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.