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Viscous Dissipation and Radiation Effects on MHD Flow and Heat
Transfer over an Unsteady Stretching Surface in a Porous Medium
Gopi Chand and R. N. Jat
Department of Mathematics, University of Rajasthan, Jaipur – 302004, India.
gcyadav87@gmail.com & Khurkhuria_rnjat@yahoo.com
Abstract
This study investigates the problem of unsteady two-dimensional laminar flow of a viscous
incompressible electrically conducting fluid past over a horizontal stretching sheet in the
presence of uniform transverse magnetic field in the porous medium. Using the similarity
transformed the governing time dependent boundary layer equations are transformed to ordinary
differential equations, before being solved numerically by standard techniques. The solutions of
non-linear differential equations are obtained for unsteady parameter, Magnetic parameter,
Radiation parameter, Permeability parameter, Eckert number and Prandtl number on the velocity
and temperature profile and discussed graphically.
Keywords: Viscous dissipation; Thermal Radiation; stretching sheet; MHD; Unsteady
Boundary layer flow; Porous Medium.
Introduction
The study of boundary layer flow over a stretching sheet has gained considerable attention due to
its applications in industrial manufacturing process, such as paper production, the aerodynamics
extrusion of plastic sheet, glass blowing metal spinning, wire and fiber coating, chemical
processing equipment etc. The production of sheeting material arises in a number of industrial
manufacturing processes and includes both metal and polymer sheets. The quality of final
product depends on the rate of heat transfer at the stretching surface. Numerous researchers have
investigated the heat transfer over stretching surface by considering the effect of magnetic field.
In the pioneering work of Crane [1], the flow of Newtonian fluid over a linearly stretching
surface was studied. Later, Laha et al. [2] studied Heat transfer characteristics of the flow of an
incompressible viscous fluid over a stretching sheet. Andersson et al. [3] investigated using a
similarity transformation the flow of a thin liquid film of a power-law fluid by unsteady
stretching surface. Subsequently, the pioneering work of Crane [1] are extended by many authors
to explore various aspects of the flow and heat transfer occurring in an infinite domain of the
fluid surrounding the stretching surface [4-12].The study of Unsteady magnetohydrodynamic
boundary layer flow over a stretching surface with viscous dissipation and Joule heating have
been investigated by Jat and Chaudhary [13]. Shateyi and Motsa [14] extended the work of ElTEPE Volume 3, Issue 3 Aug. 2014, PP. 266-278 www.vkingpub.com/journal/tepe/ © American V-King Scientific
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Aziz [11] to include mass transfer. Ellahi and Riaz [15] investigated Analytical solution for
MHD flow in a third grade fluid with variable viscosity. Analytic solution for MHD flow in an
annulus was also studied by Ellahi et al. [16]. Singh et al. [17] studied the Effects of thermal
radiation and magnetic field on unsteady stretching permeable sheet in presence of free stream
velocity. Ellahi et al. [18] discussed Analysis of steady flows in viscous fluid with heat/mass
transfer and slip effects. Numerical analysis of steady non- Newtonian flows with heat transfer
analysis, MHD and nonlinear slip effects was investigated by Ellahi and Hameed [19]. Unsteady
boundary layer flow past a stretching plate and heat transfer with variable thermal conductivity
was studied by Misra et al. [20]. Recently, Verjavelu [21] investigated unsteady convective
boundary layer flow of a viscous fluid at a vertical surface with variable fluid properties. Most
Recently, Zeeshan and Ellahi [22] studied Series solutions for nonlinear partial differential
equations with slip boundary conditions for non-Newtonian MHD fluid in porous space.
However, radiation and magnetic effects on a stretching surface for time dependent moving
viscous fluid in porous medium has been not considered so far in best of our knowledge. The
study of MHD flow of an electrically conducting fluid caused by the deformation of the wall of a
vessel containing fluid is of considerable interest in modern metallurgical and metal working
process. The present paper is concern to investigate two-dimensional unsteady flow of a viscous
incompressible flow about stagnation point on permeable stretching sheet in the presence of time
dependent free stream velocity with dissipation and radiation effect.
Formulation of the Problem
Consider the unsteady laminar two-dimensional flow of a viscous incompressible electrically
conducting fluid past a semi-infinite porous stretching sheet. Fluid is considered in the influence
of transverse magnetic field of constant strength B0 normal to the sheet. The magnetic Reynolds
number is taken to be small and therefore the induced magnetic field is neglected. The Cartesian
coordinate system has its origin located at the leading edge of the sheet with positive x-axis
extending along the sheet in the direction of flow, while y-axis is along normal to the surface of
the sheet and is positive in the direction from the sheet to the fluid (fig.1). We assume that for
time t < 0 the fluid and heat flows are steady. The unsteady fluid and heat flows start at t = 0, the
sheet is being stretched with the velocity U w ( x, t ) along the x-axis, keeping the origin fixed and
temperature Tw ( x, t ) . The thermo-physical properties of the sheet and ambient fluid are assumed
constant. Under the Boussinesq and boundary layer approximations, the governing equations of
the flow are:
u v

