urface, the no-slip
condition gives
ucx,0v at 0y, (3)
where c is a positive constant and u and v are the
velocity components along
x
and y directions, re-
spectively. In Reza & Gupta [4], stream function in the
boundary layer was assumed in the form
() (),FW


(4)
where
is the kinematic viscosity and
11
22
,
cc
xy

 

 
 
, (5)
This gives the dimensionless velocity components from
(4) and (5) as
() ()UF W

,(),VF
 (6)
where

12
Uuc
and

12.Vvc
Using (6) in
the Navier-Stokes equations it was shown in [4] that
()F
and ()W
satisfy the following equations
2
1,
F
FF Fc
 

(7)
2,
F
WFWW c
 
 (8)
where 1
cand 2
care constants. From (6), no-slip condi-
tions (3) become
(0)0,(0) 1,FF
(9)
(0) 0,(0) 0.WW
(10)
Further from (1) and (6), the boundary condition for
()F
and ()W
at infinity are
() a
Fc
; 2
() ()
a
Fd
c

as ,
 (11)
1
()2 ()
b
Wd
c

as ,
 (12)
where

1
2
22
/dc



is the dimensionless displace-
ment thickness parameter and

1
2
11
/dc



is the
dimensionless pressure gradient parameter linked to the
free stream shear flow.
Reza and Gupta [4] ignored both the constants δ1 and
δ2 in (1). While pointing out that δ2 should be taken into
account (as mentioned in the Introduction), Lok, Amin &
Pop [5] rectified this error in [4]. However, these authors
in [5] lost sight of the constant δ1 in (1) and consequently
arrived at governing equations for the velocity distribu-
tion one of which is incorrect. Hence their analysis is of
doubtful validity.
Using the boundary conditions (11) and (12) in (7) and
(8), we get
Figure 1. A sketch of the physical problem.
M. REZA ET AL.
Copyright © 2010 SciRes. ENG
707
2
12 21
22
,2( )
aab
cc dd
cc
 
. (13)
Thus the governing equations for F(η) and W(η) be-
come
2
2
2,
a
FFFFc
 
 (14)
21
2
2( ).
ab
F
WFWW dd
c
 

(15)
Note that Equation (15) derived by Lok et al. [5] does
not include 1
d. Further the boundary condition (12) in
[5] is also erroneous due to the absence of 1
d. Substitu-
tion of (14) and (15) in the
x
and ymomentum equa-
tions followed by integration gives the pressure distribu-
tion (,)
p
xy in the flow as
2
22
21
22
(, )
12 ()constant.
2
pxy
c
aab
FF dd
cc



 


