Engineering, 2010, 2, 705-709
doi:10.4236/eng.2010.29091 Published Online September 2010 (http://www.SciRP.org/journal/eng)
Copyright © 2010 SciRes. ENG
Some Aspects of Non-Orthogonal Stagnation-Point Flow
towards a Stretching Surface
Motahar Reza1, Anadi Sankar Gupta2
1Department of Mathematics, National Institute of Science & Technology, Berhampur, India
2Department of Mathematics, Indian Institute of Technology, Kharagpur, India
E-mail: reza@nist.edu
Received May 12, 2010; revised July 21, 2010; accepted August 3, 2010
Abstract
The problem of steady two-dimensional oblique stagnation-point flow of an incompressible viscous fluid
towards a stretching surface is reexamined. Here the surface is stretched with a velocity proportional to the
distance from a fixed point. Previous studies on this problem are reviewed and the errors in the boundary
conditions at infinity are rectified. It is found that for a very small value of shear in the free stream, the flow
has a boundary layer structure when/1ac, where ax and cx are the free stream stagnation-point velocity
and the stretching velocity of the sheet, respectively,
x
being the distance along the surface from the stag-
nation-point. On the other hand, the flow has an inverted boundary layer structure when /1ac. It is also
observed that for given values of /ac
and free stream shear, the horizontal velocity at a point decreases with
increase in the pressure gradient parameter.
Keywords: Oblique Stagnation-Point Flow, Stretching Surface
1. Introduction
The study of the flow of an incompressible viscous fluid
over a stretching surface has important bearing on sev-
eral technological and industrial processes. Problems
such as the extrusion of polymers in melt-spinning, glass
blowing, spinning of fibers a several metallurgical as
well as metal-working processes involve certain aspects
of flow over stretching sheets. Crane [1] obtained a simi-
larity solution in closed analytical form for steady two-
dimensional flow of an incompressible viscous fluid
caused solely by the stretching of an elastic sheet which
moves in its own plane with a velocity varying linearly
with distance from a fixed point.
Chiam [2] investigated steady two-dimensional or-
thogonal and oblique stagnation-point flow of an incom-
pressible viscous fluid towards a stretching surface in the
case when the parameterbrepresenting the ratio of the
strain rate of the stagnation-point flow to that of the
stretching surface is equal to unity. By removing this
highly restrictive assumption (1b), Mahapatra and
Gupta [3] analyzed the steady two-dimensional orthogo-
nal stagnation-point flow of an incompressible viscous
fluid to-wards a stretching surface in the general case
1b. They observed that the structure of the boundary
layer depends crucially on the value of b. Reza and
Gupta [4] generalized the problem of an oblique stagna-
tion-point flow over a stretching surface by Chiam [2] to
include surface strain rate different from that of the
stagnation flow. But since the displacement thickness
arising out of the boundary layer on the surface was ig-
nored in their boundary condition at infinity, the analysis
in [4] is of doubt full validity. This was rectified in a
paper by Lok, Amin and Pop [5]. However, these authors
[5] did not take into account the pressure gradient pa-
rameter in the boundary condition at infinity. This is a
serious omission since the pressure gradient parameter is
linked to the free stream shear in the oblique stagna-
tion-point flow (Drazin and Riley [6]). Hence the results
of the paper in [5] are also of doubtful validity.
It is noted that planar oblique stagnation-point flow of
an incompressible viscous flow of an incompressible vis-
cous fluid towards affixed rigid surface was first studied
by Stuart [7]. This problem was later independently in-
vestigated by Tamada [8] and Dorrepaal [9]. The ana-
logue of the planar oblique stagnation-point flow to stag-
nation flow obliquely impinging on a rigid circular cyl-
inder was discussed by Weidman and Putkaradze [10].
Exact similarity solutions for impingement of two vis-
cous immiscible oblique stagnation flows forming a flat
M. REZA ET AL.
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706
interface was given by Tilley and Weidman [11]. On the
other hand heat transfer in oblique stagnation-point flow
of an incompressible viscous fluid towards stretching
surface was investigated by Mahapatra, Dholey and
Gupta [12]. Further oblique stagnation-point flow of a
viscoelastic fluid towards a stretching surface was stud-
ied by Mahapatra, Dholey and Gupta [13].
The objective of the present paper is to rectify the er-
rors in [4] and [5] and give a correct solution to the
above problem. It is worth pointing out that an oblique
stagnation-point flow occurs when a separated viscous
flow reattaches to a surface.
2. Flow Analysis
Consider the steady two-dimensional flow near a stagna-
tion point when an incompressible viscous fluid im-
pinges obliquely on an elastic surface coinciding with the
plane 0y, the flow being confined to 0y. Two
equal and opposite forces are applied along the x-axis so
that the surface is stretched keeping the origin fixed, as
shown in Figure 1. The velocity components in the in-
viscid free stream along the x and y directions are
0102
2( ),()UaxbyV ay
  , (1)
respectively, where a and b are constants. Further
2
is the displacement thickness arising out of the
boundary layer on the stretching surface and1
is the
parameter which controls the horizontal pressure gradi-
ent that produces the shear flow. Note that the whole
flow field given by (1) may be viewed as being com-
posed of an orthogonal stagnation-point flow combined
with a horizontal shear flow. The corresponding stream
function for the above velocity distribution is
2
021
()()ax yby


(2)
There appears a boundary layer on the surface at high
Reynolds number. At the stretching surface, 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.
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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.
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