0
x y
(1)
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u
u
u
 2u  B02

u v
 2 
u
u
t
x
y
y

K0
(2)
2
 T
 u 
T
T 
 2T qr
cp 
u
v
      B02u 2
  2 
x
y 
y
y
 t
 y 
(3)
Fig. 1: Physical modal and coordinate system.
Where u and v are the velocities in the x- and y- directions respectively, ρ is the density of the
fluid, μ is the dynamic viscosity,  

is the kinematic viscosity, c p is the specific heat at

constant pressure,  is thermal conductivity of the fluid under consideration, qr is the radiative
heat flux, T is the temperature.
The associated boundary conditions are:
u ( x, 0)  U w ( x, t ) ,
v( x, 0)  0 ,
u ( x,  )  0 ,
T ( x,  )  0
T ( x,0)  Tw ( x, t )
(4)
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Following Shayeti and Motsa [14], the stretching velocity U w ( x, t ) is assumed U w 
where both b and
 are positive constant. We have b as the initial stretching rate
bx
(1   t )
b
and it is
(1   t )
increasing with time.
We assume the surface temperature Tw ( x, t ) of the stretching sheet to vary with the distance x
along the sheet and time in the following form:
3

 bx 2 
2
Tw ( x, t )  T  T0 
(1


t
)




(5)
where T0 is a heating or cooling reference temperature. The radiative heat flux qr is simplified by
using Rosseland approximation (Rosseland [18]) as:
qr  
4 T 4
3 y
(6)
Where  is the Stefan- Boltzmann constant and  is the Roseland mean absorption coefficient.
This approximation is valid at points optically far from the boundary surface and it is good
for intensive absorption, which is for an optically thick boundary layer. It is assumed that the
temperature difference with in the flow such that the term T 4 may be expressed as a linear
function of temperature. Hence, expanding T 4 by Taylor series about T and neglecting higherorder terms gives:
T 4  4T3T  3T4
(7)
Using equation (6) and (7), equation (3) reduces to:
 T
T
T
cp 
u
v
x
y
 t
2
 
 u 
16 T3   2T
2 2



 2       B0 u
 
3  y
 
 y 
(8)
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Analysis
We introduce the stream function  such that
u

y
v
,

x
(9)
Then the equation of continuity (1) is identically satisfied.
In order to reduce equations (2) and (8) in to set of ordinary differential equations, we introduce
the similarity variable  and the dimensionless variables f and  as:

b
xf ( )
(1   t )
3
 bx 2 
2
T  T  T0 
(1


t
)
 ( )

2



where
y
b
 (1   t )
(10)
The velocity components are then derived from the stream function expression and obtained as:
u
bx
f ' ( )
(1   t )
,
v
b
f ( )
(1   t )
(11)
Then, the momentum and energy equations (2) and (8) are transformed to:
f '''  ff ''  ( f ' ) 2 
A '' 
1
f  A M   f '  0
2
K

