Applied Mathematics, 2010, 1, 446-455
doi:10.4236/am.2010.16059 Published Online December 2010 (http://www.SciRP.org/journal/am)
Heat and Mass Transfer in MHD Visco-Elastic Fluid Flow
through a Porous Medium over a Stretching Sheet with
Chemical Reaction
Saleh M. Alharbi1, Mohamed A. A. Bazid2, Mahmoud S. El Gendy3*
1Department of Mat hemat i cs, Faculty of Science, Umm AL- Qur a U ni versi ty , Makka h, Sa udi Ara bia
2Department of Mathematics, Faculty of Science, Alkhrj University, Wadi Adawasir, Saudi Arabia
3Department of Mathematics, The University College in Makkah, Makkah, Saudi Arabia
Received June 14, 2010; revised August 13, 2010; accepted August 16, 2010
Abstract
This paper presents the study of convective heat and mass transfer characteristics of an incompressible MHD vis-
co-elastic fluid flow immersed in a porous medium over a stretching sheet with chemical reaction and thermal
stratification effects. The resultant governing boundary layer equations are highly non-linear and coupled form of
partial differential equations, and they have been solved by using fourth order Runge-Kutta integration scheme
with Newton Raphson shooting method. Numerical computations are carried out for the non-dimensional physical
parameters. Here a numerical has been carried out to study the effect of different physical parameters such as vis-
co-elasticity, permeability of the porous medium, magnetic field, Grashof number, Schmidt number, heat source
parameter and chemical reaction parameter on the flow, heat and mass transfer characteristics.
Keywords: Heat and Mass Transfer, Incompressible MHD, Visco-Elastic, Porous Medium, Chemical
Reaction
1. Introduction
In recent years, a great deal of interest has been generat-
ed in the area of heat and mass transfer of the boundary
layer flow over a stretching sheet, in view of its numer-
ous and wide-ranging applications in various fields like
polymer processing industry in particular in manufactur-
ing process of artificial film and artificial fibers and in
some applications of dilute polymer solution. Sakiadis
[1,2] was the first study of boundary layer problem as-
suming velocity of a boundary sheet as constant. This
work is followed by the pioneering work of Tsou et al. [3]
studied the flow and heat transfer developed by conti-
nuously moving surface both analytically and experi-
mentally, in which the flow is caused by an elastic sheet
moving in its own plane with a velocity varying linearly
with the distance from a fixed point studied by Crane [4].
Chakrabarti and Gupta [5] studied the temperature dis-
tribution in this MHD boundary layer flow over a stret-
ching sheet in the presence of suction. There are several
extensions to this problem, which include consideration
of more general stretching velocity and the study of heat
transfer [6-14].
In view of increasing importance of non-Newtonian
flows, a great deal of work has been carried out to find
the similarity solution of viscoelastic fluid flow over im-
pervious stretching boundary. Rajagopal et al. [15] ex-
amined for a special class of visco-elastic fluids known
as second order fluids. Siddappa et al. [16] studied the
flow of visoelastic fluids of the type Walter’s liquid B
past a stretching sheet. Abel and Veena [17] studied the
viscoelasticity on the flow and heat transfer in a porous
medium over a stretching sheet. All these studies deals
with the studies concerning non-Newtonian flows and
heat transfer in the absence of magnetic fields, but pre-
sent years we find several industrial applications such as
polymer technology and metallurgy [18], where the mag-
netic field is applied in the visco-elastic fluid flow. Sar-
pakaya [19] was mostly first researcher to investigate
MHD flows of non-Newtonian fluids, Andersson [20]
investigated the flow problem of electrically conducting
viscoelastic fluid past a flat and impermeable elastic
*Current Address: Department of Mathematics, The University College
in Makkah, Makkah, Saudi Arabia.

