convection flows and its applica-
tions could be found in Ghoshdastidar [1], Nield and
Bejan [2]. Ghoshdastidar gave various areas of applica-
tions of free convection flow such as those found in heat
transfer from pipes and transmission lines as well as
from electronic devices, heat dissipation from the coil of
a refrigerator unit to the surrounding air, heat transfer
from a heater to room air, heat transfer in nuclear fuel
rods to the surrounding coolant, heated and cooled en-
closures, quenching, wire-drawing and extrusion, at-
mospheric and oceanic circulation. Unsteady free con-
vection flows in a porous medium have received much
attention in recent time due to its wide applications in
geothermal and oil reservoir engineering as well as other
geophysical and astrophysical studies. Moreover, con-
siderable interest has been shown in radiation interaction
with convection for heat and mass transfer in fluids. This
is due to the significant role of thermal radiation in the
surface heat transfer when convection heat transfer is
small, particularly in free convection problems involving
absorbing-emitting fluids. The unsteady fluid flow past a
moving plate in the presence of free convection and ra-
diation were studied by Mansure [3], Raptis and Perdikis
[4], Das et al. [5], Grief et al. [6], Ganeasan and Loga-
nathan [7], Mbeledogu et al. [8], Makinde [9] and Ab-
dus-Sattar and Hamid Kalim [10]. All these studies have
been confined to unsteady flow in a non-porous medium.

S. DAS ET AL.
2
Israel-Cookey et al. [11] have studied the influence of
viscous dissipation and radiation on unsteady MHD free-
convection flow past an infinite heated vertical plate in a
porous medium with time-dependent suction. Radiative
and free convective effects of a MHD flow through a
porous medium between infinite parallel plates with time
dependent suction have been investigated by Alagoa et al.
[12]. Israel-Cookey et al. [13] have made an analysis on
MHD oscillatory Couette flow of a radiating viscous
fluid in a porous medium with periodic wall temperature.
Sattar and Maleque [14,15] have studied the unsteady
MHD Natural convection flow and mass transfer along
an accelerated porous plate in a porous medium. Thermal
radiation interaction with unsteady MHD flow past a
vertical porous plate immersed in a porous medium has
been analyzed by Samad and Rahman [16]. Mahanti and
Gaur [17] have studied the effects of varying viscosity
and thermal conductivity on steady free convective flow
and heat transfer along an isothermal vertical plate in the
presence of heat sink. Transient free convection past a
semi-infinite vertical plate with variable surface tem-
perature has been investigated by Takhar et al. [18].
In this present paper, we investigate the effects of ra-
diation on the free convection flow of an optically thin
incompressible viscous fluid past an infinite vertical
plate with ramped wall temperature in porous medium.
The fluid considered is a gray, radiation, absorbing,
emitting but non-scattering medium and the Rosseland
approximation is used to describe the radiative heat
transfer in the energy equation. It is seen that the velocity
1 decreases for both ramped wall temperature as well
as isothermal wall temperature with an increase in either
radiation parameter or Prandtl number . It is
also seen that the velocity 1 increases for both ramped
wall temperature as well as isothermal wall temperature
with an increase in either Grashof number or time
u
Ra Pr
Gr
u

. It is found that an increase in radiation parameter
leads to rise the temperature
Ra

for both ramped
wall temperature as well as isothermal wall temperature.
Further, it is found that an increase in Prandtl number
leads to fall the temperature for ramped temperature as
well as isothermal case.
2. Formulation of the Problem and Its
Solutions
Consider the unsteady free convection flow of an opti-
cally thin viscous incompressible fluid past an moving
infinite vertical plate coinciding with plane 0y

,
where the flow is confined to in a porous me-
dium. Choose a cartesian co-ordinates system with x-axis
along the wall in a vertically upward direction and y-axis
is normal to it into the fluid (see Figure 1). At
0y
0t

, the
Figure 1. Geometry of the problem.
plate and the surrounding fluid are at the same constant
temperature T

. At time , the temperature of the 0t
wall is raised or lowered to

0
wt
TTT
t

 when
0
0tt


l
and the constant temperature w
T is main-
tained at 0. Since the plate is infinite along x-direc-
tion, all the physical variables are the function of y an
t ony. The flow is considered optically thin gray gas
with natural convection and radiation. The radiative heat
flux in the x-direction is considered negligible in com-
parison to y-direction.
tt
d