(16)
which can be found once ()F
is known.
Equations (14) and (15) subject to the boundary condi-
tions (9)-(12) are solved numerically by finite difference
method using Thomas algorithm (Fletcher [14]).
3. Results and Discussion
Figure 2 shows the variation of (, )U
with η at a
fixed value of ξ(= 0.5) for several values of /ac
when
the pressure gradient parameter 10.5d and / bc
1.0. It can be seen that at a given value of
, Uin-
creases with increase in /ac
. Further when b/c is very
small and equal to 0.05, say, the velocity profile at a
fixed value of
0.5
for several values of
/ac
with 10.5d shows a boundary layer structure
(see Figure 3) and the thickness of the boundary layer
decreases with increase in /ac
. From a physical point
of view, this stems from the fact that increase of straining
motion in the free stream (e.g., increase in /ac
for a
fixed value of c) leads to increase in acceleration of the
free stream. This results in thinning of boundary layer.
Figure 3 shows that when the free stream shear is neg-
ligible (/0.05bc for a given value of c), the flow has
a boundary layer structure because in this case straining
motion dominates over the shear. However, this bound-
ary layer structure is affected to a great extent in the
presence of considerable shear in the free stream (see
Figure 2).
The dimensionless displacement thickness 2
d is com-
puted for different values of /ac
from the solution of
Equation (14) subject to the boundary conditions (9) and
(11) and shown in the above Table 1. It may be noticed
that for /1ac
, displacement thickness is approxi-
mately zero (numerically). This is due to fact that when
/1ac
, the stretching velocity of the plate is precisely
equal to the irrotational straining velocity. From a physi-
cal point of view, the absence of boundary layer in this
case arises from the fact that although the flow is not
frictionless in a strict sense, the friction is uniformly dis-
tributed and does not therefore affect the motion. Stuart
[7] and Tamada [8] showed that the value of the dimen-
sional displacement thickness is 0.6479 for oblique stag-
nation point flow over a rigid plate. This result can be
compared with that of our problem by considering c = 0
in the boundary condition (3) which gives F(0) = 0 and
00F
. We have found that the value of the dis-
placement thickness is 2 0.64788d
. It may be noted
that in both the studies of Stuart [7] and Tamada [8], the
pressure gradient parameter 10
.
Figure 2. Variation of (,)U
with
at 0.5
for se-
veral values of /ac when 10.5d and /bc
= 1.0.
Figure 3. Variation of (,)U
with
at 0.5
for
several values of /ac when 10.5d and /bc
= 0.05.
M. REZA ET AL.
Copyright © 2010 SciRes. ENG
708
Table 1. Values of the displacement thickness d2 for
several values of /ac.
/ac
3 2 1.5 1.0
2
d 0.235278 0.2082290 0.1548889 7
9.3602152 10
Figure 4 displays the variation of
, U
with
at a fixed location ξ(= 0.5) for several values of the pres-
sure gradient parameter 1
dwhen /3ac and/bc
1.0. It may be seen that at a given value of η, the
horizontal velocity U decreases with increase in 1
d.
The streamline patterns for the oblique stagnation-
point flow are shown in Figures 5(a) and 5(b) for very
small value of the free stream shear / 0.05bc and
1 0.4d in two cases 1) / 0.2ac, 2) / 5.0ac
.
It can be seen that for / < 1ac , the stream lines are
slightly tilted towards the left. but when / acis large (=
5), the flow almost resembles that of an orthogonal stag-
nation-point flow as long as the free stream shear is very
small (see Figure 5(b)). For moderate value of free
stream shear (/1bc), the disposition of the streamlines
is shown in Figures 6(a) and 6(b) for / ac= 0.2, and
/ ac= 2.0, respectively. It is observed from Figures 5(a)
and 6(a ) that for a given value of / ac(= 0.2), with
increase in the free stream shear, the streamlines become
more tilted towards the left. We also find that with in-
crease in the straining motion in the free stream, the
streamlines are less and less tilted to the left. This is
plausible on physical grounds because with increase in
