World Journal of Mechanics, 2013, 3, 203-214
doi:10.4236/wjm.2013.34020 Published Online July 2013 (http://www.scirp.org/journal/wjm)
Thermal Radiation, Heat Source/Sink and Work Done by
Deformation Impacts on MHD Viscoelastic Fluid over a
Nonlinear Stretching Sheet
F. M. Hady1, R. A. Mohamed2, Hillal M. ElShehabey2,3*
1Mathematics Department, Faculty of Science, Assiut University, Assiut, Egypt
2Mathematics Department, Faculty of Science, South Valley University, Qena, Egypt
3Institute of Mathematics and Scientific Computing, University of Graz, Graz, Austria
Received December 16, 2012; revised February 16, 2013; accepted February 23, 2013
Copyright © 2013 F. M. Hady et al. This is an open access article distributed under the Creative Commons Attribution License,
which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
ABSTRACT
This work is focused on the effects of heat source/sink, viscous dissipation, radiation and work done by deformation on
flow and heat transfer of a viscoelastic fluid over a nonlinear stretching sheet. The similarity transformations have been
used to convert the governing partial differential equations into a set of nonlinear ordinary differential equations. These
equations are then solved numerically using a very efficient implicit finite difference method. Favorable comparison
with previously published work is performed and it is found to be in excellent agreement. The results of this parametric
study are shown in several plots and tables and the physical aspects of the problem are highlighted and discussed.
Keywords: Flow and Heat Transfer; Second Grade Fluid; Nonlinear Stretching Sheet; Heat Source; Radiation
1. Introduction
The study of fluids is different to that of solids as there
are differences in physical structure of fluids and solids.
The nature of fluids and the way of study are two aspects
of fluid mechanics which make it different to solid me-
chanics. Furthermore, the fluids are categorized to New-
tonian and non-Newtonian fluids. The fluids of low mo-
lecular weight fall into the Newtonian class and are com-
pletely characterized by the Navier-Stokes theory. There
is a large variety of materials such as geological materi-
als, liquid foams, polymeric liquids and food products etc.
which are capable of flowing but which exhibit flow
characteristic that cannot be adequately described by the
Navier-Stokes theory. This inadequacy of the Navier-
Stokes theory has led to the development of several theo-
ries of non-Newtonian fluids.
Unlike Navier-Stokes fluids, there is not a single mo-
del which can completely describe all the properties of
the non-Newtonian fluids. They cannot be described in a
single model as for Newtonian fluids and there has been
much confusion over the classification of non-Newtonian
fluids. There are many models describing the properties
but not all of non-Newtonian fluids. These models or
constitutive equations, however, cannot describe all the
behaviors of non-Newtonian fluids, e.g., the normal stress
relaxation, the elastic effects, and the memory effects.
The constitutive equations describing the behaviors of
non-Newtonian fluids are more complicated and non lin-
ear than those of Newtonian fluids.
As non-Newtonian fluid model Rivlin-Ericksen fluids
gained much acceptance from both theorists and experi-
menters. The special cases of the model, which is the
fluid of second grade, are extensively used and a lot of
works have been done on the subject. These investiga-
tions have been for non-Newtonian fluids of the differen-
tial type [1]. In the case of fluids of differential type, the
equations of motion are an order higher than the Navier-
Stokes equations, and thus the adherence boundary con-
dition is insufficient to determine the solution completely
[2-4] for a detailed discussion of the relevant issues. The
same is also true for the approximate boundary layer ap-
proximations of the equations of motion. In the absence
of a clear means obtaining additional boundary condi-
tions, Beard and Walters [5], in their study of an incom-
pressible fluid of second grade, suggested a method for
overcoming this difficulty. They suggested a perturbation
approach in which the velocity and the pressure fields
were expanded in a series in terms of a small parame-
*Corresponding author.