The Boussinesq approximation is assumed to hold and
for the evaluation of the gravitational body force, the
density is assumed to depend on the temperature accord-
ing to the equation of reference state

01TT
 




 

, (1)
where is the fluid temperature,
T

the fluid density,


the coefficient of thermal expansion and T

and
0

being the reference temperature and the density re-
spectively.
Using Boussinesq Approximation (1), the momentum
equation in a porous medium along x-axis is

2
2
uu
g
TT u
tyk





 
, (2)
where , u
g
,


,

,

and are respectively,
fluid velocity, acceleration due to gravity, coefficient of
thermal expansion, kinematic viscosity, fluid density and
permeability of a porous media.
k
The energy equation is
2
2
1r
pp
q TkT
tc cy y

 



, (3)
where is the thermal conductivity, k
p
c the specific
heat at constant pressure and the radiative heat flux.
r
The initial and boundary conditions are
q
0u

, TT


for and ,
0y0t
0
Uu

at for t,
0y0

3 S. DAS ET AL.

0
0
0
at 0 for 0
at 0 for
0,as for 0
w
w
t
TTT Tytt
t
TTy tt
uTTy t


 

 

(4)
It has been shown by Cogley et al. [19] that in the op-
tically thin limit for a non-gray gas near equilibrium, the
following relation holds

0
0
0
4d
h re
qTT K
yT







 


, (5)
where
K

is the absorption coefficient,

is the wave
length, h
e

is the Plank’s function and subscript 0
`

indicates that all quantities have been evaluated at the
temperature which is the temperature of the wall at
time . Thus our study will be limited to small dif-
ference of wall temperature to the fluid temperature.
T
0t
On the use of (5), Equation (3) becomes

2
2
4
pp
TkT TTI
tc c y 



, (6)
where
0
0
0
d
h
e
IKT








. (7)
Introducing dimensionless variables
00
y
Ut
 ,
0
t
t
 , 1
0
u
uU
,
w
TT
TT




, (8)
Equations (2) and (6) become
2
11
1
2
1
uuuGr

 


, (9)
2
2
1
Pr Ra





, (10)
where

3
0
w
g
TT
Gr U




 is Grashof number,
Pr
p
c
k


 the Prandtl number, 2
0
4
p
I
Ra cU


 the ra-
diation parameter,
2
0
2
kU
M
aDa 

 the porosity
parameter and the Darcy number. Da
The characteristic time is defined as
0
t
02
0
tU

. (11)
The corresponding initial and boundary conditions for
and
1
u

are
10,0for 0 and 0,u

  
11at 0 for 0,
at 0 for 01,
u


 




(12)
1


at 0


for 1

, ,
10u
0

 as

for 0

.
Taking Laplace transformation of the Equations (9)
and (10), we get
2
1
1
2
d1
d
u
s
uGr
MaDa



 

 , (13)

2
2
dPr 0
dsRa



, (14)
where
 
 
11
0
0
,,
,,
s
s
us ue
se


d,
d.

 









(15)
The corresponding boundary conditions for 1 u and

are

12
1
11
,1 at
0,0 as .
s
ue
ss
u


0,

 
 
(16)
The solution of the Equations (14) and (13) subject to
the boundary conditions (16) can be easily obtained and
are given by




Pr
2
1
,
s
s
Ra
e
se
s





, (17)



1
1
1
Pr
2
1
,
1
()
sMaDa
sssRa
MaDa
us e
s
eee
ss






 












(18)
where

1Pr
Gr
  and

1
Pr
1Pr
Ra
M
aDa





. (19)
Taking the inverse Laplace transform of Equations (17)
and (18), the solution for the fluid temperature


,


and fluid velocity


1,u

 are obtained and are given
by



 
11
,,,1H  

1

, (20)

1
1
, erfc
22
erfc2
MaDa
MaDa
ue
M
aDa
e
M
aDa



 



















S. DAS ET AL.
4
,

(21)
where
 
11
,,11H  


Pr
1
Pr
1Pr
,22
Pr
erfc2
Pr Pr
erfc,
22
Ra
Ra
e
Ra
Ra
e
Ra



 