/ acfor a given value of /bc, the flow tends to re-
semble an orthogonal stagnation-point flow.
4. Summary
An exact solution of the Navier-Stokes equations is
given which represents steady two-dimensional oblique
Figure 4. Variation of (,)U
with
at 0.5
for
several values 1
d when /3.0ac
and /bc
= 1.0.
(a) when / ac= 0.2
(b) when / ac= 5.0
Figure 5. Streamline patterns for /bc
= 0.05 and d1 = 0.4
(a) when / ac
= 0.2 (b) when / ac
= 5.0.
stagnation-point flow of an incompressible viscous fluid
towards a surface stretched with velocity proportional to
the distance from a fixed point. It is shown that when the
free stream shear is negligible, the flow has a boundary
layer behaviour when the stretching velocity is less than
the free stream velocity (/1ac), and it has an inverted
boundary layer structure when just the reverse is true
(/1ac
). It is found that the obliquity of the flow to-
wards the surface increases with increase in /bc
. This
is consistent with the fact that increase in /bc
(for a
fixed value of / ac) results in increase in the shearing
motion which in turn leads to increased obliquity of the
flow towards the surface.
M. REZA ET AL.
Copyright © 2010 SciRes. ENG
709
(a) when / ac= 0.2.
(b) when / ac= 2.0
Figure 6. Streamline patterns for /bc = 1.0 and d1 = 0.4
(a) when / ac= 0.2; (b) when / ac= 2.0.
5. Acknowledgements
One of the authors (A. S. G) acknowledges the financial
assistance of Indian National Science Academy, New
Delhi for carrying out this work. Authors would also like
to acknowledge the use of the facilities and technical
assistance of the Center of Theoretical Studies at Indian
Institute of Technology, Kharagpur.
6. References
[1] L. J. Crane, “Flow Past a Stretching Plate,” Zeitschrift für
angewandte Mathematik und Physik, Vol. 21, 1970, pp.
645-657.
[2] T. C. Chiam, “Stagnation-Point Flow towards a Stretch-
ing Plate,” Journal of Physical Society of Japan, Vol. 63,
No. 6, 1994, pp. 2443-2444.
[3] T. R. Mahapatra and A. S. Gupta, “Heat Transfer in Stag-
nation-Point Flow towards a Stretching Sheet,” Heat and
Mass Transfer, Vol. 38, No. 6, 2002, pp. 517-521.
[4] M. Reza and A. S. Gupta, “Steady Two-Dimensional
Oblique Stagnation Point Flow towards a Stretching Sur-
face,” Fluid Dynamics Research, Vol. 37, No. 5, 2005, pp.
334-340.
[5] Y. Y. Lok, N. Amin and I. Pop, “Non-Orthogonal Stag-
nation Point towards a Stretching Sheet,” International
Journal of Non-Linear Mechanics, Vol. 41, No. 4, 2006,
pp. 622-627.
[6] P. G. Drazin and N. Riley, “The Navier-Stokes Equations:
A Classification of Flows and Exact Solutions,” Cam-
bridge University Press, Cambridge, 2006.
[7] J. T. Stuart, “The Viscous Flow near a Stagnation Point
when External Flow has Uniform Vorticity,” Journal of
the Aero/Space Sciences, Vol. 26, 1959, pp. 124-125.
[8] K. Tamada, “Two-Dimensional Stagnation-Point Flow
Impinging Obliquely on a Plane Wall,” Journal of Physical
Society of Japan, Vol. 46, No. 1, 1979, pp. 310-311.
[9] J. M. Dorrepaal, “An Exact Solution of the Navier-Stokes
Equation which Describes Non-Orthogonal Stagnation-
Point Flow in Two Dimensions,” Journal of Fluid Me-
chanics, Vol. 163, 1986, pp. 141-147.
[10] D. Weidman and V. Putkaradzeb, “Axisymmetric Stag-
nation Flow Obliquely Impinging on a Circular Cylin-
der,” European Journal of Mechanics - B/Fluids, Vol. 22,
No. 2, 2003, pp. 123-131.
[11] B. S. Tilley, P. D. Weidman, “Oblique Two-Fluid Stag-
nation-Point Flow,” European Journal of Mechanics -
B/Fluids, Vol. 17, No. 2, 1998, pp. 205-217.
[12] T. R. Mahapatra, S. Dholey and A. S. Gupta, “Heat Transfer
in Oblique Stagnation-Point Flow of an Incompressible
Viscous Fluid towards a Stretching Surface,” Heat and
Mass Transfer, Vol. 43, No. 8, 2007, pp. 767-773.
[13] T. R. Mahapatra, S. Dholey and A. S. Gupta, “Oblique
Stagnation-Point flow of an Incompressible Visco-Elastic
Fluid towards a Stretching Surface,” International Jour-
nal of Non-Linear Mechanics, Vol. 42, No. 3, 2007, pp.
484-499.
[14] C. A. J. Fletcher, “Computational Techniques for Fluid
Dynamics,” Vol. 2, Springer-Verlag, Berlin, 1988.