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Global Journal of Pure and Applied Mathematics.
ISSN 0973-1768 Volume 13, Number 6 (2017), pp. 1593-1610
© Research India Publications
http://www.ripublication.com
MHD Stagnation Point Flow and Heat Transfer Due
to Nano Fluid over Exponential Radiating Stretching
Sheet
Punnam. Rajendar1*, L. Anand Babu2* and T.Vijaya Laxmi2
1,2Department of Mathematics, Osmania University, Hyderabad, Telangana, India. 2* Department Of Mathematics, School of Engineering, Anurag Group of Institutions
Ghatkeshar (Mdl), Medchal (Dist), 500088, Telangana, India. 2M.V.S. Govt. Arts & Science College, Mahabubnagar-509001, Telangana, India.
Abstract
This paper deals with the MHD stagnation point flow and heat transfer due to
nano fluid over exponential radiating stretching sheet. The basic governing
partial diffential equations are converted into nonlinear ordinary partial
differential equations by employing suitable similarity transformations. The
resulting equations are successfully solved by using implicit finite difference
scheme known as Keller box method. The results are very closely agree with
the existing results in the literature. The major effect of governing parameters
namely, magnetic and radiation parameters on the flow field skin friction
coefficient, nusselt number and Sherwood numbers for several values of
governing parameters examined and are presented in the form of tables and
figure.The results indicate that the local Nusselt number decreases and
Sherwood number increases with an increase in velocity ratio parameter
A.The local Nusselt number and Sherwood number increases with an increase
in Lewis number Le and radiation parameter R. Besides, it is found that the
heat transfer rate at the surface increases with the magnetic parameter when
the free stream velocity exceeds the stretching velocity, i.e. A > 1, and it
decreases when A < 1.
Keywords: MHD, stagnation point, exponential stretching sheet, nano fluid,
radiation parameter, heat transfer, Keller Box Method.
INTRODUCTION
The heat transfer of a viscous fluid over flat surfaces have been investigated in several
technological processes such as hot rolling, metal extrusion, continuous stretching of
1594 Punnam. Rajendar, L. Anand Babu and T.Vijaya Laxmi
plastic films and glass-fibre, polymer extrusion, wires drawing and metal spinning.
Various researchers are engaged in this rich area. Numerous physical phenomena
related to stretched sheet moving with constant velocity under various thermal
conditions have been investigated by Carragher and Crane (1982)[1], Grubka and
Bobba (1985)[2]. The thermal boundary layer of a power law fluid over a stretching
surface was studied by Ali (1995)[3]. A good effort has been made to gain insight
information regarding the stretching flow problem in various situations include
considerations of porous surfaces, MHD fluids, heat and mass transfer, slip effects
etc. Magyari and Keller (1999)[4]investigated the steady boundary layers on an
exponentially stretching continuous surface with an exponential temperature
distribution. A new dimension is added to this investigation by Elbashbeshy
(2001)[5]who examined the flow and heat transfer characteristics over an
exponentially stretching permeable surface.Mukhopadhyay (2013)[6]examined slip
effects on MHD boundary layer flow over an exponentially stretching sheet. Some
recent attempts in this direction are described in wang (2008)[7],Hayat(2008)[8],El-
Aziz (2009)[9], Khan and Pop (2011)[10],Ishak (2011)[11], Cortell (2012)[12].
The flow near the stagnation point has attracted the attention of many investigators for
more than a century because of its wide applications. Some of the applications are
cooling of electronic devices by fans, cooling of nuclear reactors during emergency
shutdown, solar central receivers exposed to wind currents, and many hydrodynamic
processes in engineering applications. The study of a stagnation point flow towards a
solid surface in moving fluid traced back to Hiemenz in 1911. He was the pioneer to
analyse two-dimensional stagnation point flow on stationary plate using a similarity
transformation to reduce the Navier–Stokes equations to non-linear ordinary
differential equations. Since then many investigators have extended the idea to
different aspect of the stagnation point flow problems. Stagnation point flow is
continuing to be an interesting area of research among scientists and investigators due
to its importance in a wide variety of applications both in industrial and scientific
applications. Many researchers have been working still on the stagnation- point flow
in various ways. Accordingly, Mahapatra and Gupta [13]numerically analyzed two
dimensional boundary layer flow, stagnation point flow and heat transfer over a
stretching sheet.