S. M. ALHARBI ET AL.
447
sheet and later his work is extended by many authors
[21-25].
Chemical reactions usually accompany a large amount
of exothermic and endothermic reactions. These charac-
teristics can be easily seen in a lot of industrial processes.
Recently, it has been realized that it is not always per-
missible to neglect the convection effects in porous con-
structed chemical reactors [26]. The reaction produced in
a porous medium was extraordinarily in common, such
as the topic of PEM fuel cells modules and the polluted
underground water because of discharging the toxic sub-
stance, etc.
Fourier’s law, for instance, described the relation be-
tween energy flux and temperature gradient. In other
aspects, Fick’s law was determined by the correlation of
mass flux and concentration gradient. Moreover, it was
found that energy flux can also be generated by compo-
sition gradients, pressure gradients, or body forces. The
energy flux caused by a composition gradient was dis-
covered in 1873 by Dufour and was correspondingly
referred to the Dufour effect. It was also called the diffu-
sion-thermo effect. On the other hand, mass flux can also
be created by a temperature gradient, as was established
by Soret. This is the thermal-diffusion effect. In general,
the thermal-diffusion and the diffusion-thermo effects
were of a smaller order of magnitude than the effects
described by Fourier’s or Fick’s law and were often neg-
lected in heat and mass transfer processes. There were
still some exceptional conditions. The thermal-diffusion
effect has been utilized for isotope separation and in
mixtures between gases with very light molecular weight
(H2, He) and of medium molecular weight (N2, air), the
diffusion-thermo effect was found to be of a magnitude
such that it may not be neglected in certain conditions
[27]. In recent years, Kandasamy et al. studied the heat
and mass transfer under a chemical reaction with a heat
source [28,29]. Seddeek studied the thermal radiation
and buoyancy effect on MHD free convection heat gen-
eration flow over an accelerating permeable surface with
the influence temperature dependent viscosity [30], and
later the chemical reaction, variable viscosity, radiation,
variable suction on hydromagnetic convection flow
problems were included [31-33].
Although there are numerous widely practical applica-
tions in industrial processes, few previous published pa-
pers discussed the combined relation. In the present pa-
per, we make an attempt to investigate the problem of
convective heat and mass transfer of incompressible
MHD visco-elastic fluid embedded in a porous medium
over a stretching sheet under a chemical reaction. The
presence of combined buoyancy effects leads to the mo-
mentum, heat and mass transfer equations in the coupled
form of highly non-linear partial differential equations.
To deal with the coupling and non-linearity, a numerical
shooting technique for three unknown initial conditions
with Runge-Kutta fourth order integration scheme has
been developed. The results are analyzed for various
physical parameters such as visco-elasticity, permeability
of the porous medium, magnetic field, Grashof number,
Schmidt number, Prandtl number, heat source parameter
and chemical reaction parameter on the flow, heat and
mass transfer characteristics.
2. Mathematical Formulation
We consider a free convective, laminar boundary layer
flow and heat and mass transfer of viscous incompressi-
ble and electrically conducting visco-elastic liquid due to
a stretching sheet. The sheet lies in the plane 0y

with the flow being confined to 0y. The coordinate
x
is being taken along the stretching sheet and y is
normal to the surfaced, two equal and opposite forces are
applied along the x-axis, so that the sheet is stretched,
keeping the origin fixed. A uniform transverse magnetic
field of strength 0 B is applied parallel to the y-axis
and the chemical reaction is taking place in the flow. The
viscous dissipation effect and Joule heat are neglected on
account of the fluid is finitely conducting. It is assumed
that the induced magnetic field, the external electric field
and the electric field due to the polarization of charges
are negligible. The density variation and the effects of
the buoyancy are taken into account in the momentum
equa- tion (Boussinesq’s approximation) and the concen-
tration of species far from the wall is infinitesimally
small and the viscous dissipation term in the energy equ-
ation is neglected (as the fluid velocity is very low). Un-
der these assumptions, the governing boundary layer
equations of momentum, energy and diffusion under
Boussinesq approximations could be written as follows:
0
u
xy
 


 (1)
 
23322
0
2232
2
*0
uu uuuuuuu
uku
x
yxyxy
yxyyy
B
ug TTgCCu
 





  
 






 
(2)

2
2
pp
TTkTQ
uTT
xyC C y


 

  (3)

2
1
2
CC C
uDKCC
xy y


 
 
  (4)
where u, υ are velocity components, T and C are,
respectively, the temperature and concentration of che-
mical species in the fluid,

is the kinematic viscosity,
0
kis the non-Newtonian visco-elastic parameter, εis

S. M. ALHARBI ET AL.
448
the permeability coefficient of porous medium,
g
is the
acceleration due to gravity,

is the volumetric coeffi-
cient of thermal expansion, *

is the volumetric con-
centration coefficient, 0 B is the magnetic induction,

is the fluid density,

is the fluid electrical conduc-
tivity, k is the thermal conductivity, p C is the specific
heat at constant pressure, Qis the dimensional heat
generation/absorption coefficient,
D
is the mass diffu-
sivity and 1
K
is the chemical reaction parameter.
The boundary conditions governing the flow are:



0,,0,Α,
,,, ,
ww
y
x
x
yubx CCTTB
ll
yuu TTCC


 
 
(5)
To take into account the effect of stretching of the
boundary sheet, and the effects due to temperature and
concentration gradients, we prescribe the wall boundary
conditions in the form of (5). In order to study the heat
transfer analysis we consider two general cases of non-
isothermal temperature boundary conditions, namely
boundary with prescribed power law surface temperature.
The subscript y denotes the differentiation w.r.t. y.
Now, we introduce the following dimensionless va-
riables:
 
 
',,
,
ww
b
ubxf bvfy
v
TT CC
TT CC
 
 


 



(6)
where




,
ww
x
x
TT BCCA
ll

  (7)
With these changes of variables Equation (1) is iden-
tically satisfied and Equations (2)-(4) are transformed to


2'''''' '''''''2'
12
'2''
'
f
fffkfffffk f
GrGc Mf 
 
 (8)

''' '0Pr ff
  (9)


''' '0Sc ff
 
(10)
The corresponding boundary conditions take the form:
'
'
0,0,1,1, 1
,'0,' 0,0,0
ff
ff


 
 (11)
where subscript ' denotes the differentiation with respect
to

. 12 ,kk are the viscoelastic and porosity parame-
ters, Gr and Gc are the free convection parameters,
M
magnetic field parameter,

is the heat generation
or absorption coefficient,

is the Chemical reaction
parameter, and
P
r, Sc denote Prandtl number and
Schmidt number respectively. These dimensionless phy-
sical parameters are defined as:




*
0
12 22
2
0 1
,, ,,
Pr,,,
ww
p
p
gTTgC C
kb v
kk GrGc
vb bx bx
CB
K Q
ScandM
kDbCbb








 
(12)
where expressions for


w
TT

 and


w
CC


are given in Equation (6). The important physical quanti-
ties of our interest are the local skin friction w

, Nusselt
number Nuand Sherwood number Sh and they are
defined in the sequel:

*
'' *
0
0
w
y
u
fwhere y
b
bx v




 
 




'0
y
w
h
Nu T
tT


 

'(0)
y
w
h
Sh C
CC


 

3. Numerical Solution
Equations (8)-(10) constitute a highly non-linear coupled
boundary value problem of fourth and second order. So
we develop most effective numerical shooting technique
with fourth-order Runge-Kutta integration scheme with
Newton Raphson method. To select

we begin with
some initial guess value and solve the problem with
some particular set of parameters to obtain ''(0),f
'(0)

and


'0

. The solution process is repeated with
another larger value of


until two successive values
of ''(0),f '(0)

and


'0

differ only after desired
digit signifying the limit of the boundary along

. The
last value of


is chosen as appropriate value for that
particular set of parameters.
Equations (8)-(10) of fourth order in
f
and second
order in


and


has been reduced to a system of
eight simultaneous equations of first order for eight un-
knowns following the method of superposition [34]. To
solve this system we require eight initial conditions
whilst we have only two initial conditions '(0)f and
(0)
f
on
f
, two initial conditions on each on

and

. The third initial condition on '''(0)f has been de-
duced by applying initial conditions of (11) in Equation
(8). Still there are three initial conditions ''(0)f, '(0)

and '(0)

which are not prescribed. Now, we employ
numerical shooting technique where these two ending
boundary conditions are utilized to produce two known
initial conditions at 0


. In this calculation, the step
size Δ0.001


is used while obtaining the numerical

S. M. ALHARBI ET AL.
449
solution with max 7

 and five-decimal accuracy as
the criterion for convergence.
4. Results and Discussion
The numerical computations have been carried out for
various values of visco-elastic parameter 1 k, porosity
parameter 2
k, Grashof number Gr , modified Grashof
number Gc , Prandtl number
P
r and Schmidt number
Sc using numerical scheme discussed in the previous
section. In order to illustrate the results graphically, the
numerical values are plotted in Figures 1-18. These fig-
ures depict the horizontal velocity, temperature and con-
centration profiles for power law surface temperature.
Values of local skin friction w