 R a














 
  

 
 

 

(22)





1
12
1
Pr
Pr
1
,2
1
erfc2
1
erfc2
Pr
erfc2
Pr
erfc2
MaDa
MaDa
Ra
Ra
ee
MaDa
eMaDa
eRa
eRa
 



 





















 



























Pr
11
2
erfc2
11
2
11 Pr
erfc2
2
Pr1 1Pr
erfc22
MaDa
MaDa
R
a
MaDa e
MaDa
MaDa e
e
MaDa Ra
Ra Ra











 
































Pr Pr
,
2
Ra
eerfc Ra






 




(23)
where

erfc
x

1
is the complementary error function and
is the unit step function.
2.1. Solution in Case of Unit Prandtl Number
Prandtl number is a measure of the relative stren
the viscosity and thermal conductivity of the fluid. So the
H  
gth of
case Pr 1

us a
order
t solutio
co
visconess are of the
ameof magnitude. Setting in Equation
the
xacn for the fluid temp
rresponds to those fluids for which both
nd thermal boundary layer thick
sPr 1
erature
(14) and following the same procedure as before,
e


,

 and
fluid velocity


1,u

 is obtained and is expressed in
the following form



 
22
,,,11H   

, (24)

 
1
1
, erfc
22
MaDa
Ma
ue MaDa
e


 











22
erfc 2
,,11,
Da MaDa
H


 














(25)
where

2
1
,erfc
222
erfc,
22
Ra
Ra
eR
Ra
eR
Ra



a
a

 






 





 






 
(26)



2
1
,erfc
222
erfc
22
erfc
22
erfc,
22
Ra
Ra
MaDa
MaDa
eRa
Ra
eRa
Ra
MaDa eMaDa
MaDa eMaDa





 











 



 
 
 
 

 





 









  

  
and 1 Gr Ra
M
aDa





.
2.2. Solution for Isothermal Case
In order to highlight the effects of the ramped tempera-
ture distribution near a vertical plate, it may be important
mpare the effects of thal temperature dis-
tribution for the fluid flow. The temperature and the ve-
city for the fluid flow near an isothermal plate can be
to coe isotherm
lo
expressed as

Pr
Pr
1Pr
, erfc
22
Pr
erfc,
2
Ra
Ra
eRa
eRa




 



















(27)

5 S. DAS ET AL.

 
1
11 12
1
, erfc
22
erfc2
,,,
MaDa
MaDa
ue
M
aDa
e
M
aDa
uu



 


 


















(28)
where

1
11
1
1
,2
1
erfc2
1
erfc2
erfc2
erfc2
MaDa
MaDa
MaDa
MaDa
uee
MaDa
eMaDa
eMaDa
e





 
































 










 








Pr
12
Pr
Pr
1
,2
Pr
erfc2
Pr
erfc2
Pr
erfc2
Ra
Ra
Ra
MaDa
uee
Ra
e
eRa





 



R a












 




































Pr Pr
erfc.
2
Ra
eRa







 







(29)
When , the Solutions (27) and (28) become Pr 1

1
,erfc
22
erfc,
2
Ra
Ra
eR
eR



 
















a
a
(30)
 
113 1314
, ,,uuu u

,







  

, (31)
where

13
1
,erfc
22
MaDa
ue
M
aDa


 






erfc ,
2
MaDa
eMaDa






 




(32)

14
1
,erfc
22
erfc.
2
Ra
Ra
ue
eR



Ra
a


















(33)
3. Results and Discussion
We have plotted the non-dimensional velocity and tem-
perature for several values of radiation para
Prandtl number , Grashof number , Darcy
ber and tim
meter Ra ,
Pr
e
Gr num-
Da

in Figures 2-9 2-6
resen e ve against
. Figures rep-
t thlocity 1 u

foral valu
and
r sevees of
Ra , Pr , Gr , Da

.
a
ram
mp
creases
mb
Figure
rameter
p
eratu
for
er Pr
ow
n pads
hperat
re 3 dis
e ramped
eratur
tl nuysically, th
2 sh
le
tem
Figur
th
mp
in t
am
s that an
to fall in in
th
at
n
enh
crease in th
e velo
well as iso
the
in
luid.
a
e radi
rmal wall te
ty u
buo
u
atio
for bot
in
Ra
ed wall
e.
bo
. Ph
r
city 1 u
the
veloci
crease in Prand
nt i
l
ure as
plays
wall
e with
is is
the
th 1
temperature as well as isothermal wall te
d
a