The study of magneto-hydrodynamic (MHD) flow of an electrically conducting fluid
is of considerable interest in modern metallurgical and metal-working processes. The
process of fusing of metals in an electrical furnace by applying a magnetic field and
the process of cooling of the wall inside a nuclear reactor containment vessel are good
examples of such fields (Ibrahim et al., 2013)[14]. Some important applications of
radiative heat transfer include MHD accelerators, high temperature plasmas, power
generation systems and cooling of nuclear reactors. Many processes in engineering
areas occur at high temperatures and knowledge of radiation heat transfer becomes
very important for the design of pertinent equipment (Seddeek, 2003)[15]. In
controlling momentum and heat transfers in the boundary layer flow of different
fluids over a stretching sheet, applied magnetic field may play an important
role(Turkyilmazoglu, 2012)[16].Kumaran et al. (2009)[17]investigated that magnetic
MHD Stagnation Point Flow and Heat Transfer Due to Nano Fluid… 1595
field makes the streamlines steeper which results in the velocity boundary layer being
thinner. The heat transfer analysis of boundary layer flow with radiation is also
important in electrical power generation, astrophysical flows, solarpower technology,
space vehicle re-entry and other industrial areas. Raptis et al.(2004)[18] reported the
effect of thermal radiationon the MHD flow of a viscous fluid past a semi-infinite
stationary plate.
Rеnuka Devi et al.[19] presentеd analysis of the radiation and mass transfеr effеcts on
MHD boundary layer flow due to an exponentially stretching sheet with hеat sourcе.
Recently, Poornima and Bhaskar Rеddy [20] presentеd an analysis of the radiation
effеcts on MHD freе convective boundary layеr flow of nanofluids ovеr a nonlinеar
strеtching sheеt. Howevеr, the intеraction of radiation with mass transfеr due to a
strеtching sheеt has receivеd littlе attention. The present paper provides an analytical
solution of MHD boundary layer flow over an exponentially stretching sheet in the
presence of radiation, which has not been considered before. The effects of
controlling parameters on MHD flow and heat transfer characteristics are discussed
and shown graphically.
FORMULATION OF THE PROBLEM:
Consider a two-dimensional stagnation point flow of a nanofluid towards a stretching
sheet kept at a constant temperature Tw and concentration Cw. The ambient
temperature and concentration are T∞ and C∞ respectively. The flow is subjected to a
constant transverse magnetic field of strength B = B0 which is assumed to be applied
in the positive y-direction, normal to the surface. The induced magnetic field is
assumed to be small compared to the applied magnetic field and is neglected. It is
further assumed that the base fluid and the suspended nanoparticles are in thermal
equilibrium and no slip occurs between them. Where Tw, T∞, Cw, C∞ and B0 are
temperature at the surface of the sheet, ambient temperature of the fluid, concentration
at the surface of the sheet, ambient concentration and magnetic field strength
respectively. We choose the coordinate system such that x-axis is along the stretching
sheet and y-axis is normal to the sheet. The flow configuration and coordinate system
is shown in Fig. 1
Figure-1. Flow configuration and coordinate system.
1596 Punnam. Rajendar, L. Anand Babu and T.Vijaya Laxmi
A variable magnetic field B(x) is applied normally to the sheet surface while the
induced magnetic field is negligible, which can be justified for MHD flow at small
magnetic Reynolds number. Under boundary layer approximations, the flow and heat
transfer with radiation effects are governed by the following dimensional form of
equations.
0 (1)u v
x y
22
0
2( ) (2)
f
BUu u uu v U U u
x y x y
22
2
1(3)r T
B
p
q DT T T C T Tu v D
x y T c y y y T y
2 2
2 2(4)T
B
DC C C Cu v D
x y y T y
where u and v are the components of the velocity in the x and y directions
respectively, is the kinematic viscosity, is thermal diffusivity, T is the fluid
temperature in the boundary layer, is fluid density, qr is the radiative heat flux, Cp
is the specific heat at constant pressure. By the use of Rosseland approximation for
radiation, we
have4
*
4(5)
3r
Tq
k y
where is Stefan–Boltzman constant, and *k is the absorption coefficient.
