International Journal of Engineering Research (ISSN : 2319-6890)

Volume No.2, Issue No.2, pp : 113-118 01 April 2013

IJER@2013 Page 113

Effects of Radiation and free Convection Currents on Unsteady Couette Flow
between two Vertical Parallel Plates with Constant Heat flux and Heat Source
Through Porous Medium

Damala Ch Kesavaiah
1
, P V Satyanarayana
2
and A Sudhakaraiah
3
1
Department of H & BS, Visvesvaraya College of Engineering & Technology, Greater Hyderabad, A.P, India
chennakesavaiah@gmail.com
2
Fluid Dynamics Division, School of Advanced Science, VIT University, Vellore - 632 014, T N, India
3
Department of Future Studies, S V University, Tirupati - 517 502, A.P, India
ABSTRACT
The present study the free convection in unsteady Couette flow of a viscous incompressible fluid confined between two vertical
parallel plates in the presence of thermal radiation with heat source in the presence of uniform magnetic field is presented. The flow is
induced by means of Couette motion and free convection currents occurring as a result of application of constant heat flux on the wall
with a uniform vertical motion in its own plane while constant temperature on the stationary wall. The fluid considered here is a gray,
absorbing-emitting but non-scattering medium, and the Rosseland approximation is used to describe the radiative heat flux in the
analysis. The dimensionless governing partial differential equations are solved by using regular perturbation technique. The results for
the velocity, temperature and the skin-friction are shown graphically. The effects of different parameters are discussed.

Keywords: Couette flow, Natural convection, Vertical plate, Constant heat flux, Radiation and heat source.

1 INTRODUCTION
Unsteady free convection 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,
considerable 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 radiation were studied by
Grief et al. [8] and Makinde [13]. From the previous literature
survey about unsteady fluid flow, we observe that little papers
were done in porous medium. The effect of radiation on MHD
flow and heat transfer must be considered when high
temperatures are reached. El-Hakiem [7] studied the unsteady
MHD oscillatory flow on free convection-radiation through a
porous medium with a vertical infinite surface that absorbs the
fluid with a constant velocity. Cookey et al. [6] researched the
influence of viscous dissipation and radiation on unsteady
MHD free-convection flow past on infinite heated vertical
plate in a porous medium with time-dependent suction.

The fluid flow between parallel plates by means of Couette
motion is a classical fluid mechanics problem that has
applications in magnetohydrodynamic (MHD) power
generators and pumps, accelerators, aerodynamics heating,
electrostatic precipitation, polymer technology, petroleum
industry, purification of crude oil, and also in many material
processing applications such as extrusion, metal forming,
continuous casting, wire and glass fiber drawing, etc. This
problem has received considerable attention in the case of
horizontal parallel plates, Attia, Choi et.al Kumar et. al and
Soundalgekar [2, 4, 12, 16] than vertical parallel plates.

In many engineering applications such as cooling of electronic
equipments, design of passive solar systems for energy
conversion, cooling of nuclear reactors, design of heat
exchangers, chemical devices and process equipment,
geothermal systems, and others. However, very few papers
deal with free convection in Couette motion between vertical
parallel plates. Jha [11] Fully-developed laminar free
convection Couette flow between two vertical parallel plates
with transverse sinusoidal injection of the fluid at the
stationary plate and its corresponding removal by constant
suction through the plate in uniform motion has been analyzed
by Jain and Gupta [10]. In their study, the moving wall is
thermally insulated and the wall at rest is kept at a uniform
temperature. Narahari [14] Effects of thermal radiation and
free convection currents on the unsteady couette flow between
two vertical parallel plates with constant heat flux at one
boundary.

International Journal of Engineering Research (ISSN : 2319-6890)
Volume No.2, Issue No.2, pp : 113-118 01 April 2013

IJER@2013 Page 114

The study of heat generation/absorption effects in moving
fluids is important in view of several physical problems, such
as fluids undergoing exothermic or endothermic chemical
reactions. Hosssain et al. [9] studied the problem of natural
convection flow along a vertical wavy surface with uniform
surface temperature in the presence of heat generation /
absorption. Alam et al. [1] studied the problem of free
convection heat and mass transfer flow past an inclined semi-
infinite heated surface of an electrically conducting and steady
viscous incompressible fluid in the presence of a magnetic
field and heat generation. Chamkha [3] investigated unsteady
convective heat and mass transfer past a semi-infinite porous
moving plate with heat absorption.