, Nusselt number Nu
and Sherwood number Sh are recorded in Table 1-3 for
various values of 1
kvisco-elastic, 2 kporosity para-
meter, Gr, Gc free convection parameters,
M
mag-
netic field parameter,

heat generation or absorption
coefficient,

Chemical reaction parameter,
P
r Prandtl
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0246
k2= 0. 0
k2= 0. 5
k2= 1. 0
Figure 1. Effect of k2 on the velocity f׳(η) profiles for k1 =
0.1 , M = Gr = Gc = 0.5, Pr = 1, Sc = 0.96, δ = –0.5 and γ =
0.5.
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0123456
k2 =0. 0
k2 =0. 5
k2 =1
Figure 2. Effect of k2 on the temperature θ(η) profiles for k1
= 0.1, M = Gr = Gc = 0.5, Pr = 1, Sc = 0.96, δ= –0.5 and γ =
0.5.
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0123456
k2=0 .0
k2=0 .5
k2=1
Figure 3. Effect of k2 on the concentration φ(η) profiles for k1
= 0.1, M = Gr = Gc = 0.5, Pr = 1, Sc = 0.96 δ = –0.5 and γ =
0.5.
number and Sc Schmidt number.
Figures 1-3 display results for the velocity, tempera
ture and concentration distributions. As shown, the tem-
perature and concentration are increasing with increa-
singthe dimensionless porous medium parameter 2 kand
the velocity decreases as 2 kincreases. The effect of the
dimensionless porous medium parameter 2 kbecomes
smaller as 2 k increases.
Figures 4-6 illustrate the influence of the magnetic
parameter
M
on the velocity, temperature and concen-
tration profiles in the boundary layer, respectively. Ap-
plication of a transverse magnetic field to an electrically
conducting fluid gives rise to a resistive-type force called
the Lorentz force. This force has the tendency to slow
down the motion of the fluid in the boundary layer and to
increase its temperature and concentration. Also, the ef-
fects on the flow and thermal fields become more so as
the strength of the magnetic field increases.
Figures 7-12 show the effects of Grashof number Gr
and modified Grashof number Gc on the velocity, tem-
perature and concentration respectively. As shown, the
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0123456
M=0.0
M=0.5
M=1
Figure 4. Effect of M on the velocity f׳(η) profiles for k1 = 0,
k2 = 1, Gr = Gc =0.5, Pr = 1, Sc = 0.96, δ= –0.5 and γ = 0.5.
()f


k2 = 0.0
k2 = 0.5
k2 = 1.0
()


k2 = 0.0
k2 = 0.5
k2 = 1
()f


M = 0.0
M = 0.5
M = 1
)(


k2 = 0.0
k2 = 0.5
k2 = 1

S. M. ALHARBI ET AL.
450
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0123456
M=0.0
M=0.5
M=1.0
Figure 5. Effect of M on the temperature θ(η) profiles for k1
= 0.1, k2 = 1, Gr = Gc = 0.5, Pr = 1, Sc = 0.96, δ = –0.5 and γ
= 0.5.
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0123456
M=0.0
M=0.5
M=1.0
Figure 6. Effect of Mon the concentration φ(η) profiles for
k1 = 0.1, k2 = 1, Gr = Gc = 0.5, Pr = 1, Sc = 0.96, δ = –0.5, and
γ = 0.5.
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0123456
Gr=0 . 0
Gr=0 . 5
Gr=1
Figure 7. effect of Gr on the velocity f׳(η) profiles for k1 =
0.1, k2 = 1, M = Gc = 0.5, Pr = 1, Sc =0.96, δ = –0.5 and γ =
0.5.
temperature and the concentration are decreasing with in-
creasing Gr and ,Gc,but the velocity increases as Gr
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0123456
Gr=0.0
Gr=0.5
Gr=1
Figure 8. Effect of Gr on the temperature θ(η) profiles for
k1 = 0.1, k2 = 1, M = Gc = 0.5, Pr= 1, Sc = 0.96, δ = –0.5 and γ
= 0.5.
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0123456
Gr=0.0
Gr=0.5
Gr=1
Figure 9. Effect of Gr on the concentration φ(η) profiles for k1
= 0.1, k2 = 1, M = Gc = 0.5, Pr = 1, Sc = 0.96, δ = –0.5 and γ =
0.5.
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0246
Gc=-0 . 5
Gc=0.0
Gc=0.5
Gc=1
Gc=2
Figure 10. Effect of Gc on the velocity f׳(η) profiles for k1 =
0.1, k2 = 1, M = Gr = 0.5, Pr = 1, Sc = 0.96, δ = –0.5 and γ =
0.5.
and Gc increases. Physically Gr> 0 means heating of
the fluid or cooling of the boundary surface, Gr> 0
)(