We assume the temperature difference within the flow such that 4T may be expanded
in a Taylor series about T, (the free stream temperature) and neglecting terms of
higher order, wehave
4 3 44 3T T T T Then 2
3
* 2
16
3
rq TT
y k y
Eq (3) becomes
22 23
2 * 2
1 16(6)
3
TB
p
DT T T T C T Tu v T D
x y T c k y y y T y
MHD Stagnation Point Flow and Heat Transfer Due to Nano Fluid… 1597
The hydrodynamic boundary conditions are
( ), 0 0 (7)wu U x v at y
0 (8)u as y
Where ( )x
LwU x ae is stretching velocity, ‘a’ is the reference velocity, L is the
characteristic Length. The thermal and solutal boundary conditions are
20 0 (9)
x
LwT T T T e at y and T T as y
20 0 (10)
x
LwC C C C e at y and C C as y
Where wT is the variable temperature at the sheet with T0 being a constant and
wC is
the variable concentration at the sheet with C0 being constant. It is assumed that the
magnetic field B(x) is of the form 20( )
x
LB x B e where B0 is a constant magnetic field.
The continuity Eq (1) is satisfied by introducing a stream function such that
u and vy x
For non dimensionalized form of momentum and energy equations as well as
boundary conditions, the following transformations are introduced.
2 2
2 20 0
, '( ), ( ) '( )2 2
( ) ( ) (11)
x x x
L L L
x x
L L
a ay e u ae f v e f f
L L
T T T e and C C C e
Where is the similarity variable, f( ) is the dimensionless stream function, ( ) is
the dimensionless temperature, ( ) is the dimensionless concentration and prime
denote differentiation with respect to .
Using Eq. (11), the momentum, energy and concentration equations can be reduced
into ordinary differential equations:
2 2
2
''' '' 2( ') ( ') 2 0 (12)
41 '' Pr ' Pr ' Pr ' ' Pr ( ') 0 (13)
3
'' '' ' ' 0 (14)
f ff f M A f A
R f f Nb Nt
NtLef Lef
Nb
The transformed boundary conditions of the problem are:
(0) 0, '(0) 1, '( ) 0, (0) 1, (0) 1, ( ) 0, ( ) 0 (15)f f f
1598 Punnam. Rajendar, L. Anand Babu and T.Vijaya Laxmi
Where 'f is the dimensionless velocity, is the temperature, is the particle
concentration,2
02 B LM
a
is the magnetic parameter ,
bA
a is velocity ratio,
Pr
is the prandtl number,
B
LeD
is the Lewis number,
( )p B w
f
c DNb
c
is
the Brownian motion parameter,( )p T w
f
c D T TNt
c T
is the thermoporesis parameter
and
3
*
4 TR
kk
is the radiation conduction parameter.
The important physical quantities of interest in this problem are the skin friction
coefficient Cf , local Nusselt number Nux and the local Sherwood number Shx are
defined as
2
0 0 0
, ,( ) ( )
, , (16)
w w mf x x
w w B w
w w m B
y y y
xq xC Nu Sh
U k T T D
u T Cwhere q k D
y y y
Here the skin friction w ,wall heat flux
wq and wall mass flux m .
By using the above equations we get
Re ''(0), '(0), '(0)Re Re
x xf x
x x
Nu NuC f
where ’Rex’ local Reynolds number.
NUMERICAL METHOD
The higher order ordinary differential equations with the boundary conditions are
solved numerically by using implicit finite difference scheme known as Keller-Box
method, the following steps are involved to achieve the Numerical solution.
Reduce the non-linear higher order ordinary differential equations into a system of
first order ordinary differential equations.
Write the finite differences for the first order equations.
Linearize the algebraic equations by Newton’s method, and write them in matrix–
vector form. Solving the linear system by the block tri-diagonal elimination
technique.
In order to solve the above differential equations numerically, we adopt Mat lab
software which is very efficient in using the well known Keller box method.
For getting accuracy of this method to choose appropriate initial guesses.