true because the increase in the Prandtl number is due to
increase in the viscosity of the fluid which makes
fluid thick and hence causes a decrease he velocity of
the f It is observed from Figure 4 that an increase in
Gr , leads to a rise the values of velocity 1 u due to
ncemen yancy force. Figure 5 reveals that
the velocity 1 u increases for both ped wall tem-
perature as wel as isothermal wall temperature with an
increase in Darcy number Da. It is seen from Figure 6
that the velocity 1 increases for both ramped wall
temperature as well as isothermal wall temperature with
an increase in time

. It is observed from Figure 7 that
the temperature

decreases as the radiation parameter
Ra increases for both ramped wall temperature as well
as isothermal wall temperature. This result qualitatively
agrees with expectations, since the effect of radiation is
ecrease the rate of energy transport to the fluid,
thereby decreasing the temperature of the fluid. It is seen
from Figure that the temperature
to d
8

decreases for
both ramped wall temperature as well as isothermal wall
temperature with an increas in Prandtl number Pr.
This implies that anncrease in Prandtl number leads to
fall the thermal boundary layer flow for ramped tem-
perature as well as isothermal wall temperature. The ef-
fect of the Prandnumber is very important in the tem-
ture field. A fall in temperature occurs due to an in-
creasing value of the Prandtl number. This is in agree-
ment with the physical fact that the thermal boundary
e
i
tl
decrease
pera
layer thicknesssFigure with increase in Pr . 9

S. DAS ET AL.
6
and 0.04Da .
R
a when Pr0.71

, 25Gr

, 0.1


Figure 2. Velocity profiles for variations in
Figure 3. Velocity profiles for variations in when Pr 0.04Da

, 25Gr

, 0.1


and
shows that the temperature
2Ra .

increases for both ramped
wall temperature as well as isothermal wall temperature
with an increase in time

.
From the physical point of view, it is necessary to
know the shear stress and the rate of heat transfer (or the
Nusselt number) at the wall


0  . We have presen ted
the expression for the rate heat transfer and
shear stress
of Nu
0

at the wall 0


emp
erature
in the following form
for both themped wall terature and isothermal
wall temp
For the ramed wall temp
ra
erature.
p

, 
 
33
,
 

 
0
11Nu H

 

 
, (34)





S. DAS ET AL. 7
Figure 4. Velocity profiles for variations in when GrPr 0.71

, 0.04Da

, 0.1


and 2Ra .
D
a when Pr0.71

, 25Gr

, 0.1


and 2Ra . Figure 5. Velocity profiles for variations in
 
1
0
0
33
11
erf
π
,,11,
MaDa
ue
M
aDa MaDa
H




 





 






 

(35)



32
1
11
,erf
1
Prerf
π
MaDa
e
MaDa MaDa
eRaRa


 










 






 
where

S. DAS ET AL.
8
Figure 6. Velocity profiles for variations in time

when Pr0.71

, 25Gr

, 2Ra

and 0.04a. D
R
aFigure 7. Temperature profiles for variations in when Pr0.71

and 0.5

.


Pr1 erf
π2
11
erf
11 11Pr
erf
2
π
Ra
MaDa
MaDa
eMaDa
MaDa
MaDa
eR
Ra

a









 


 
 












1Pr1
Pr erf,
π
Ra
Ra Rae




 

 

 

 

(36)

 

 
 


 
and for the isothermal wall temperature

0
Pr
Pr erfπ
R
a
NuRa Rae








 

, (37)

S. DAS ET AL. 9
Figure 8. Temperature profiles for variations in whenPr 2Ra

and 0.5

.
and 2Ra . Figure 9. Temperature profiles for variations in time

when Pr0.71




1
0
0
1
1erf
1
Prerf
π
MaDa
u
MaDa
MaDa
eeRa Ra
















 






11
erf
Pr erf.
eMaDa MaDa
Ra Ra






 









(38)