F. M. HADY ET AL. 204
ter ε. Danberg and Fansler [6] studied the solution for the
boundary layer flow past a wall that is stretched with a
speed proportional to the distance along the wall. Ra-
jagopal et al. [7] independently examined the same flow
as in [5] and obtained similarity solutions of the bound-
ary layer equations numerically for the case of small vis-
coelastic parameter λ. It is shown that skin-friction de-
creases with increase in λ. Dandapat and Gupta [8] ex-
amined the same problem with heat transfer, where Hady
and Gorla [9] studied the effect of uniform suction or
injection on flow and heat transfer from a continuous
surface in a parallel free stream of viscoelastic second-
order fluid. The effect of radiation on viscoelastic boun-
dary-layer flow and heat transfer problems can be quite
significant at high operating temperature. Very recently,
researches in these fields have been conducted by many
investigators [10-14].
On the other hand, another physical phenomenon is the
case in which the sheet stretched in a nonlinear fashion.
On this domain, Mahdy and Elshehabey [15] studied the
flow and heat transfer in a viscous fluid over a nonlinear
stretching sheet utilizing nanofluid where, effects of vis-
cous dissipation and radiation on the thermal boundary
layer over a nonlinearly stretching sheet were studied by
Cortell [16]. Vajravelu [17] studied viscous flow over a
nonlinearly stretching sheet, where viscous flow and heat
transfer over a nonlinearly stretching sheet were obtained
by Cortell [18] then, series solution of flow over non-
linearly stretching sheet with chemical reaction and mag-
netic field was investigated by employing the Adomian
decomposition method by Kechil and Hashim [19] where,
Ziabakhsh et al. [20] used homotopy analysis method to
present flow and diffusion of chemically reactive species
over a nonlinearly stretching sheet immersed in a porous
medium. Muhaimin et al. [21] studied the effect of che-
mical reaction, heat and mass transfer on nonlinear boun-
dary layer past a porous shrinking sheet in the presence
of suction and, Robert [22] discussed high-order nonlin-
ear boundary value problems admitting multiple exact
solutions with application to the fluid flow over a sheet.
Cortell [23] studied heat and fluid flow due to non-line-
arly stretching surfaces where, existence and uniqueness
results for a nonlinear differential equation arising in vis-
cous flow over a nonlinearly stretching sheet were ob-
tained by Robert et al. [24]. Finally, Vajravelu et al. [25]
studied the diffusion of a chemically reactive species of a
power-law fluid past a stretching surface.
In this paper, as motivated by the previous studies and
the study of Cortell [26] which investigated the effects of
heat source/sink, radiation and work done by deforma-
tion on flow and heat transfer of a viscoelastic fluid over
a stretching sheet, we consider viscoelastic fluid with an-
other physical phenomenon in which the sheet stretched
in a nonlinear fashion. Also, the effects of work due to
deformation on viscoelastic flows and heat transfer in the
presence of radiation, viscous dissipation and heat source/
sink have been studied.
2. Problem Formulation
Consider a steady two-dimensional flow of an incom-
pressible second grade fluid through a porous medium
over a wall coinciding with the plane , the flow
being confined to . Two equal and opposite forces
are applied along the x-axis so that the wall is stretched
keeping the origin fixed. Thus, the basic boundary layer
equations, governing the flow and heat transfer in pres-
ence of radiation, with a temperature-dependent heat
source/sink in the flow region, viscous dissipation, and
taking into account the work due to deformation, are
given in usual notation by
0y
0y
0,
uv
xy



 (1)
2
2
00
2
223
1
23
,
B uu u
uv u
xy y
uuu u
uv
xyxy yy




 
 
 



 


 
 





(2)
2
2
2
1
( 1
.
r
pp
p
TT
uv
xy
qQTT Tu
cy cyc y
uuu
uv
cyy xy




)
p




 

 





 

 
 


(3)
where, the power-law heat flux on the wall is considered
in the form


0
,,
at 0,
n
w
s
w
w
ux bxvv
T
qk Dxy
y




 



(4)
0,0,as ,
u
uTT
y



 

