American Journal of Computational Mathematics, 2011, 1, 129-133
doi:10.4236/ajcm.2011.12013 Published Online June 2011 (http://www.scirp.org/journal/ajcm)
Copyright © 2011 SciRes. AJCM
Magneto Hydrodynamic Orthogonal Stagnation Point
Flow of a Power-Law Fluid toward a Stretching Surface
Manisha Patel1, Munir Timol2
1Department of Mat hem at ic s, Sarvajanik College of Engineering & Technology, Surat, India
2Department of Mat hem at ic s, Veer Narmad South Gujarat University, Surat, India
E-mail: {manishapramitpatel, mgtimol }@gmail.com
Received April 27, 2011; revised May 25, 2011; accepted June 5, 2011
Abstract:
Steady two dimensional MHD stagnation point flow of a power law fluid over a stretching surface is inves-
tigated when the surface is stretched in its own plane with a velocity proportional to the distance from the
stagnation point. The fluid impinges on the surface is considered orthogonally. Numerical and analytical
solutions are obtained for different cases.
Keywords: Stagnation Point Flow, Galerkin’s, Finite Difference Method, Stretching Surface.
1. Introduction
The stagnation point is a point on the surface of a body
submerged in a fluid flow where the fluid velocity is zero.
Stagnation flow, describing the fluid motion near the
stagnation region, exists on all solid bodies moving in a
fluid. The stagnation region encounters the highest pres-
sure, the highest heat transfer, and the highest rates of
mass deposition. The stud y of flow over a stretching sur-
face has generated much interest in recent years in view
of its numerous industrial applications such as extrusion
of polymer sheets, continuous stretching, rolling and
manufacturing plastic films and artificial fibers. The flow
near a stagnation point has attracted many investigations
during the past several decades because of its wide ap-
plications such as cooling of electronic devices by fans,
cooling of nuclear reactors, and many hydrodynamic
processes [1-5].
The two-dimensional flow of a fluid near a stagnation
point was first examined by Hiemenz [6], who demon-
strated that the Navier-Stokes equations governing the
flow can be reduced to an ordinary differential equation
of third order using similarity transformation. Later the
problem of stagnation point flow was extended in nu-
merous ways to include various physical effects. The
results of these studies are of great technical importance,
for example in the prediction of skin-friction as well as
heat/mass transfer near stagnation regions of bodies in
high speed flows and also in the design of thrust bearings
and radial diffusers, drag reduction, transpiration cooling
and thermal oil recovery. Axisymmetric three-dimen-
sional stagnation point flow was studied by Homann [7].
Either in the two or three-dimensional case Navier-
Stoke’s equations governing the flow are reduced to an
ordinary differential equation of third order using a simi-
larity transformation. In hydromagnetics, the problem of
Hiemenz flow was chosen by Na [8] to illustrate the so-
lution of a third-order boundary value problem using the
technique of finite differences. An approximate solution
of the same problem has been provided by Ariel [9]. At-
tai [1] has made an analysis of the steady lamin ar flow in
a porous medium of an incompressible viscous fluid im-
pinging on a permeable stretching surface with heat gen-
eration. The steady magneto hydrodynamic (MHD)
mixed convection stagnation point flow towards a verti-
cal surface immersed in an incompressible micropolar
fluid with prescribed wall heat flux was investigated by
Bachok et al. [4]. They have transformed the governing
partial differential equations into a system of ordinary
differential equations, which is then solved numerically
by a finite-difference method. Hayd et al. [2] have stud-
ied the boundary layer equations for axisymmetric point
flow of power-law electrically conducting fluid through a
porous medium with transverse magnetic field. McLeod
and Rajagopal [10] have discussed the uniqueness of the
exact analytical solution of the flow of a Newtonian
fluid due to a stretching boundary. On the other hand,
Rajagopal et al. [10,11] obtained an approximate ma-
thematical solution of the viscoelastic boundary layer
flow over a stretching plastic sheet and studied the flow
M. PATEL ET AL.
Copyright © 2011 SciRes. AJCM
130
behaviors.
Chiam [12] and Mahapatra and Gupta [13] have inves-
tigated the steady two-dimensional stagnation point flow
of an incompressible viscous fluid over a flat deformable
sheet when the sheet is stretched in its own plane with a
velocity proportional to the distance from the stagnation
point. It is shown that a boundary layer is formed near
the stretching surface and that the structure of this boun-
dary layer depends on the ratio of the velocity of the
stretching surface to that of the frictionless potential flow
in the neighborhood of the stagnation point. Recently,
Patel et al. [5] have discussed the numerical solution for
steady two-dimensional MHD forward stagnation point
flow introducing the Crocco’s independent variable with
Galerkin’s. In stagnation point flow, a rigid wall or a
stretching surface occupies the entire horizontal x-axis,
the fluid domain is 0y and the flow impinges on the
wall either orthogonal or at an arbitrary angle of inci-
dence.
In this paper we investigate steady two dimensional
stagnation point flow of a power law fluid over a stret-
ching surface, when the surface is stretched in its own
plane with a velocity proportional to the distance from
the stagnation point. The fluid impinges on the surface is
considered orthogonally. Numerical and analytical solu-
tions are obtained for different flow geometries along
with graphical presentation.
2. Flow Analysis
The well-known Ostwald-de-Wale model of power-law
fluid is purely phenomenological; however, it is useful in
that approximately describes a great number of real
non-Newtonian fluids. This model behaves properly un-
der tensor deformation. Use of this model alone assumes
that the fluid is purely viscous. Mathematically it can be
represented in the form
1
2
1:
2
n
m