MHD Stagnation Point Flow and Heat Transfer Due to Nano Fluid… 1599
( ) (1 )(1 ), ( ) , ( )f Ae A e e e
The step size 0.01 is used to obtain numerical solution with four decimal place
accuracy as criterion of convergence.
RESULT AND DISCUSSION
The theme of this section is to discuss the effect of various physical parameters such
as magnetic parameter M, velocity ratio A, prandtl number Pr, the radiation parameter
R, Brownian motion parameter Nb, themoporesis parameter Nt, lewis number Le.
The transformed momentum, energy, concentration Eqs.(12) to (14)and boundary
conditions (15) were numerically solved by using Keller-Box method. The present
results are compared with the existing results in absence of velocity ratio A, Le, Nb,
Nt for local nusselt number '(0) of several values A, M and Pr presented in Table-1
The results so obtained in this vary fairly agree with the previous
results(Mukhopadhyay, Fazle Mabood). Skin friction coefficient ''(0)f for different
values of radiation parameter R and magnetic parameter M calculated and presented
in Table-2 . It is observed that increases R and M, the skin friction coefficient
increased but where as increase in A, the skin friction coefficient slowly decreases.
Calculated the local Nusselt number '(0) and local Sherwood number '(0) for
various parameters presented in in Table-3.
Table-1: Comparison of '(0) for several values of Magnetic, Radiation Parameters
and Pr
R M Pr Mukhopadhyay
(2013)
Fazle Mabood Present Result
0 0 1 0.9547 0.95478 0.954705
2 1.4714 1.47151 1.471551
3 1.8691 1.86909 1.86958
5 2.5001 2.50012 2.50221
10 3.6603 3.66039 3.670012
1 0 1 0.5312 0.53121 0.53108
0 1 1 0.8610 0.86113 0.86096
0.5 0 2 1.0734 1.07352 1.073455
3 1.3807 1.38075 1.38078
1 0 3 1.1213 1.12142 1.12137
1 1 1 0.4505 0.45052 0.45045
1600 Punnam. Rajendar, L. Anand Babu and T.Vijaya Laxmi
We obtained velocity, temperature, concentration profile graph for different values of
governing parameters.The obtained results are displayed through figures.2-4,5-8 and
9-11 for velocity, temperature and concentration profile respectively. Moreover skin
friction coefficient, local nusselt number and local Sherwood number are given in
figures 12-14 respectively. Figures. 2–4show the velocity graphs for different values
of magnetic parameter M and velocity ratio parameter A, when other parameters
remain fixed.
Figure 2 .Effect of several values of A on velocity profile.
Figure. 2 illustrates the influence of velocity ratio parameter A on velocity graph.
When the free stream velocity exceeds the velocity of the stretching sheet, the flow
velocity increases and the boundary layer thickness decreases with increase in A.
Moreover, when the free stream velocity less than stretching velocity, the flow field
velocity decreases and boundary layer thickness also deceases. When A> 1, the flow
has a boundary layer structure and boundary layer thickness decreases as values of A
increases. On the other hand, when A< 1, the flow has an inverted boundary layer
structure, for this case also, as the values A decrease the boundary layer thickness
decreases.
0 1 2 3 4 50
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
f'(
)
A = 1.8
A = 1.0
A = 0.2
A = 1.4
A = 0.6
MHD Stagnation Point Flow and Heat Transfer Due to Nano Fluid… 1601
Figure.3 Effect of several values of M on velocity profile
The magnetic parameter M represents the importance of magnetic field on the flow
field. The presence of transverse magnetic field sets in Lorentz force, which results in
retarding force on the velocity field. Therefore, as the values of M increase, so does
the retarding force and hence the velocity decreases. It is shown in Figure. 3.When A
= 3.3 i.e. 1b
a the flow has boundary layer structure and the boundary layer
thickness decreases as the values of M increase.
Figure.4. Effect of several values of M and A on velocity profile.