S. DAS ET AL.
10
Numerical results of shear stress at the wall


0 

es radia-
of num-
thermal wall temperature with an increase in Prandtl
number
Numerical results of the rate of heat transfer at the
are presented in Tables 1 to 4 for various valu
tion parameter , Prandtl number , Grash
ber , Dnumber and time
Ra
arcy
Pr
4GrDa

.
0
Table 1
shows thegnitude that ma of shear stress

decreases
thermal
ber Da
is reversed
r fixed
that the
for bop temp
erature with an in
fo ande th
with i ram fo
valuserom
mas
th ram
wall temp
r fixed value
an in
es of
gnitude of
ed wall
s of
crease in
. It i
hear st
erature
crease in
whil
ation pa
ved fr
as well as
Darcy
e resu
eter
Ta
iso
num
lt
Ra
ble 2
Ra
rad
s ob
res
Da
s0

y num
d
decreases
wall term
with iDab
nu from
mas
er
Ta
for both ram
m
or
ble 3
ped
perature
Grashof
that the
mp
ncreas
bmer Gr
gnitude of
erature as well as isothe
r
. It is also observe
res
al wall te
Da e
s
in eithe
hear st
rc
0

decreases
wall term
withe
for both ram
m
ped
perature mp
an in
erature as well as isothe
tim
al wall te
crease in

. Table 4
creases for both ramped wall temperature as well as iso-
Table 1. Shear stress
displays that for
0

 for Pr0.71, 10Gr

and
1

.
Ramped temrature peratue Isothermal temper
DaRa 25 30 35 25 30 35
0.040
0.045
0.050
0.055
3.93632
3.61771
3.34675
3.11244
3.97912
3.66324
3.39462
3.16245
4.01576
3.70195
3.43524
3.20487
3.91459
3.59386
3.32075
3.08436
3.95998
3.64215
3.37170
3.13779
3.99849
3.68301
3.41473
3.18282
Table 2. Shear stress 0

 for Pr0.71, 25Ra

and
1

.
Ramped temature perature Isothermal temper
Da Gr 10 15 20 10 15 20
0.040
0.045
0.050
0.055
3.9363
3.6177
3.3467
3.1124
3.4044
3.0696
2.7840
2.5366
2.8726
2.5214
2.2214
1.9608
3.91459
3.59386
3.32075
3.08436
3.37188
3.03377
2.74506
2.49454
2.82917
2.47368
2.16937
1.90471
Table 3. Shear stress 0

 for and Pr 0.7110Gr

.
Ramped temperature Isothermal temperature
R
a  25 30 35 25 30 35
0.5
1.0
1.5
2.0
4.93191
4.80524
4.69611
4.58754
4.93417
4.81177
4.70716
4.60313
4.93632
4.81752
4.71677
4.61657
3.96181
3.91696
3.91474
3.91460
4.00034
3.96170
3.96007
3.95998
4.03430
3.99986
3.99855
3.99849
Table 4. Shear stress 0

 for 25Ra , 0.4Da

.
Ramped temperature Isothermal temperature
Pr

Pr .


0 

are presented in Tables 5 to 6 for various val-
ation parameter , Prandtl number and ues radi
time
Ra Pr

.
Nu inc
as isot
tion pa
increases
isotherm
Table 5 shows at the rate of heat fer
reases for both ramed wall temperature as well
hermal wall tempwith an increase in radia-
rameter . Further, the rate of heatr
for raed wall teperature while it decreases
al wallwith an increase in time
th
p
erature
m
erature
trans
transfe Ra
mp
temp

at for fixe
for fixed
d valuesbserved from Ta th
value
of
of tim
Ra .
e
It is oble 6

, th
ed wa
w
e rate of heat tran
for ll temperature
al wallith an increase in Pran
4. Conclusions
An analysis is made to study the radiation effects on free
convection flow past an impulsively started infinite ver-
tical wall with ramped wall temperature in a porous me-
dium. The velocity field and temperature distribution are
ntental parameters graphically. It
is observed that the velocity profiles decrease with an
increase in Prandtl number for ramped wall tem-
perature as well as isotherm wall temperature. An in-
crease in Grashof number ads to a rise in the val-
ues of velocity due to enhent in buoyancy force.
The velocity field is accelerated due to increase in Darcy
number . The effect Prandtl number is very
importan in the temperatu. A fall in temperature
occurs due to an increasing of the Prandtl number.
It is fouat the temperature decreases as the radiation
parameter increases for boted wall temperature as
well as isormal wall temre. Further, the absolute
value of ear stress
sfer
as well as for
Nu
dtl
increases
isotherm
number
both ram
temp
p
erature
Pr .
prese d for differe physic
Pr
al
le
cem
he
field
value
ramp
eratu
Gr
an
of t
re
h
p
Da
t
nd th
the
sh 0