(5)
where


,
x
y denotes the Cartesian coordinates along
the sheet and normal to it, u and v are the velocity com-
ponents of the fluid in the
x
 and directions, re-
spectively, b and n are parameters related to the surface
stretching speed, v is the kinematic viscosity,
y

is the
thermal diffusivity,
p
c is the specific heat at constant
pressure, r is the radiative heat flux and Q the volu-
metric rate of heat generation/absorption. The radiative
heat flux term by using the Rosseland approximation is
given by [27]
q
4
4,
3
r
T
qy
k




  (6)

F. M. HADY ET AL. 205


sid
where and are the Stefan
and the mean abs rption coeffici
ation (3) reduces to
k
o
-Boltzmann constant
ent, respectively. We
can coner that the temperature differences within the
flux are sufficiently small such that the term 4
T may be
expressed as a linear function of temperature by expand-
ing 4
T in a Taylor series about T and neglecting
higher-order terms we get [28]

443 34
.443 .TTTT TTT
 
 (7)
using Equations (6) and (7), Equ
T


2
32
2
1
16
3
,
p
p
pp
TT
uv
xy
TTu
cy
ck y
QT Tuuu
uv
ccyyxy











 
 







 
 
 

 
 



(8)
where it can be seen that the effect of radiation is to en-
hance the thermal diffusivity.
Defining the following dimensionless function u, v and
g, which related to the similarity variable

as


1
2
,
2n
s
TT
gDv
xx




 (9)
1sb n

1
2
11,
21
n
bn n
vxf
n




  



f
(10)
where,

is the free stream function that
Equation (1) with
satisfies
,,uv
yx


 (11)

Then we have the transformed mom
equations together with the boundary c
Equ
entum and energy
onditions given by
ations (5) and (11) in the form


2
22
11
nM
ffff f
nn
 


2
31 1
20
22
iv
nn
ff
f ff




   


(12)


512
2
2
33
(2 1)
34 341
3
2
134
3
34
311 0
22
RR
RR
R
R
ns
R
c
R
NnN
g
gfsn gf
NNn
NNg
nN
NEx
N
nn
ffff ff







 
 













   
 



(13)
If 51 ,
2
n
s

 we find from (13) that


2
33
4
3434 1
3
2
13 4
331 1
34 22
0
RR
RR
R
R
R
c
R
NN
n
ggf gf
NNn
NNg
nN
Nnn
Ef fffff
N





  








 


  
  




(14)
e transformThed boundary conditions are

,1,(0)1at0
0,0,0as
fRf g
ffg


  



 

 (15)
Here the prime denotes differentiation with respect to
the independent similarity variable Moreover,
.

is the viscoelastic parameter,
02
0
1n
B
Mbx




1
1
n
bx 





is the magnetic parameter,

01
2
1n
Rv nbx


 i
suction parameter,
s the




is the Prandtl number,

21
2
c
p
Dc v
bn
kb
E  is the Eckert number,
1n
p
Q
N

cbx


 is the heat source/sink parameter,
3
4
R
kk
NT




 is the radiation parameter and
0
3R
R
N
01k

34
kN  (with thermal radiation);

(with-
n).
The shear stress at the stretched surface is defined as
out thermal radiatio
w
w
u


y




, (16)
using Equations ( 9) and (10) we have,
 
31
2
10
n
bn
bxf




2
wv
 (17)
3. Results a
s-
fer-
rous stretching sheet, in the pr
is examined in this paper. Str
dary, viscous dissipation, temperature dependent heat
source/sink and thermal radiation are taken into consid-
nd Discussion
A boundary layer problem for momentum and heat tran
in a viscoelastic fluid flow over a non-isothermal po
esence of thermal radiation,
etching of the porous boun-