The aim of the paper is to provide an exact analysis of
unsteady free convection in Couette motion between two
vertical parallel plates in the presence of thermal radiation in
the presence of uniform magnetic field where the moving plate
is subject to constant heat flux and the plate at rest is
isothermal. These solutions are useful to gain a deeper
knowledge of the underlying physical processes and it
provides the possibility to get a benchmark for numerical
solvers with reference to basic flow configurations.
2 MATHEMATICAL ANALYSIS
Consider the unsteady free-convective Couette flow of an
incompressible viscous radiating fluid between two infinite
vertical parallel plates in the presence of uniform magnetic
field separated by a distance h . The x' axis is taken along
one of the plates in the vertically upward direction and the
y' axis is taken normal to the plate. Initially, at time 0 t' s ,
the two plates and the fluid are assumed to be at the same
temperature
h
T' and stationary. At time 0 t' > , the plate at y
= 0 starts moving in its own plane with an impulsive velocity
U and is heated by supplying heat at constant rate whereas the
plate at y h ' = is stationary and maintained at a constant
temperature
h
T' . It is also assumed that the radiative heat flux
in the x' direction is negligible as compared to that in the
y' direction. As the plates are infinite in length, the
velocity and temperature fields are functions of y' and
t' only. Then under the usual Boussinesqs approximation, the
flow of a radiating fluid is shown to be governed by the
following system of equations:
( )
2 2
0
2 h
B u u
g T T u u
t y K
o v
v

' ' c c
'
' ' ' = +
' ' c c

(1)
( )
2
0 2
r
p h
q T T
C k Q T T
t y y

' ' c c c
'
' =
' ' ' c c c

(2)
With the following boundary conditions
0: 0, 0
0: , 0
0,
h
h
t u T T for y h
T q
t u U at y
y k
u T T at y h
' ' ' ' ' s = = s s
' c
' ' ' > = = =
' c
'
' ' ' = = =

(3)
where g is the acceleration due to gravity, the volumetric co-
efficient of thermal expansion, v the kinematic viscosity,
the density, k the thermal conductivity,
p
C the specific heat at
constant pressure, q the constant heat flux,
r
q the radiative
heat flux in y' direction, T' the fluid temperature, and u'
is the fluid velocity.
The radiative heat flux term is simplified by making use of the
Rosseland approximation Siegel and Howell [15] as
4
4
3
r
T
q
k y
o
-
' c
=
' c

(4)
Where o is the Stefan - Boltzmann constant and k
-
is the
mean absorption coefficient. It should be noted that by using
the Rosseland approximation we limit our analysis to optically
thick fluids. If temperature differences within the flow are
sufficiently small such that
4
T' may be expressed as a linear
function of the temperature, Then the Taylor series for
4
T'
about
h
T' , after neglecting higher order terms, is given by
4 3 4
4 3
h h
T T T T ' ' ' ' ~
(5)
It is emphasized here that equation (5) is widely used in
computational fluid dynamics involving radiation absorption
problems Chung [5] in expressing the term
4
T' as a linear
function.
In view of Equations (4) and (5), Equation (2) reduces to
3 2 2
2 2
16
3
h
p
T T T T
C k
t y k y y
o

-
' ' ' ' c c c c
= +
' ' ' ' c c c c

(6)
In order to solve the governing equations in dimensionless
form, we introduce the following non-dimensional quantities:
International Journal of Engineering Research (ISSN : 2319-6890)
Volume No.2, Issue No.2, pp : 113-118 01 April 2013

IJER@2013 Page 115

( )
3
2
2 2 2 2
0 0
3
, , , ,
/
, , Pr ,
4
h
p
p h
T T y t u g h q
y t u Gr
h h U hq k Uk
C
B h Q h h kk
M K R
k C k T
v |
u
v

o
|
v v o
-
'
' ' ' '
= = = = =
= = = = =
'
(7)
where Gr is the thermal Grashof number, Pr the Prandtl
number, R the radiation parameter, | heat source parameter, t
the dimensionless time, u the dimensionless velocity, y the
dimensionless coordinate axis normal to the plate, the
coefficient of viscosity and is the dimensionless temperature.
Then in view of Equations (7), Equations (1), (6) and (3)
reduces to the following non-dimensional form of equations:
2
2
1 u u
Mu u Gr
y t K
u
c c
+
c c