increases for both ramped wall
Table 5. Rate of heat transfer for NuPr 0.71

.
Ramped temperature Isothermal temperature
R
a

0.1 0.2 0.3 0.1 0.2 0.3
2
4
6
8
0.32032
0.33924
0.35748
0.37509
0.47976
0.53046
0.57787
0.62246
0.61917
0.70774
0.78852
0.86297
1.79436
2.06757
2.32489
2.56804
1.46199
1.81588
2.13371
2.42239
1.34228
2.08819
2.91893
2.39419
Tale 6. Rateransfer fb of heat t orNu 25Ra

.
Ramped temperature Isothermal temperature
0.5 1.0 1.5 0.5 1.0 1.5 Pr

0.1 0.2 0.3 0.1 0.2 0.3
0.71
2.0
5.0
7.0
4.93191
4.95270
4.96972
4.97534
4.80524
4.85151
4.88940
4.90190
4.69611
4.76807
4.82698
4.84643
3.96181
4.21229
4.41735
4.48504
3.91696
4.17355
4.38361
4.45295
3.91474
4.17170
4.38207
4.45151
0.71
2.0
5.0
7.0
0.32032
0.53761
0.85004
1.00578
0.47976
0.80520
1.27314
1.50640
0.61917
1.03919
1.64310
1.94414
1.79436
3.01159
4.76174
5.63417
1.46199
2.45375
3.87972
4.59055
1.34228
2.25283
3.56204
4.21466