)(


()f


()


()


()f


M = 0.0
M = 0.5
M = 1.0
M = 0.0
M = 0.5
M = 1.0
Gr = 0.0
Gr = 0.5
Gr = 1
Gc = －0.5
Gc = 0.0
Gc = 0.5
Gc = 1
Gc = 2
Gr = 0.0
Gr = 0.5
Gr = 1.
Gr = 0.0
Gr = 0.5
Gr = 1

S. M. ALHARBI ET AL.
451
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0246
Gc=-0.5
Gc=0.5
Gc=2
Figure 11. effect of Gc on the temperature θ(η) profiles fvor k1
= 0.1, k2 = 1, M = Gr = 0.5, Pr = 1, Sc = 0.96, δ = –0.5 and γ =
0.5.
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0246
Gc=-0.5
Gc=0.5
Gc=2
Figure 12. Effect of Gc on the concentration φ(η) profiles
for k1 = 0.1, k2 =1, M = Gr = 0.5, Pr = 1, Sc = 0.96, δ = –0.5
and γ = 0.5.
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0123456789
δ=1
δ=-0.5
δ=0.0
δ=0.3
Figure 13. Effect of δ on the velocity f׳(η) profiles for k1 =
0.1, k2 = 1, M = Gr = Gc = 0.5, Pr = 1, Sc = 0.96 and γ = 0.5.
means cooling of the fluid or heating of the boundary sur-
face and Gr= 0 corresponds to the absence of free con-
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0246810
δ=-1
δ=-0.5
δ=0.0
δ=0.3
Figure 14. Effect of δ on the temperature θ(η) profiles for
k1 = 0.1, k2 = 1, M = Gr = Gc = 0.5, Pr = 1, Sc = 0.96 and γ =
0.5.
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0123456
δ=-1
δ=-0.5
δ=0.0
δ=0.3
Figure 15. Effect of δ on the concentration φ(η) profiles for
k1 = 0.1, k2 = 1, M = Gr = Gc = 0.5, Pr = 1, Sc = 0.96 and γ =
0.5.
vection current.
Figures 13-15 present typical profiles for the velocity,
temperature and concentration for various values of a
heat source (0

) or a heat sink (0

), respectively.
As shown, the velocity and the temperature are increas-
ing with increasing

, but the concentration decreases
as

increases. In the event that the strength of the heat
sink (0


) is relatively large, the maximum fluid tem-
perature does not occur at the wall but rather in the fluid
region close to it. Conversely, the presence of a heat
source (0

) effect causes a reduction in the thermal
state of the fluid, thus producing lower thermal boundary
layers.
Figures 16-18 illustrate the influence of the Chemical
reaction parameter

on the velocity, temperature and
concentration profiles in the boundary layer, respectively.
As shown, the velocity and the concentration are decr-
easing with increasing

, but the temperature increases
Gc = －0.5
Gc = 0.5
Gc = 2
Gc = －0.5
Gc = 0.5
Gc = 2
δ = －1
δ = －0.5
δ = 0.0
δ = 0.3
δ = －1
δ = －0.5
δ = 0.0
δ = 0.3
δ = 1
δ = －0.5
δ = 0.0
δ = 0.3

S. M. ALHARBI ET AL.
452
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
02468
γ=-0.2
γ=0.0
γ=0.5
γ=1
Figure 16. Effect of γ on the velocity f׳(η) profiles for k1 =
0.1, k2 = 1, M = Gr = Gc = 0.5, Pr = 1, Sc = 0.96 and δ = –0.5.
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0123456
γ=-0.2
γ=0.0
γ=0.5
γ==1
Figure 17. Effect of γ on the temperature θ(η) profiles for k1 =
0.1,k2 = 1, M = Gr = Gc = 0.5, Pr = 1, Sc = 0.96 and δ = –0.5.
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0246810
γ=-0.2
γ=0.0
γ=0.5
γ=1
Figure 18. Effect of γ on the concentration φ(η) profiles for
k1 = 0.1, k2 = 1, M = Gr = Gc = 0.5, Pr = 1,Sc = 0.96 and δ =
–0.5.
as

increases, this is due to the fact that destructive
chemical reduces the solutal boundary layer thickness
and increases the mass transfer.
Tables 1-3 represents values of w