(8)
( )
2
2
3 4 3 Pr 3 Pr R R R
y t
u u
|u
c c
+
c c

(9)
The initial and boundary conditions are
0: 0, 0 0 1
0: 1, 1 0
0, 0 1
t u for y
t u at y
y
u at y
u
u
u
s = = s s
c
> = = =
c
= = =

(10)
3 SOLUTION OF THE PROBLEM
Equation (8) (9) are coupled, non linear partial differential
equations and these cannot be solved in closed form using
the initial and boundary conditions (10). However, these
equations can be reduced to a set of ordinary differential
equations, which can be solved analytically. This can be done
by representing the velocity, temperature and concentration of
the fluid in the neighbourhood of the fluid in the
neighbourhood of the plate as
( ) ( ) ( )
2
0 1
0
nt
f f y e f y c c = + + (11)
Substituting (11) in Equation (8) (9) and equating the
harmonic and non harmonic terms, and neglecting the higher
order terms of
( )
2
0 c , we obtain
0 1 0 0
u u Gr | u '' = (12);
1 4 1 5 1
u u | | u '' =
(13)
0 9 0
0 u | u '' = (14);
1 10 1
0 u | u '' = (15)
The corresponding boundary conditions can be written as
0 1
0 1
0 1 0 1
1, 0, 1, 0, 0
0, 0, 0, 0, 1
u u at y
y y
u u as y
u u
u u
c c
= = = = =
c c

(16)
The solutions of Equations (12) - (15) under the initial and
boundary conditions (16) by perturbation technique is given
by
( )
6 2 1 2
1 2 3 4
,
m y m y m y m y
u y t Z e Z e Z e Z e = + + +
( )
2 1
1 2
,
m y m y
y t De De u = +
APPENDIX
The constant not given Brevity of the space
4 RESULTS AND DISCUSSION
An exact solution to the problem of natural convection in
unsteady Couette flow between two long vertical parallel
plates in the presence of constant heat flux and thermal
radiation have been presented in the preceding section. In
order to get the physical insight into the problem, the
numerical values of the temperature field, the velocity field,
the skin-friction, the Nusselt number, the volume flow rate and
the vertical heat flux are computed for different values of the
system parameters such as Radiation parameter (R), Grashof
number
( ) Gr , Prandtl number (Pr) and heat source
parameter
( ) | . Figure (1) presents the velocity profiles for
both air and water
( ) Pr 7.0 = in the case of pure convection
( ) 100 R for different values of Gr . It is seen that the
velocity of air and water increases with increasingGr . At a
smaller Gr the velocity distribution is monotonic, but at a
higher time it passes through a maximum near the moving
plate when the buoyancy effect partly suppresses the inertial
effects of the plate velocity. Moreover, the velocity of air is
greater than the velocity of water. Physically this is possible
because fluids with high Prandtl number have greater
viscosity, which makes the fluid thick and hence move slowly.
Figure (2) presents the velocity profiles of air for different
values of K . It is observed that the velocity increases with
increasing K . Physically this is possible because as the
Grashof number or time increases, the contribution from the
buoyancy force near the moving hot plate become more
significant and hence a small rise in the fluid velocity near the
plate is observed. Figure (3) illustrate the influences of M
International Journal of Engineering Research (ISSN : 2319-6890)
Volume No.2, Issue No.2, pp : 113-118 01 April 2013