Figure. 4 shows the variation of flow field velocity with magnetic field and velocity
ratio A. Here consider three different values of A i.e. A = 3.3, A = 1.0 , A = 0.2 and A
= 0.3. When A = 3.3 i.e. 1b
a , the flow has boundary layer structure and the
boundary layer thickness is decreasing with M.When A = 1, the velocity graph for
different values of M are coincide, this indicate that the velocity graph is not
0 1 2 3 4 51
1.5
2
2.5
3
f'(
)
M = 0.0, 1.0, 5.0, 10.0
0 1 2 3 4
0.5
1
1.5
2
2.5
3
f'(
)
A =3.3
A = 0.2
A = 0.3
M = 0, 1, 5, 10A = 1.0
M = 0.0
M = 1.0
M = 5.0
M = 10. 0
1602 Punnam. Rajendar, L. Anand Babu and T.Vijaya Laxmi
influenced by the magnetic field. On the other hand, when A < 1, the flow has an
inverted boundary layer structure, which results from the fact that when A < 1, the
stretching velocity “ax’ of the surface exceeds the velocity “bx” of the free stream
velocity. For this case also the thickness of the boundary layer decrease with M,
which implies increasing manner of the magnitude of the velocity gradient at the
surface.
Figure.5 Effect of several values of A on temperature profile
Figure. 5 shows the variation of temperature profile in response to a change in the
values of velocity ratio parameter A. It shows that as velocity ratio parameter
increases the thermal boundary layer thickness decreases. Moreover, the temperature
gradient at the surface increase (in absolute value) as A increases. As a result,
temperature profile decreases.
Figure.6 Effect of several values of Pr on temperature profile
0 5 10 150
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
(
)
A = 0.0
A = 0.3
A = 0.6
A = 0.9
0 2 4 6 8 100
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
(
)
Pr = 0.5
Pr = 1.0
Pr = 1.5
Pr = 2.0
MHD Stagnation Point Flow and Heat Transfer Due to Nano Fluid… 1603
Figure.6. The graph depicts that the temperature decreases when the values of Prandtl
number Pr increase at a fixed value of . This is due to the fact that a higher Prandtl
number fluid has relatively low thermal conductivity, which reduces conduction and
thereby the thermal boundary layer thickness and as a result, temperature decreases.
Figure.7 Effect of several values of Nb on temperature profile
Figure.7.shows the influence of the change of Brownian motion parameter Nb on
temperature profile graph. It is noticed that as Brownian motion parameter increases
the thermal boundary layer thickness increases and the temperature gradient at the
surface decrease (in absolute value) as Nb. Consequently, temperature on the surface
of a plate increases.
Figure.8 Effect of several values of R on temperature profile when Pr = 2.2
0 2 4 6 80
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
(
)
Nb = 0.1
Nb = 0.3
Nb = 0.6
Nb = 1.0
0 1 2 3 4 5 60
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
(
)
Pr = 2.2
R = 0.0, 0.5, 1.5, 2.0
1604 Punnam. Rajendar, L. Anand Babu and T.Vijaya Laxmi
It is also noticed from Figure. 8 that the dimensionless temperature increases with an
increase in radiation parameter R.The Lorentz force has the tendency to increase the
temperature and consequently, the thermal boundary layer thickness becomes thicker
for stronger magnetic field.
Figure.9 Effect of several values of A on concentration profile
Figure.9 depicts the influence of velocity ratio parameter A on concentration graph.
As the values of A increase, the concentration boundary layer thickness decreases.
Moreover, it is recognize from the graph that the magnitude of temperature gradient
on the surface of a plate increases as A increases.
Figure.10 Effect of several values of Nb on concentration profile
Figure.10 reveals variation of concentration graph in response to a change in
Brownian motion parameter Nb. The influence of Brownian motion on concentration
0 2 4 6 8 10 12 14 160
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
(
)
A = 0.0
A = 0.2
A = 0.4
A = 0.8
0 1 2 3 4 50
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
(
)
Nb = 0.5
Nb = 1.0
Nb = 1.5
Nb = 2.0
MHD Stagnation Point Flow and Heat Transfer Due to Nano Fluid… 1605
profile graph is as the values of Brownian motion parameter increase, the
concentration boundary layer thickness is decreasing. The graph also reveals that the
thermal boundary layer thickness does not change much when the values of Nb
increases.