 



(1)
where m and n are called the consistency and flow
behavior indices respectively. If 1n, the fluid is called
pseudo plastic power law fluid and if 1n, it is called
dilatants power law fluid since the apparent viscosity
decreases or increases with the increase shear of rate ac-
cording as 1n or 1n, if 1n the fluid will be
Newtonian.
Consider the steady two-dimensional stagnation-point
flow of power-law fluid flowing towards a flat surface
coinciding with the plane 0y
, the flow being con-
fined to the region 0y. Two equal and opposing
forces are applied on the stretching surface along the
x-axis so that the surface is stretched keeping the origin
fixed as shown in Figure 1. The MHD equations for
steady two-dimensional stagnation-point flow in the
boundary layer towards the stretching surface are, in the
usual notation,
0
uv
xy

 (2)
2
0()
xy
B
uu U
uvU uU
xy xy
 
 
  (3)
Here the magnetic Reynolds number is assumed to be
very small so that the induced magnetic field is neglected.
Here u and v are the velocity components along the x and
y direction, respectively. Further
x
y
is stress tensor in
the direction of Y-axis perpendicular to X-axis. ()Ux
stands for the stagnation-point velocity in the inviscid
free stream. The stress tensor is defined by Equation (1).
In the present problem we have
and 0
au
ay when 1
a
c. (4)
Therefore the shear stress will convert as:
n
xy u
ky



when 1
a
c (5)
Figure 1. A sketch of the physical problem.
Figure 2. Variation of
(0)F with o
S
for = 1.1a/c .
M. PATEL ET AL.
Copyright © 2011 SciRes. AJCM
131
Table 1.
n 0
o
S 0.5
o
S 1.0
o
S 1.5
o
S 2.0
o
S
0.4 0.1123 0.1252 0.1445 0.1564 0.1601
0.8 0.1512 0.1668 0.1748 0.1799 0.1869
1.2 0.1924 0.2214 0.2356 0.2455 0.2662
1.5 0.2358 0.2514 0.2636 0.2741 0.2852
2.0 0.2897 0.2999 0.3121 0.3255 0.3412
Table 2.
n 0
o
S 0.5
o
S 1.0
o
S 1.5
o
S 2.0
o
S
0.4 1.2216 1.3625 1.4886 1.6289 1.7865
0.8 1.0023 1.0423 1.0858 1.1821 1.3226
1.2 0.8984 0.9672 1.0115 1.1000 1.2101
1.5 0.8864 0.9355 1.0021 1.0552 1.1001
2.0 0.8876 0.9301 0.9864 0.9945 1.0477
Table 3.
n 0
o
S 0.5
o
S 1.0
o
S 1.5
o
S 2.0
o
S
0.4 3.8771 4.1332 4.4755 4.8163 5.2445
0.8 2.3854 2.5446 2.5845 2.8354 2.8714
1.2 1.9328 1.9454 2.0023 2.1426 2.1735
1.5 1.7001 1.7537 1.8817 1.9583 1.9966
2.0 1.5481 1.5841 1.6120 1.6526 1.6998
Table 4. Results of finite difference method of Case-I.
No. of iteration 0
o
S 0.5
o
S1.0
o
S 1.5
o
S2.0
o
S
1 1.02326 1.2326 1.2326 1.2326 1.2326
2 1.1688 1.3593 1.5393 1.7882 1.8385
3 1.1743 1.3637 1.5407 1.7890 1.8339
4 1.1715 1.3618 1.5392 1.7662 1.8328
5 1.1717 1.3621 1.5394 1.7914 1.8330
6 1.1717 1.3620 1.5394 1.7914 1.8330
Now the momentum Equation (3) become, in non-di-
mensional fo r m, when 1
a
c
2
0()
nB
uu UKu
uvU uU
xy xyy