, Nu and Sh for
various values of 1 ,,,,,Pr, ,
M
Gr GcSck

 and 2 k. It
is clear that, with increasing Gr and Gc, ,Nu Sh
and w

increases, whereas with increasing 1,k 2 kand
M
, ,Nu Shand w

decreases. Also, w

and Sh increase
as Sc and

increases and Nu decrease, whereas,
Nu and w

decreases and Sh increase as

increases,
Also, Sh and w

increases and Nu decrease as
P
rin-
crease.
Table 1. The effect of parameter on(0).f


(0)f


k
1 = 0.3
k
1 =0.2
k
1 = 0.1
k
2M G
r
Gc P
r
δ Sc γ
–
0.9233
–
0.8908
–
0.86260.00.50.50.51.0
–
0.5 0.96 0.5
–
1.1978
–
1.1344
–
1.08380.50.50.50.51.0
–
0.5 0.96 0.5
–
1.4345
–
1.3476
–
1.27911.00.50.50.51.0
–
0.5 0.96 0.5
–
1.1978
–
1.1344
–
1.08381.00.00.50.51.0
–
0.5 0.96 0.5
–
1.4345
–
1.3476
–
1.27911.00.50.50.51.0
–
0.5 0.96 0.5
–
1.6446
–
1.5387
–
1.45521.01.00.50.51.0
–
0.5 0.96 0.5
–
1.6639
–
1.556
–
1.46921.00.50.00.51.0
–
0.5 0.96 0.5
–
1.4345
–
1.3476
–
1.27911.00.50.50.51.0
–
0.5 0.96 0.5
–
1.1994
–
1.1400
–
1.09291.00.51.00.51.0
–
0.5 0.96 0.5
–
1.8913
–
1.7696
–
1.66881.00.5 0.5
–
0.5 1.0
–
0.5 0.96 0.5
–
1.6654
–
1.5578
–
1.47121.00.50.50.01.0
–
0.5 0.96 0.5
–
1.4345
–
1.3476
–
1.27911.00.50.50.51.0
–
0.5 0.96 0.5
–
1.4345
–
1.3476
–
1.27911.00.50.50.51.0
–
0.5 0.96 0.5
–
1.4622
–
1.379
–
1.31201.00.50.50.52.0
–
0.5 0.96 0.5
–
1.5010
–
1.4201
–
1.35311.00.50.50.55.5
–
0.5 0.96 0.5
–
1.4457
–
1.3602
–
1.29221.00.50.50.51.0
–
1.0 0.96 0.5
–
1.4345
–
1.3476
–
1.27911.00.50.50.51.0
–
0.5 0.96 0.5
–
1.4165
–
1.3268
–
1.25711.00.50.50.51.00.0 0.96 0.5
–
1.4034
–
1.3110
–
1.24021.00.50.50.51.00.2 0.96 0.5
–
1.4345
–
1.3476
–
1.27911.00.50.50.51.0
–
0.5 0.96 0.5
–
1.4637
–
1.3808
–
1.31401.00.50.50.51.0
–
0.5 2.0 0.5
–
1.5025
–
1.4219
–
1.35511.00.50.50.51.0
–
0.5 5.0 0.5
–
1.4032
–
1.3106
–
1.23971.00.50.50.51.0
–
0.5 0.96
–
0.2
–
1.4164
–
1.3267
–
1.25691.00.50.50.51.0
–
0.5 0.96 0.0
–
1.4345
–
1.3476
–
1.27911.00.50.50.51.0
–
0.5 0.96 0.5
–
1.4457
–
1.3603
–
1.29231.00.50.50.51.0
–
0.5 0.96 1.0
γ = －0.2
γ = 0.0
γ = 0.5
γ = 1
γ = －0.2
γ = 0.0
γ = 0.5
γ = 1
γ = －0.2
γ = 0.0
γ = 0.5
γ = 1