F. M. HADY ET AL. 206
dary layer partial
differential equations, which are highly non-linear, have
been converted into a set of nonlinear ordi
tial equations by applying suitable similarity transforma-
ns are obtained with a
ements the three-stage
mation is taken in account and this is also true for the
second case, where ,,Rn

and

have an opposite
behavior.
The plots in Fures 1(a) and (b) show the temera-
ture distribution
eration in this study. The basic boun
nary differen- ig p


g

different values of the heat
source/sink parameith two cases of the radiation; in
the case of existence of the thermal radiation and absence
of the thermal radiatio and whout work done by
deformatio
for
r w
n with
viou
pos
tions and their numerical solutio
finite difference code which implte
Lobatto IIIa formula is used to solve that system [29-31].
In order to verify the accuracy of the present numerical
method, the results are compared with those reported
earlier by [26] for the case of linear stretching sheet. The
results of these comparisons are shown in Table 1. It can
be seen from this table that excellent agreement between
it
n. It is obs that, the effect of increasing
th
op
e strength of the heat sink is to increase the temperature
profile, and the ite behavior is seen for a heat
source. In contrast of thermal radiation, the existence the
work done by deformation is to decrease the temperature
profile.
From Figures 2(a) and (b), we can see that the effect
of increasing values of Prandtl number

is to decrease
temperature at a point in the flow field, as there would be
a thinning of the thermal boundary layer as a result of
reduced thermal conductivity. On the other hand Figures
3(a) and
the results exists. This lends confidence to the numerical
results.
The effects of viscous dissipation, work due to defor-
mation, internal heat generation/absorption and thermal
radiation are considered in the energy equation and the
variations of dimensionless surface temperature, dimen-
sionless velocity profiles as well as the heat transfer cha-
racteristics with various values of non-dimensional vis-
coelastic parameter

, nonlinear stretching parameter
n, heat source/sink parameter N

, magnetic parameter
M, suction parameter R, Prandtl number

, Eckert
number c E and radiation parameter
R
N shown in
Figures 1-10.
Also, the values of wall temperature g(0) for various
va
(b), demonse effect non linear pa-
ra
trate thof the
meter n and it is evident that, the temperature profile
decreases with increasing the values of the non linear
stretching parameter. Moreover, we can see the effect of
increasing the Eckert number c E from Figu res 4(a) and
(b) for the same cases disused in Figures 1(a) and (b),
which is to increase the temperature distribution.
Figures 5(a) and (b) depict the effect of the magnetic
field
M
, by analyzing these graphs, we see that the ef-
fect of increasing values of
M
is to increase the tem-
perature distribution in the bodary layer. This is be-
cause of the fact that the introduction of transverse mag-
netic field to an electrically conducting fluid give
lues of physical parameters are shown in Tables 2 and
3 with and without taking the work done by deformation
at the energy equation, respectively. There are many re-
sults which can be obtained from those tables. For more
details, Increasing the heat source parameter N

or, the
Eckert number c E, or the magnetic parameter
M
, or
the radiation paramete
un
s rise to
a
en
f σ
r
R
N leads to an increasing in the
wall temperature

0g when twork done by defor-
resistive force, known as Lorentz force. This force has
a tendcy to slow down the motion of the fluid in the
, NR and Ec wi th n = 0.0 (Non lin ear stretching sheet).
g(0)
he
Table 1. Comparison results of g(0) for various values o
λ E
c NR σ N β Cortell [26] Present Result
0.2 0.02 1.0 3.0 0.05 0.671732 0.6548280
0.5 0.651127 0.6421121
0.2
0.4 0.757167 0.7573403
1 1
7
3. −05 0.
−05
0.5
0.25 0.716127 0.7079965
0.0 0.646621 0.6494326
0.02 5.0 0.462286 0.4647484
8.0 0.441677 0.4443301
.0.01.291859 1.2864327
.00.406628 0.4087103
0 .2597791 0.5987765
.10.614045 0.6153823
0.00.641036 0.6440087
10.693218 0.6790583