A


(3R  4) ''  3R Pr  f  '  2 f '  (3   ' )  MEc( f ' ) 2  Ec( f '' ) 2   0
2


(12)
(13)
The corresponding boundary conditions are:
  0:
 :
f  0’
f ' 0
f ' 1,
 1
 0
(14)
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Where prime (´) denote the differentiation with respect to η and dimensionless parameters are:
M
 B02
(1   t )
c
(Magnetic parameter)
Ec 
u2
c p (Tw  T )
(Eckert number)
Pr 
c p

(Prandtl number)
K
K 0b
R

4 T3
A

(1   t )
(Permeability parameter)
(Radiation parameter)

(15)
(Unsteady parameter)
b
The physical quantities of interest are the skin-friction coefficient c f and heat transfer rates i.e.
the Nusselt number Nu are:
cf 
w

U w 2
 u 

 y  y 0

U w 2
2
 cf  2
2
 (1   t )
bx 2
f '' (0) 
2
f '' (0)
Re
(16)
and
 T 
x

 y  y o
Nu  
Tw  T
 Nu  
bx 2
 ' (0)   Re ' (0)
 (1   t )
(17)
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where
Re 
Uwx

(Reynolds number)
(18)
Result and Discussion
The boundary layer equations of momentum and energy are solved numerically using Runge Kutta forth order algorithm with a systematic guessing of f ′′(0) and θ′(0) by the shooting
technique until the boundary conditions at infinity are satisfied. The step size ∆𝜂 = 0.001 is used
while obtaining the numerical solution and accuracy up to the seventh decimal place i.e.1 ×
10−7 , which is very sufficient for convergence. In this method, we choose suitable finite values
of η→∞, say η∞ , which depend on the values of the parameter used. The computations were
done by a program which uses a symbolic and computational computer language Matlab. The
computation through employed numerical scheme has been carried out for various values of the
parameters such as Unsteadiness parameter A, Permeability parameter K, Magnetic parameter
M, Radiation parameter R, Prandtl number Pr and Eckert number Ec. The velocity profile f ' ( )
for different values of the unsteady parameter A is shown in fig.2. It is observed that the velocity
decreases with the increasing values of unsteady parameter A. It is interesting to note that the
thickness of boundary decreases with increasing values of A. This is due to the fluid flow caused
solely by the stretching sheet. The velocity profile f ' ( ) for different values of the magnetic
parameter M is shown in fig.3. It is observed that velocity decreases with the increasing values of
magnetic parameter M. As M increases, the Lorentz force, which opposes the flow, also
increases and leads to enhanced deceleration of the flow, and from fig.4, we observed that
velocity increases for increasing values of permeability parameter K. The temperature profiles
for different values of A, M, R, Pr and Ec are presented in figure 5 to figure 9. It is observed
from the figures that the boundary conditions are satisfied asymptotically in all the cases, which
supporting the accuracy of the numerical results obtained. From fig.6, we observed that
temperature of the fluid is decreases with the increasing values of unsteady parameter A.
Temperature at a point of surface decreases significantly with the increases of A i.e. rate of heat
transfer increases with increasing values of A. Physically, it means that the temperature gradient
at the surface increases as A increases, which imply the increases of heat transfer rate  ' ( ) at
the surface. From fig.5 and fig.8, we observed that temperature of the fluid is increases as
magnetic parameter M and Eckert number Ec increases. It is observed from the fig.7 and fig.9,
temperature of the fluid is decreases with the increasing values of Prandtl number Pr and
Radiation parameter R, this is because of the increase in Prandtl number Pr, indicates the
increase of the fluid heat capacity or the decrease of the thermal diffusivity hence cause a
diminution of the influence of the thermal expansion to the flow. This implies momentum
boundary layer is thicker than thermal boundary layer.
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Acknowledgements

The authors wish to express their sincere appreciation to the learned referee for careful
reading of the manuscript and valuable suggestions.