S. M. ALHARBI ET AL.
453
Table 2. The effect of parameter on (0).



(0)



k1 = 0.3 k1 = 0.2 k1 = 0.1k2MGr Gc Pr δ Sc γ
1.2609 1.2663 1.2711 0.00.50.50.51.0 –0.5 0.96 0.5
1.2282 1.2360 1.2425 0.5 0.5 0.5 0.5 1.0 –0.5 0.96 0.5
1.2013 1.2106 1.2184 1.0 0.5 0.5 0.5 1.0 –0.5 0.96 0.5
1.2282 1.2360 1.2425 1.00.00.50.51.0 –0.5 0.96 0.5
1.2013 1.2106 1.2184 1.0 0.5 0.5 0.5 1.0 –0.5 0.96 0.5
1.1785 1.1889 1.1976 1.0 1.0 0.5 0.5 1.0 –0.5 0.96 0.5
1.1724 1.1841 1.1941 1.00.50.00.51.0 –0.5 0.96 0.5
1.2013 1.2106 1.2184 1.0 0.5 0.5 0.5 1.0 –0.5 0.96 0.5
1.2278 1.2346 1.2404 1.0 0.5 1.0 0.5 1.0 –0.5 0.96 0.5
1.1390 1.1529 1.1650 1.00.50.5–0.5 1.0–0.5 0.96 0.5
1.1719 1.1835 1.1935 1.0 0.5 0.5 0.0 1.0 –0.5 0.96 0.5
1.2013 1.2106 1.2184 1.0 0.5 0.5 0.5 1.0 –0.5 0.96 0.5
1.2013 1.2106 1.2184 1.00.50.50.51.0 –0.5 0.96 0.5
1.7734 1.7849 1.7946 1.0 0.5 0.5 0.5 2.0 –0.5 0.96 0.5
2.9327 2.9463 2.9578 1.0 0.5 0.5 0.5 5.5 –0.5 0.96 0.5
1.4134 1.4207 1.4268 1.00.50.50.51.0 –1.0 0.96 0.5
1.2013 1.2106 1.2184 1.0 0.5 0.5 0.5 1.0 –0.5 0.96 0.5
0.9181 0.9321 0.9442 1.0 0.5 0.5 0.5 1.00.0 0.96 0.5
0.7494 0.7656 0.7827 1.0 0.5 0.5 0.5 1.0 0.2 0.96 0.5
1.2013 1.2106 1.2184 1.00.50.50.51.0 –0.5 0.96 0.5
1.1922 1.2016 1.2097 1.0 0.5 0.5 0.5 1.0 –0.5 2.0 0.5
1.1836 1.1934 1.2020 1.0 0.5 0.5 0.5 1.0–0.5 5.0 0.5
1.2141 1.2232 1.2305 1.0 0.5 0.5 0.5 1.0 –0.5 0.96 –0.2
1.2083 1.2174 1.2249 1.00.50.50.51.0 –0.5 0.96 0.0
1.2013 1.2106 1.2184 1.0 0.5 0.5 0.5 1.0 –0.5 0.96 0.5
1.1975 1.2069 1.2149 1.0 0.5 0.5 0.5 1.0–0.5 0.96 1.0
Table 3. The effect of parameter on (0).



(0)