Figure.11 Effect of several values of Le on concentration profile
As it is noticed from Figure.11 as Lewis number increases the concentration graph
decreases. Moreover, the concentration boundary layer thickness decreases as Lewis
number increases. This is probably due to the fact that mass transfer rate increases as
Lewis number increases. It also reveals that the concentration gradient at surface of
the plate increases.
Table-2: Calculation of skin friction coefficient - ''(0)f for various
parameters M, R, A.
M R A - ''(0)f
1.0 0.5 0.2 1.4381
1.5 1.5454
2.0 1.6456
1.0 1.4381
1.5 1.4880
2.0 1.8298
0.4 1.1707
0.6 0.8373
0.8 0.4452
0 1 2 3 4 5 6 7 8 90
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
(
)
Le = 0.5
Le = 1.0
Le = 2.0
Le = 5.0
1606 Punnam. Rajendar, L. Anand Babu and T.Vijaya Laxmi
Table-3: Calculation of Nusselt number & Sherwood number for various values of
M, R, A, Le, Pr, Nb, Nt.
M R A Le Pr Nb Nt '(0) '(0)
0 0.5 0.2 2.0 1.0 0.5 0.5 0.5789 1.2952
1.0 0.5560 1.2508
5.0 0.5069 1.1492
1.0 0.4354 1.1892
1.5 0.3862 1.2155
2.0 0.3498 1.2343
0.4 0.6126 1.3471
0.6 0.6613 1.4337
0.8 0.7055 1.5142
3.0 0.5441 1.6803
4.0 0.5364 2.0383
5.0 0.5309 2.6513
2.0 0.6534 2.3236
3.0 0.6985 2.3252
4.0 0.7134 2.3368
0.2 0.5973 2.0444
0.3 0.5743 2.2161
0.4 0.5521 2.3010
0.2 0.5831 1.3731
0.3 0.5738 1.3303
0.4 0.5647 1.2896
MHD Stagnation Point Flow and Heat Transfer Due to Nano Fluid… 1607
0.6 0.8 1.0 1.2 1.4
0.30
0.35
0.40
0.45
0.50
0.55
0.60
0.65
R
Pr = 0.5
Pr = 1.0
Pr = 1.5
'(0)
Figure. 12 represents a variation of local Nusselt number with respect to Radiation
Parameter R and Prandtl number Pr.
Figure.13 Demonstrate the variation of local Sherwood number with respect to
Brownian motion parameter Nb and Lewis number Le.
0.20 0.25 0.30 0.35 0.40
0.8
0.9
1.0
1.1
1.2
1.3
1.4
1.5
1.6
Nb
Le = 2.0
Le = 2.5
Le = 3.0
1608 Punnam. Rajendar, L. Anand Babu and T.Vijaya Laxmi
1.0 1.2 1.4 1.6 1.8 2.00.8
0.9
1.0
1.1
1.2
1.3
1.4
1.5
-f''(0
)
M
A = 0.6
A = 1.0
A = 1.5
Figure.14. Illustrate the influence of Magnetic parameter M and velocity
ratio A on skin friction co efficient.
CONCLUSION
In this work, the MHD stagnation point flow of a nano fluid and heat transfer over a
exponential radiating stretching sheet was analysed. The effects of magnetic
parameter M, Radiation parameter R, velocity ratio A, Prandtl number Pr, Lewis
number Le, Brownian motion parameter Nb, thermophoresis parameter Nt on the
fluid flow and heat transfer characteristics of the MHD stagnation point flow of a
Nano fluid over exponential radiating stretching sheet were investigated. The
numerical results obtained are agreed very well with the previously published data in
limiting condition and for some particular cases of the present study.
An increase in velocity ratio A is to stagnate in the velocity decreases in concentration
and temperature profiles. As Lewis number Le increases, concentration profile
decreases. As magnetic parameter M increases, velocity profile increases.As
Radiation parameter R increases, the temperature of nano fluid increases. As Nb
increases, the temperature of nano fluid increases, and the concentration decreases. As
Pr increases, the temperature of nano fluid decreases.
REFERENCES
[1] Carragher, P., Crane, L.J., “Heat transfer on a continuous stretching sheet”
Zeit.Angew.Math. Mech. 62, 564–573(1982).
[2] Grubka, L.J., Bobba, K.M., “Heat transfer characteristics of a continuous
stretching surface with variable temperature”. J. Heat Transfer 107, 248–
250(1985).