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(Magnetic parameter) on velocity profiles respectively. It is
found that the velocity decreases with increase of magnetic
parameter M. It is also found that the velocity decreases away
from the plate and becomes minimum and finally takes
asymptotic value. Finally here we also see that point of
separation takes place for different values of magnetic
parameter. Figure (4) present typical profiles for the velocity,
for various values of a heat source
( ) | respectively. As
shown, the velocity decreasing with increasing| . In the
event that the strength of the presence of a heat source
( ) |
effect causes a reduction in the thermal state of the fluid, thus
producing lower thermal boundary layers. The effect of the
Prandtl number on the velocity shown in figure (5). As the
Prandtl number increases, the velocity decreases. Further,
figure (6) it is observed that the fluid velocity decreases with
increasing value of R. This result may be explained by the fact
that an increase in the radiation parameter
4
h
kk
R
T o
-
| |
=
|
'
\ .
for
fixed k and
h
T
'
means an increase in the Rosseland mean
absorption coefficient k
-
. When radiation is present, the
momentum boundary layer was found to be thicken, which is
in agreement with the observation made earlier with regard to
the temperature variations of air. Figure (7) present typical
profiles for the temperature for various values of a heat source
( ) | respectively. As shown, the temperature decreasing with
increasing. Figure (8) shows that the temperature profiles for
different values of Prandtl number
( ) Pr . It is observed that an
increase in the Prandtl number results a decrease of the
thermal boundary layer thickness and in general lower average
temperature within the boundary layer. The reason is that
increasing values of Prandtl number equivalent to increase the
thermal conductivities and therefore heat is able to diffuse
away from the heated plate more rapidly. Hence in the case of
increasing Prandtl numbers, the boundary layer is thinner and
the heat transfer is reduced. Boundary layer suction is the
technique in which air pumps is used to extract the boundary
layer at the wing. Further, increment of suction parameter
decreases the fluid temperature. Figure (9) shows the
temperature profiles for different values of the Radiation
parameter R, it is noticed that an increase in the radiation
parameter results decrease in the temperature with in boundary
layer, as well as decreased the thickness of the temperature
boundary layers. Figure (10) presents the skin-friction
variation with M in the pure convection case for different
values of Gr at the moving plate. It is observed that the skin-
friction increases with increasing Gr .
5 CONCLUSIONS
The temperature of the fluid increases with increasing time
whereas it decreases due to an increase in the value of
radiation parameter. In the case of pure convection (i.e. in the
absence of radiation), the velocity of the fluid increases with
increasing Grashof number, but falls owing to an increase in
the Prandtl number. The velocity of the fluid increases with
increasing Grashof number and time but it decreases owing to
an increase in the value of the radiation parameter. The skin-
friction at the moving plate increases with increasing values of
Grashof number and time for air flows.
We may conclude therefore, that the interaction between
the radiation, buoyancy forces and the applied shear induced
by a uniform vertical motion of the hot wall can affect the
configuration of the flow field significantly.
REFERENCES
[1] Alam M S, Rahman M M and Sattar M A: MHD Free
convection heat and mass transfer flow past an inclined
surface with heat generation, Thamasat. Int. J. Sci. Tech.
11 (4) (2006), pp. 1 8.
[2] Attia H A: The Effect of Variable Properties on the
Unsteady Couette Flow with Heat Transfer Considering
the Hall Effect, Communications in Nonlinear Science
and Numerical Simulation,13, (2008), pp. 1596- 1604.
[3] Chamkha A J: Unsteady MHD convective heat and mass
transfer past a semi- infinite vertical permeable moving
plate with heat absorption, Int. J. Eng. Sci. 24 (2004), pp.
217 230.
[4] Choi C K, Chung T J and Kim M C: Buoyancy Effects in
Plane Couette Flow Heated Uniformly From Below with
Constant Heat Flux, International Journal of Heat and
Mass Transfer, Vol. 47, (2004), pp. 2629-2636.
[5] Chung T J, Computational Fluid Dynamics, Cambridge
University Press, 2002.
[6] Cookey C I, Ogulu A and Omubo-Pepple V M: 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, Int. J.
Heat Mass Transfer, 46, (2003), pp. 2305 2311.
[7] El-Hakiem M A: MHD oscillatory flow on free-convection
radiation through a porous medium with constant
velocity, J. Magn. And Magnetic Mater. 220 (2, 3),
(2000), pp. 271 276.
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Volume No.2, Issue No.2, pp : 113-118 01 April 2013