S. DAS ET AL.
11
temperature as well as isothermal wall temperature with
an increase in Darcy number for fixed values of
and while the result is rev with an increase in
ion parameter for fivalues of. The
heat transfer increases for both raed wall
erature as well as isothermal wall tempe with
rease in radiation rameter
5. References
[1] P. S. Ghoshdastidar, “Heat Transfer,” Oxford
Press, Oxford, 2004.
[2] D. A. Nield and A. Bejan, “Convection in Porous Me-
dia,” Springer, New York, 2006, pp. 94-97.
[3] Effects
ce S
275. 98
Da
ersed
xed
Ra
Ra
radiat
rate of
temp
an inc
Ra
Nu
pa
Da
mp
eratur
.
University
M. A. Mansour, “Radiation and Free Convection
on the Oscillating Flow past a Vertical Plate,” Astrophys-
ics and Spacience, Vol. 166, No. 2, 1990, pp. 269-
doi:10.1007/BF010948
F
ka
id
[4] A. Raptis and C. Perdikis, “Radiation andree Convec-
tion Flow past a Moving Plate,” Applied Mechanics and
Engineering, Vol. 4, No. 4, 1999, pp. 817-821.
[5] U. N. Das, R. De and V. M. Soundalgekar, “Radiation
Effect on Flow past an Impulsively Started Vertical Plate:
An Exact Solutions,” Journal of Applied Mathematics
and FluMechanics, Vol. 1, No. 2, 1966, pp. 111-115.
[6] R. Grief, I. S. Habib and J. C. Lin, “Laminar Convection
of Radiating Gas in a Vertical Channel,” Journal of Fluid
Mechanics, Vol. 46, 1991, pp. 513-520.
doi:10.1017/S0022112071000673
[7] P. Ganesan and P. Lonathan, “Radiation and Mass
Transfer Effects on Flow of an Incompressible Viscous
Fluid past a Moving Vertical Cylinder,” International
Journal of Heat and Mass Transfer, Vol. 45, No. 21,
2002, pp. 4281-4288.
ga
doi:10.1016/S0017-9310(02)00140-0
lu, “Un-
ady MHD Free Convection Flow of a Compressible
al Plata-
diaveransfer,” Inatioournf Hend
-331.
:1
[8] I. U. Mbeledogu, A. R. C. Amakiri and A. Ogu
ste
Fluid pst a Moving Verticae in the Presence of R
ti Heat Tnternal Jal oat a
Mass Transfer, Vol. 50, No. 9-10, 2007, pp. 326
doi 0.1016/j.ijheatmasstransfer.2006.10.032
[9 D
d r
rous Plate,” International Communications in Heat and
32, No. 10, 2005, pp411-141
asstransfer.2005.07 05
] O.. Makinde, “Free-Convection Flow with Thermal
Raiation and Mass Transfer past a Moving Ve tical Po-
Mass Transfer, Vol. . 19.
doi:10.1016/j.icheatm .0
[10] M. D. Abdus-Sattar and M. D. Hamid Kalim, “Unsteady
ree-Co with a
Boundarea
Jona MaatiPhysVol.96, pp.
25-37.
[11] C. Israel-Cookey, A. Ogulu and V. B. Omubo-Pepple,
“Influence of Viscous Dissipation-
steadHD e-Cotion Flow past an Infinite
lat aum
im nct
nV3 .
n and Radiation on U
y MFrenvec
Heated
T
Vert
e-Depe
ical P
dent Su
e in
tion,” Int Porous ernationa Medi l Journa with l of Hea
ad Mass Transfer, ol. 46, No. 13, 2003, pp. 205-2311
doi:10.1016/S0017-9310(02)00544-6
[12] an. Abbey
cts HD Flow through a Po-
rous Medium Infinite
Time,” Astro ci-
ence l. 2699, p5-46
doi:10.1016 9
K. D. Alagoa, G. Tayd T. M, “Radiative and
Free Convective Effeof a M
between
-Dependent Suction
Parallel Plates with
physics and Space S
, Vo0, 19p. 458.
/S0017- 310(02)00544-6
[1 C C
O s
a Porous Medium with Periodic Wall Temp
1,. 2, 2010, pp. 326-331.
3] . Israel-ookey, E. Amos and C. Nwaigwe, “MHD
scillatory Couette Flow of a Radiating Viscou Fluid in
erature,”
American Journal of Scientific and Industrial Research,
Vol. No
doi:10.5251/ajsir.2010.1.2.326.331
. A. Sattar and M. A. Maleque, “Unsteady MHD Natu-
ral Convection Flow along an Accelerated Porous Plate
with Hall Current and Mass Transfer in a Rotating Porous
Medium,” Journal of Energy, Heat and Mass Transfer,
Vol. 22, 2000, pp. 67-72.
[15] M. A. Sattar, M. M. Rahman and M. M. Alam, “Free
Convection Flow and Heat Transfer through a Porous
Vertical Flat Plate Immersed in a Porous Medium,”
Journal of Energy Research, Vol. 22, No. 1, 2000, p.
17-21.
[16] M. D. Abdus Samad and M. Mansur Rahman, “Thermal
Radiation Interaction with Unsteady MHD Flow past a
Vertical Porous Plate Immersed in a Porous Medium,”
Jourl o
Vol. 3
[14] M
p
naf Naval Architecture and Marine Engineering,
, No. 1, 2006, pp. 7-14.
“The Effects of Varying Vis-
Conductivity on Steady Free Convec-
[17] N. C. Mahanti and P. Gaur,
cosity and Thermal
tive Flow and Heat Transfer along an Isothermal Vertical
Plate in the Presence of Heat Sink,” Journal of Applied
Fluid Mechanics, Vol. 2, No. 1, 2009, pp. 23-28.
[18] H. S. Takhar, P. Ganesan, K. Ekambavanan and V. M.
Soundalgekar, “Transient Free Convection past a Semi
Infinite Vertical Plate with Variable Surface Tempera-
ture,” International Journal of Numerical Methods for
Heat and Fluid Flo w, Vol. 7, No. 4, 1997, pp. 280-296.
doi:10.1108/09615539710165804
[19] A. C. L. Cogley, W. G.ncenti and E. S. Gilles, “Dif-
ferential Approximation for Radiative Heat Transfer in a
Non Grey Gas near Equilibrium,” American Institute of
Aeronautics and Astronautics, Vol. 6, No. 3, 1968, pp.
551-553.
F nvection Intraction Thermal Radiation in
y Lay
l ofr Flow past them Vertical Porous Plate,” ics, urcal 30, 19
Vi