 
 

 

(6)
The boundary conditions are:
, 0 y0ucxv at  (7)
(), , yuU xaxvayat  (8)
where a and c are positive constants.
Introducing stream function (, )
Y
, where
uy
and =v
x
(9)
Following Labropulu [3], we assume that
()cxF y
(10)
Using Equations (9) and (10) into the Equation (6),
we obtained

12
()()() ()
1
nn
nF yFyFyF y
n

 


2
2
2
() ()0
oo
aa
FySFyS
cc


(11)
with the boundar y conditions:
(0) 0F
, (0) 1F
, ()a
Fc
 (12)
where 2
0
o
B
Sc
is the magnetic parameter.
Case-I: Newtonian fl uid: cons ider 1n and /1ac
Equation (11) is converted in the following equation
2
()()()()()1 0
oo
FyFyFy FySFy S
 
  (13)
with the boundar y conditions:
(0) 0F
, (0) 1F
, ()a
Fc
 (14)
Case-II: if we consider case for 1n, and let 0U
(i.e. 0a
) then the Equation (11) is converted in
2
()()()()() 0
o
Fy FyFyFySFy
 
 
(15)
Subject to the boundary conditions;
(0)0,(0)1,( )0FFF
 (16)
It is interesting to note that the above BVP Equation (15)
has a simple analytic solution of the form
Figure 3. Variation of
(0)F with o
S
for = 1.5a/c .
Figure 4. Variation of
(0)F with o
S
for = 2.0a/c .
M. PATEL ET AL.
Copyright © 2011 SciRes. AJCM
132
Figure 5. Variation of (0)F with o
S
for =1a/c ,
=1n.
Figure 6. Variation of ()
y with
y
.
1
( )[1exp()]
F
ypy
p
 (17)
where 1/2
0
(1 )pS
Method of solution: The transformed momentum
Equation (11) is solved using Galerkin’s method after
introducing Crocco’s variables [5,14]. The equation for
Case-I (Equation (13)) is solved by T. Y. Na [8] using
Finite difference method the graphical presentation of
those tabular values ( Table 3) is shown in Figu re 2. And
the equation of Case-II (Equation (15)) is solved analyti-
cally.
Results and discussion: The computed variation of