F. M. HADY ET AL. 207
Table 2. Wall temperature (0) with work dormation.
g(0)
gne by defo
Nσ n Ec M R λ iation
β With Radiation Without Rad
−08.0 1/3 0.02 1.0 1 0.15 4259812 5 .2 0.0.0.264870
0. 0.4802939
0. 5710018 1
−04.0 6397695 0
7.0 0.2850470
10.0 0.3750452 0.2345641
1/3 0.
0.
0.
0. 0.
0.
1.
1.0.
0.
1.
1.
0.0.
0.
0.
0.
0
2
0.2925684
0.3321800.
.2 0.0.390038
0.4600843
8.0 1.0 0.3496030
0.2987571
0.2190434
0.1891062 3.0
5.0 0.2848280 0.1811730
0 0.4116075 0.2478815
05 0.4475416 0.2903539
1 0.4834757 0.3328264
025 0.4139565 0.2572655
8 0.4214058 0.2619547
5 0.4364144 0.2716140
0 05 0.4438452 0.2808115
5 0.3132555 0.1747897
0 0.2270719 0.1181463
5 0.1753241 0.0886570
1 050.4271187 0.2653735
1 0.4265433 0.2651201
3 0.4243760 0.2641472
7 0.4206671 0.2624275
Table 3. Wall tempre gith wor by de
g(0)
eratu (0) wk doneformation.
Nβ W Without Radiation σ n Ec M R λ ith Radiation
−2 81/3 0. 1.0. 0..0 020 10.15 0.5083353 0.2642551
0
0
−0.2 4.0 0.3894706
7.0 0.2844400
1 0.4613835 0.2339356
8.0 1.0
5.
1/3 0.
0.
0.
0. 0.
0.
1.
1.0.
0.
1.
1.
0.0.
0.
0.
0.
.0 0.5798098 0.2918004
.2 0.7017604 0.3311588
0.7102198
0.5400828
0.0
0.4018430 0.2172851
3.0 0.3274262 0.1856562
0 0.3050635 0.1767095
0 0.4116075 0.2478815
05 0.6534269 0.2888155
1 0.8952464 0.3297495
025 0.4755726 0.2568651
8 0.4957535 0.2614253
5 0.5374969 0.2707835
0 05 0.5272657 0.2803169
5 0.3862716 0.1732892
0 0.2889979 0.1157326
5 0.2281186 0.0854866
1 050.5126295 0.2651608
1 0.5023476 0.2629805
3 0.4888003 0.2600536
7 0.5083353 0.2642551

F. M. HADY ET AL. 208
00.5 11.5 22.5
0
0
0.8
.2
0.4
0.6

( a )
g (  )
00.5 11.5 22.5
0
0.2
0.4
0.6
0.8

( b )
g (  )
- - - - - Withermal radi0,
- - - - - With thermal radiation N
R
= 1.0,
.... Without thermal
........... Without thermal radiation.
 = 0.
M = 1.
 = 8.
 = 0.15, R = 0.1,
M = 1.0, n = 1/3,
 = 8.0, E
c
= 0.02.
thation N
R
= 1.
....... radiation.
15, R = 0.1,
0, n = 1/3,
0, E
c
= 0.02.
N = 0.2, 0. -0.2.

0,
N

= 0.2, 0.0, -0.2.
Figure 1. Temperature profiles for several values of Nβ (a) with and (b) without work done by deformation.
00.5 11.5 22.5
0
0.1
0.2
0.3
0.4
0.5

( a )
g (  )
00.5 11.5 22.5
0
0.1
0.2
0.3
0.4
0.5
0.6

( b )
g (  )
- - - - - With thermal radiation N
R
= 1.0,
- - - - - With thermal radiation N
R
= 1.0,
........... Without thermal radiation.
 = 0.15, R = 0.1,
 = 0.15, R = 0.1,
M = 1.0, n = 1/3,
N

= -0.2, E
c
= 0.02.
........... Without thermal radiation.
 = 7.0, 8.0,10.0.
M = 1.0, n = 1/3,
N