This work has been carried out with the financial support of CSIR in the form of J.R.F
awarded to one of the author (Gopi Chand).
References
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unsteady stretching surface. International Journal of Heat and Mass Transfer. 43(1), 69–
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Journal of Heat and Mass Transfer. 51, 1024-1033, (2008).
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11. El-Aziz, M. A.: Radiation effect on the flow and heat transfer over an unsteady stretching
sheet. International Communications in Heat and Mass Transfer. 36(5), 521–524, (2009).
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over a stretching surface with viscous dissipation and Joule heating. IL NUOVO
CIMENTO. 124(3), (2009).
14. Shateyi, S., Motsa, S.S.: Thermal radiation effects on heat and mass transfer over an
unsteady stretching surface. Mathematical Problem in Engineering. (2009).
15. Ellahi, R. and Riaz, A.: Analytical solution for MHD flow in a third grade fluid with
variable viscosity. Mathematical and Computer Modelling. 52, 1783-1793, (2010).
16. Ellahi, R., Hayat, T., Mahomed, F. M., and Zeeshan, A.: Analytic solution for MHD flow
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23. S. Rosseland, Theoretical Astrophysics. New York: Oxford University, (1936).
1
0.9
0.8
0.7
f ' ()
0.6
0.5
K = 0.1, 0.2, 0.3, 0.5
0.4
0.3
0.2
0.1
0
0
0.5
1
1.5
2
2.5

Fig. 2: Velocity profile against  for various values of Permeability parameter K for
A = 1.0 and M = 1.0.
1
0.9
0.8
0.7
f ' ()
0.6
0.5
0.4
M = 0, 0.5, 1.0, 2.0
0.3
0.2
0.1
0
0
0.5
1
1.5
2
2.5

Fig. 3: Velocity profile against  for various values of Magnetic parameter M for
A= 1.0 and K = 0.5.
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1
0.9
0.8
0.7
f ' ()
0.6
0.5
0.4
A = 0, 1.0, 2.0
0.3
0.2
0.1
0
0
0.5
1
1.5
2
2.5

Fig. 4: Velocity profile against  for various values of Unsteady parameter A for
M = 1.0 and K = 0.5.
1
0.9
0.8
0.7
()
0.6
0.5
M = 0, 0.5, 1.0, 2.0
0.4
0.3
0.2
0.1
0
0
0.5
1
1.5
2
2.5
3
3.5
4

Fig. 5: Temperature distribution against  for various values of Magnetic parameter M
for Pr = 7.0, Ec = 0.5, R = 0.5, K = 0.5, A = 1.0.
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1
0.9
0.8
0.7
()
0.6
0.5
A = 0, 0.5, 1.0, 2.0
0.4
0.3
0.2
0.1
0
0
0.5
1
1.5
2
2.5
3
3.5
4

Fig. 6: Temperature distribution against  for various values of Unsteady parameter A
for Pr = 7.0, Ec = 0.5, R = 0.5, K = 0.5, M = 1.0.
1
0.9
0.8
0.7
()
0.6
0.5
R = 0.5, 1.0, 1.5, 2.0
0.4
0.3
0.2
0.1
0
0
0.5
1
1.5
2
2.5
3
3.5
4

Fig. 7: Temperature distribution against  for various values of Radiation parameter R
for Pr = 7.0, Ec = 0.5, K = 0.5, M = 1.0, A = 1.0.
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1.4
1.2
1
()
0.8
Ec = 0, 0.5, 1.0, 2.0
0.6
0.4
0.2
0
0
0.5
1
1.5
2
2.5
3
3.5
4

Fig. 8: Temperature distribution against  for various values Eckert number Ec
for Pr = 7.0, R = 0.5, K = 0.5, M = 1.0, A = 1.0.
1
0.9
0.8
0.7
()
0.6
Pr = 1.0, 2.0, 5.0, 7.0
0.5
0.4
0.3
0.2
0.1
0
0
0.5
1
1.5
2
2.5
3
3.5
4

Fig. 9: Temperature distribution against  for various values Prandtl number Pr
for R = 0.5, Ec = 0.5, K = 0.5, M = 1.0, A = 1.0.
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