k1 = 0.3 k1 = 0.2 k1 = 0.1k2MGr Gc Pr δ Sc γ
1.2322 1.2375 1.2423 0.00.50.50.51.0 –0.5 0.96 0.5
1.1999 1.2075 1.2140 0.5 0.5 0.5 0.5 1.0 –0.5 0.96 0.5
1.1733 1.1824 1.1902 1.0 0.5 0.5 0.5 1.0 –0.5 0.96 0.5
1.1999 1.2075 1.2140 1.00.00.50.51.0 –0.5 0.96 0.5
1.1733 1.1824 1.1902 1.0 0.5 0.5 0.5 1.0 –0.5 0.96 0.5
1.1508 1.1610 1.1697 1.0 1.0 0.5 0.5 1.0 –0.5 0.96 0.5
1.1448 1.1563 1.1661 1.00.50.00.51.0 –0.5 0.96 0.5
1.1733 1.1824 1.1902 1.0 0.5 0.5 0.5 1.0 –0.5 0.96 0.5
1.1994 1.2062 1.2119 1.0 0.5 1.0 0.5 1.0 –0.5 0.96 0.5
1.1118 1.1255 1.1374 1.00.50.5 -0.51.0 –0.5 0.96 0.5
1.1442 1.1557 1.1655 1.0 0.5 0.5 0.0 1.0 –0.5 0.96 0.5
1.1733 1.1824 1.1902 1.0 0.5 0.5 0.5 1.0 –0.5 0.96 0.5
1.1733 1.1824 1.1902 1.00.50.50.51.0 –0.5 0.96 0.5
1.1648 1.1741 1.1821 1.0 0.5 0.5 0.5 2.0 –0.5 0.96 0.5
1.1563 1.1660 1.1744 1.0 0.5 0.5 0.5 5.5 –0.5 0.96 0.5
1.1696 1.1789 1.1868 1.00.50.50.51.0 –1.0 0.96 0.5
1.1733 1.1824 1.1902 1.0 0.5 0.5 0.5 1.0 –0.5 0.96 0.5
1.1801 1.1891 1.1965 1.0 0.5 0.5 0.5 1.00.0 0.96 0.5
1.1858 1.1948 1.2020 1.0 0.5 0.5 0.5 1.0 0.2 0.96 0.5
1.1733 1.1824 1.1902 1.00.50.50.51.0 –0.5 0.96 0.5
1.7729 1.7844 1.7940 1.0 0.5 0.5 0.5 1.0 –0.5 2.0 0.5
2.9322 2.9458 2.9573 1.0 0.5 0.5 0.5 1.0–0.5 5.0 0.5
0.7266 0.7420 0.7589 1.0 0.5 0.5 0.5 1.0 –0.5 0.96 –0.2
0.8935 0.9074 0.9194 1.00.50.50.51.0 –0.5 0.96 0.0
1.1733 1.1824 1.1902 1.0 0.5 0.5 0.5 1.0 –0.5 0.96 0.5
1.3820 1.3891 1.3952 1.0 0.5 0.5 0.5 1.0–0.5 0.96 1.0
5. Conclusions
In this study, a numerical analysis is presented to inves-
tigate the influence of chemical reaction of first-order
and magnetic field on the heat and mass transfer of an
electrically conducting viscoelastic fluid flow through a
porous medium over a stretching sheet. The non-linear
and coupled governing equations are solved numerical
by using fourth order Runge-Kutta integration scheme-
with Newton Raphson shooting method. Velocity, tem-

S. M. ALHARBI ET AL.
454
perature and concentration profiles are presented graphi-
cally and analyzed. The fundamental parameters found to
effect the problem under consideration are the chemical
reaction parameter, magnetic field parameter, viscoelas-
tic parameter, porosity parameter, Grashof number, mo-
dified Grashof number, Prandtl number, Schmidt number
and heat absorption parameter. It is found that, the tem-
perature as well as concentration increases with increas-
ing the visco-elastic parameter, porosity parameter and
magnetic parameter whereas reverse trend is seen with
Grashof number and modified Grashof number increas-
ing. Additionally, the velocity temperature is increased in
the presence of heat absorption parameters and decreased
with chemical reaction.
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Nomenclature
,
A
B constants T

temperature of the ambient fluid
b stretching rate, positive constant w
T surface temperature
0
B magnetic induction ,u

velocity components along x and y direction
p
C specific heat at constant pressure
x
coordinate along the stretching sheet
D mass diffusivity ydistance normal to the stretching sheet
f
dimensionless stream function Greek symbols
Gr temperature Grashof of number

dimensionless temperature function
Gc mass Grashof of number

dimensionless concentration function
g
acceleration due to gravity

dimensionless space variable
1
K first order chemical reaction rate

kinematic viscosity
k thermal conductivity

fluid density
0
k non-Newtonian visco-elastic parameter

electrical conductivity
1
k visco-elastic parameter

coefficient of viscosity
2
k porosity parameter

coefficient of thermal expansion
l characteristic length *

volumetric concentration coefficient
M
magnetic field parameter

heat generation or absorption coefficient
Nu Nusselt number w

skin friction
P
r Prandtl number

permeability coefficient of porous medium
Q dimensional heat generation/absorption coefficient

Chemical reaction parameter
Sc Schmidt number Subscripts
Sh Sherwood number w properties at the plate
T temperature of the fluid

free stream condition