= -0.2, E
c
= 0.02.
 = 7.0, 8.0,10.0.
Figure 2. Temperature profiles for several values of σ (a) with and (b) without work done by deformation.
boundary layer and to increase the temperature distribu-
tion, where the effect of the suction parameter is plotted
in Figures 6(a) and (b) and the effect of the viscoelastic
parameter

is clearly shown in Figures 7(a) and (b)
with all cases.
Velocity profiles with for various values
of
0.5,3.0n
M
is shown in Figure 8(a) where the same values is
pln Figure 8(b) but fo Also, the same
val of Figures 8(a) and (b) otted for different
vaes of the viscoelastic paraFigures 9(a) and
(b) We can see from those figufaster motion is
red when the viscincreases
otted i
ues
lu
.
conside
r1.0.R
are Pl
meter in
res that, a
oelastic parameter

F. M. HADY ET AL. 209
00.5 11.5 22.5
0
0.1
0.2
0.3
0.4
0.5

( a )
g (  )
00.5 11.5 22.5
0
0.1
0.2
0.3
0.4
0.5
0.6

g (  )
( b )
- - -
....
- -t
- - - - - With thermal radiation N
R
= 1.0,
....
........... Without thermal radiation.
With thermal radia ion N
R
= 1.0,
... Without thermal radiation.
 = 0.15, R = 0.1,
M = 1.0,  = 8.0,
N

= -0.2, E
c
= 0.02.
n = 1/3, 1.0, 3.0.
n = 1/3, 1.0, 3.0. = 0.15, R = 0.1,
M = 1.0,  = 8.0,
N

= -0.2, E
c
= 0.02.
Figure 3. Temperature profiles for several values of n (a) with and (b) without work done by deformation.
00.5 11.5 22.5
0
0.1
0.2
0.3
0.4
0.5

( a )
g (  )
00.5 11.5 22.5
0
0.2
0.4
0.6
0.8
1

( b )
g (  )
- - - - - With thermal radiation N
R
= 1.0,
- - - - - With thermal radiation N
R
= 1.0,
........... Without thermal radiation.
........... Without thermal radiation.
 = 0.15, R = 0.1,
M = 1.0,  = 8.0,
N

= -0.2, n = 1/3.
 = 0.15, R = 0.1,
M = 1.0,  = 8.0,
N

= -0.2, n = 1/3.
E
c
= 0.1, 0.05, 0.0.
E
c
= 0.1, 0.05, 0.0.
Figure 4. Temperature profiles for several values of Ec (a) with and (b) without work done by deformation.
whereas it is slower when the suction parameter and
magnetic parameter increase. Finally, Figure 10 shows
variation of skin fraction coefficient against R for differ-
ent values of ,,,Mn

from which we can say that,
with an increasing in the nonlinear stretching parameter
or the viscoelastic parameter
n

t
repr
ends to a decreasing
the wall shear stress whichesented in terms of in


0f


profile
as defined by Equatio7) i.e. the velocity
increases, but the opposite effect is seen for the
magnetic parameter M.
n (1

F. M. HADY ET AL. 210
00.5 11.5 2
0
0.1
0.2
0.3
0.4
0.5

( a )
g
(

)
00.5 11.5 2
0
0.1
0.2
0.3
0.4
0.5
0.6

( b )
g
(

)
- - - - - With thermal radiation N
R
= 1.0,
- - - - - With thermal radiation N
R
= 1.0,
on.
........... Without thermal radiation.
 = 0.15, R = 0.1,
n = 1/3,  = 8.0,
N

= -0.2, E
c
= 0.02.
 = 0.15, R = 0.1,
n = 1/3,  = 8.0,
N

= -0.2, E
c
= 0.02.
........... Without thermal radiati
M = 1.5, 0.5.
M = 1.5, 0.5.
M ( Figure 5. Temperature profiles for several values of a) with and (b) without work done by deformation.
00.5 11.5
0
0.1
0.2
0.3
0.4

( a )
g (  )
00.5 11.5
0
0.1
0.2
0.3
0.4
0.5
0.6

( b )
g (  )
- - - - - With thermal radiation N
R
= 1.0,
........... Without thermal radiation.
- - - - - With thermal radiation N
R
= 1.0,
........... Without thermal radiation.
R = 0.05, 0.5, 1.5.
R = 0.05, 0.5, 1.5.
 = 0.15, N

= -0.2,
M = 1.0, n = 1/3,
 = 8.0, E
c
= 0.02.
 = 0.15, N

= -0.2,
M = 1.0, n = 1/3,
 = 8.0, E
c
= 0.02.
Figure 6. Temperature profiles for several values of R (a) with and (b) without work done by deformation.