0F with o
S and n is summarized in Tables 1 and 2
for /1.1, 1.5ac and 2.0, respectively. It can be con-
clude (from the above Tables 1-3, Figures 2-4) that fo r a
fixed value of o
S,
0F increases with increase in n
in a small neighbourhood of /1ac then it decreases
for other val ues of a/c. Fr om Tables 1-3, it is clear that for
a fixed value of n and /ac, the value of
0F
de-
creases with increase in o
S. This behaviour may perhaps
be attributed to the change in the character of the flow as
/ac changes its values.
It is interesting to note that when the velocity of the
stretching surface is equal to the velocity of the inviscid
stream
ac
, Equation (16) ad mit to the ex act analytic
solution ()
F
yy
. From this we can infer that when
ac
, the velocity distribution near the stretching sur-
face is the same as athat of a flow away from the surface
so that no boundary layer is formed near the surface. It
should be mentioned here th at when , the flow is not fric-
tionless in a strict sense. In fact in this case the friction is
uniformly distributed and does not, therefore, affect the
motion.
If we consider 1n
then the entire flow geometry is
reduced in Newtonian fluid, which is discussed as a
Case-I with ac
(i.e. /1ac, Table 4, Figure 5).
For that case
0F
increases with increase in the val-
ue of o
S for a fixed value of 1n. If we consider
stream velocity is zero
1n
then the flow is treated
as a uniform stagnation point flow, in this case
F
y
decreases with increase in y for a fixed value of o
S.
The graphical representation is shown in Figure 6.
3. References
[1] H. A. Attai, “Stagnation Point Flow towards a Stretching
Surface through a Porous Medium with Heat Generation,”
Turkish Journal of Engineering & Environmental Scien-
ces, Vol. 30, No. 5, 2006, pp. 299-306.
[2] F. M. Hayd and I. A. Hassanien, “Magnetohydrodynamic
and Constant Suction/Injection Effects of Axisymmetric
Stagnation Point Flow and Mass Transfer for Power-Law
Fluids,” Indian Journal of Pure and Applied Mathematics,
Vol. 17, No. 1, 1986, pp. 108-120.
[3] F. Labropulu and D. Li, “Stagnation Point Flow of a
Second-Grade Fluid with Slip,” International Journal of
Non-Linear Mechanics, Vol. 43, No. 9, 2008, pp. 941-947.
doi:10.1016/j.ijnonlinmec.2008.07.004
[4] B. Norfifah and I. Anuar, “MHD Stagnation-Point Flow
of a Micropolar Fluid with Prescribed Wall Heat Flux,”
European Journal of Scientific Research, Vol. 35 No. 3,
2009, pp. 436-443.
[5] M. Patel and M. G. Timol, “Numerical Solution of Steady
Two-Dimensional MHD forward Stagnation Point Flow,”
Applied Mathematical Science, Vol. 3, No. 4, 2009, pp.
187-193.
[6] K. Hiemenz, “Die Grenzschicht an Einem in Den Gleich-
formingen Flussigkeitsstrom Eingetauchten Graden Krei-
szylinder,” Dingler’s Polytechnic Journal, Vol. 326, 1911,
pp. 321-324.
[7] F. Homann, “Der Einfluss Grosser Zahigkeit bei der
Stromung um den Zylinder und um die Kugel,” Journal
of Applied Mathematics and Mechanics/Zeitschrift für
Angewandte Mathematik und Mechani, Vol. 16, No. 3,
1936, pp. 153-164. doi:10.1002/zamm.19360160304
[8] T. Y. Na, “Computational Methods in Engineering
Boundary Value Problems,” Academic Press, New York,
1979.
M. PATEL ET AL.
Copyright © 2011 SciRes. AJCM
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[9] P. D. Ariel, “Hiemenz Flow in Hydromagnetics,” Acta
Mechanica, Vol. 103, No. 1-4, 1994, pp. 31-43.
[10] K. R. Rajagopal, T. Y. Na and A. S. Gupta, “A Non Sim-
ilar Boundary Layer on a Stretching Sheet in a
Non-Newtonian Fluid with Uniform Free Stream,” Jour-
nal of Mathematical Physics, Vol. 21, No. 2, 1987, pp.
189-200.
[11] K. R. Rajagopal, T. Y. Na and A. S. Gupta, “Flow of a
Viscoelastic Fluid over a Stretching Sheet,” Rheologica
Acta, Vol. 23, No. 2, 1984, pp. 213-215.
doi:10.1007/BF01332078
[12] T. C. Chiam, “Stagnation-Point Flow towards a Stretch-
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63, No. 6, 1994, pp. 2443-2444.
doi:10.1143/JPSJ.63.2443
[13] T. R. Mahapatra and A. S. Gupta, “Heat Transfer in
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doi:10.1007/s002310100215
[14] T. C. Chiam, “Solution for the Flow of a Conducting
Power-Law Fluid in a Transverse Magnetic Field and
with a Pressure Gradient Using Crocco Variables,” Acta
Mechanica, Vol. 137, No. 3-4, 1999, pp. 225-235.
Nomenclature:
u, v – velocity components in X, Y directions respec-
tively
U – main stream velocity in X direction
a, c – positive constants
,, ij
– usual shear stress tensor
, ij
e – usual rate of deformation tensor/ Strain rate
component
y
x
– stress tensor in the direction of X-axis perpen-
dicular to Y-axis.
K – kinematic Viscosity
m – Physical constant
n – flow behavior indices
0
B – Imposed magnetic field
M
HD – Magneto hydro dynamics
– field density
– Electrical conductivity
p
C – Specific heat
– Stream function
F
– Similarity function
o
S – Magnetic parameter – 2
0
B