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[8] Grief R, Habib I S and Lin J C: Laminar convection of
radiating gas in a vertical channel, J. Fluid Mech. 46,
(1971), pp. 513 520.
[9] Hossain M A, Molla M M and Yaa L S: Natural
convection flow along a vertical wavy surface
temperature in the presence of heat generation/absorption,
Int. J. Thermal Science, 43, (2004), pp. 157 163.
[10] Jain N C and Gupta P: Three Dimensional Free
Convection Couette Flow with Transpiration Cooling,
Journal of Zhejiang University SCIENCE A, 3, (2006),
pp. 340-346.
[11] Jha B K: Natural Convection in Unsteady MHD Couette
Flow, Heat and Mass Transfer, 37, (2001), pp. 329-331.
[12]Kumar J, Lakshmana Rao C and Massoudi M: Couette
Flow of Granular Materials, International Journal of Non-
Linear Mechanics, 38, (2003), pp. 11-20.
[13] Makinde O D: Free-convection flow with thermal
radiation and mass transfer past a moving vertical porous
plate, Int. Comm., Heat Mass Transfer, 32, (2005), pp.
1411 1419.
[14] Narahari M: Effects of Thermal Radiation and Free
Convection Currents on the Unsteady Couette Flow
Between Two Vertical Parallel Plates with Constant Heat
Flux at one Boundary, Wseas transactions on heat and
mass transfer, 1 (5), (2010)
[15] Siegel R and Howell J R, Thermal Radiation Heat
Transfer, 4th Edition, Taylor & Francis, (2002).
[16] Soundalgekar V M: Hall Effects in MHD Couette Flow
with Heat Transfer, IEEE Transactions on Plasma
Science, Vol. PS-14, (5), (1986), pp. 579-583.
0 0.2 0.4 0.6 0.8 1
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
1.4
y
Figure 1: Velocity profiles for different values of Gr (Pure convection case)
u


Gr=5.0,10.0,15.0,20.0
R=100,|=1.0,K=1.0,M=1.0
Pr=7.0
Pr=0.71

0 0.2 0.4 0.6 0.8 1
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
y
Figure 2: Velocity profiles for different values of K
u


K=1.0,2.0,3.0,4.0
R=100.0,|=1.0,Gr=10.0,M=1.0,Pr=0.71

0 0.2 0.4 0.6 0.8 1
0
0.2
0.4
0.6
0.8
1
1.2
1.4
y
Figure 3: Velocity profiles for different values of M
u


R=5.0,|=2.0,K=1.0,Gr=15.0,Pr=0.71
M=1.0,2.0,3.0,4.0

0 0.2 0.4 0.6 0.8 1
0
0.2
0.4
0.6
0.8
1
1.2
1.4
y
Figure 4: Velocity profiles for different values of |
u


|=1.0,2.0,3.0,4.0
R=2.0,M=1.0,K=1.0,Gr=15.0,Pr=0.71

0 0.2 0.4 0.6 0.8 1
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
y
Figure 5: Velocity profiles for different values of Pr
u


Pr=1.0,2.0,3.0,4.0
R=100.0,|=1.0,K=1.0,M=1.0,Gr=10.0

0 0.2 0.4 0.6 0.8 1
0
0.2
0.4
0.6
0.8
1
1.2
1.4
y
Figure 6: Velocity profiles for different values of R
u


R=1.0,2.0,3.0,4.0
|=2.0,M=1.0,K=1.0,Gr=15.0,Pr=0.71

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Volume No.2, Issue No.2, pp : 113-118 01 April 2013

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0 0.2 0.4 0.6 0.8 1
0
0.2
0.4
0.6
0.8
1
y
Figure 7.: Temperature profiles for different values of Pr
u


Pr=0.71,R=1.0
|=1.0,2.0,3.0,4.0


0 0.2 0.4 0.6 0.8 1
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
y
Figure 8: Temperature profiles for different values of Pr
u


|=20.0,R=1.0
Pr=0.7,0.8,0.9,1.0




















0 0.2 0.4 0.6 0.8 1
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
y
Figure 9: Temeprature profiles for different values of R
u


Pr=0.71,|=30.0
R=1.0,2.0,3.0,4.0


1 1.5 2 2.5 3 3.5 4
-2.2
-2
-1.8
-1.6
-1.4
-1.2
-1
-0.8
-0.6
-0.4
M
Figure 10: Skin friction for different values of Gr versus M
t
0


R=1.0,Pr=0.71,|=1.0,K=1.0
Gr=1.0,2.0,3.0,4.0