F. M. HADY ET AL. 211
00.5 11.5
0
0.1
0.2
0.3
0.4
0.5

( a )
g (  )
00.5 11.5
0
0.1
0.2
0.3
0.4
0.5
0.6

( b )
g (  )
- - - - - With thermal radiation N
R
= 1.0,
- - - - - With thermal radiation N
R
= 1.0,
........... Without thermal radiation.
........... Without thermal radiation.
 = 0.05, 0.7.
R = 0.1, N

= -0.2,
M = 1.0, n = 1/3,
 = 8.0, E
c
= 0.02.
R = 0.1, N

= -0.2,
M = 1.0, n = 1/3,
 = 8.0, E
c
= 0.02.
 = 0.05, 0.7.
Figure 7. Temperature profiles for several values of λ (a) with and (b) without work done by deformation.
01 2 3 4 5
0
0.2
0.4
0.6
0.8
1

( a )
f ' (  )
01 2 3 4 5
0
0.2
0.4
0.6
0.8
1

( b )
f ' (  )
........... n = 3.0.
........... n = 3.0.
R = 0.05,  = 0.15.
M = 0.8,
R = 1.0,  = 0.15.
1.5.
M = 0.8, 1.5.
- - - - - n = 0.5,
- - - - - n = 0.5,
Figure 8. Velocity profiles with n = 0.5, 3.0 for various values of M and R.

F. M. HADY ET AL. 212
01 2 3 4 5
0
0.2
0.4
0.6
0.8
1

( a )
f ' (  )
01 2 3 4 5
0
0.2
0.4
0.6
0.8
1

( b )
f ' (  )
- - - - - n = 0.5,
- - - - - n = 0.5,
........... n = 3.0.
R = 0.05, M = 1.0.
R = 1.0, M = 1.0.
........... n = 3.0.
 = 0.3, 0.1.
 = 0.3, 0.1.
Figure 9. Velocity profiles with n = 0.5, 3.0 for various values of λ and R.
00.5 11.5
1
1.5
2
2.5
R
- f '' ( 0 )
 = 0.01
 = 0.3
M = 0.5
.............. M = 1.0, n = 1/3,
 = 0.2, M = 1.0 .
- - - - - - -  = 0.2, n = 1/3,
M = 1.0
n = 3.0
n = 0.2
Figure 10. Variation of skin fraction coefficient against R for different values of M, n, and λ.
4. Conclusions
The effects of heat source/sink, radiation and work done
by deformation on flow and heat transfer of a viscoelastic
fluid over a nonlinear stretching sheet have been investi-
gated. Numerical solutions for momentum and heat
transfer are obtained. In the light of the numerical results
the following conclusions may be drawn:
 A faster motion is considered when the viscoelastic
parameter or the nonlinear stretching parameter in-
creases whereas it is slower when the suction pa-
e magnetic force increase.
 The effect of increasing values of magnetic parameter
is to increase the temperature distribution in the boun-
dary layer and so does the Eckert number.
 The influence of the work due to deformation has sig-
nificance effect of the temperature distribution.
 The presence of the thermal radiation term in the en-
ergy equation yields an augment in the fluid’s tem-
perature.
 The internal heat generation/absorption enhances or
damps the heat transport. Finally, increasing suction
rameter and th

F. M. HADY ET AL. 213
parameter or viscoela
ber is to decrease the temperature distribution in the
flow region.
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