MHD Stagnation Point Flow and Heat Transfer Due to Nano Fluid… 1609
[3] Magyari, E., Keller, B., “Heat and mass transfer in the boundary layers on an
exponentially stretching continuous surface”. J. Phys.D: Appl. Phys. 32, 577–
585(1999).
[4] Magyari, E., Keller, B., Exact Solutions for self similar boundary layer flows
induced by permeable stretching walls. Eur. J. Mech. BFluids19, 109–
122(2000).
[5] Elbashbeshy, E.M.A.,”Heat transfer over an exponentiallystretching
continuous surface with suction”. Arch. Mech. 53 (6),643–651(2001).
[6] Mukhopadhyay, S.,”Slip effects on MHD boundary layer flow over an
exponentially stretching sheet with suction/blowing and thermal radiation”.
Ain Shams Eng. J. 4, 485–491(2013).
[7] C.Y. Wang, “Stagnation point flow towards a shrinking sheet”, Int. J. Non-
Linear Mech. 43 pp377–382(2008).
[8] Sajid, M., Hayat, T., “Influence of thermal radiation on the boundary layer
flow due to an exponentially stretching sheet”. Int.Commun. Heat Mass
Transfer 35, 347–356(2008).
[9] El-Aziz, M.A., “Viscous dissipation effect on mixed convection flow of a
micropolar fluid over an exponentially stretching sheet” Can. J. Phys. 87, 359–
368(2009).
[10] Khan, W.A., Pop, I., “Flow and heat transfer over a continuously moving flat
plate in a porous medium” J. Heat Transfer 133 (5)(2011).
[11] Ishak, A., “MHD boundary layer flow due to an exponentially stretching sheet
with radiation effect”.Sains Malaysiana 40 (4), 391–395(2011).
[12] Cortell, R., “Heat transfer in a fluid through a porous medium over a
permeable stretching surface with thermal radiation and variable thermal
conductivity”. Can. J. Chem. Eng. 90, 1347–1355(2012).
[13] T.R. Mahapatra, A.G. Gupta, “Heat transfer in stagnation point flow towards a
stretching sheet”, Heat Mass Transfer 38 517–521(2002).
[14] Ibrahim, W., Shankar, B., Nandeppanavar, M.M., “MHD stagnation point
flow and heat transfer due to nanofluid towardsa stretching sheet” Int. J. Heat
Mass Transfer 56, 1–9(2013).
[15] Seddeek, M.A., “Effects of radiation and variable viscosity on a MHD free
convection flow past a semi-infinite flat plate with an aligned magnetic field in
the case of unsteady flow” Int. J. Heat Mass Transfer 45, 931–935(2003).
[16] Turkyilmazoglu, M., “Exact analytical solutions for heat and mass transfer of
MHD slip flow in nanofluids” Chem. Eng. Sci. 84,182–187(2012).
[17] Kumaran, V., Banerjee, A.K., Kumar, A.V., Vajravelu, K., “MHD flow past a
stretching permeable sheet” Appl. Math. Comput. 210, 26–32(2009).
1610 Punnam. Rajendar, L. Anand Babu and T.Vijaya Laxmi
[18] Raptis, A., Perdikis, C.,Takhar, H.S., “Effects of thermal radiation on MHD
flow” Appl. Math. Comput. 153, 645–649(2004).
[19] Renuka devi and Basker reddy.N. “Radiation in mass transfer effect on MHD
boundary layer flow due to an exponentially stretching sheet with heat source”
IJEIT vol.3(8) pp 33-39(2014).
[20] Poornima, T. and BhaskarRеddy, N.,”Radiation effеcts on MHD freе
convectivе boundary layеr flow of nano fluids ovеr a non linеar strеtching
sheеt”, Advancеs in Appliеd Sciencе Resеarch, Pеlagia Resеarch Library, Vol.
4(2), pp. 190-202(2013).
[21] L.Anandbabu,G.vijayalaxmi,”Radiation and mass transfer effects MHD
boundary layer flow due to an exponentially stretching sheet with heat source”
IJITE.vol(22) ISSN: 2395-2946(2016).
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