We present a purely analytic solution to the steady three-dimensional viscous stagnation point flow of second grade fluid over a heated flat plate moving with some constant speed. The analytic solution is obtained by a newly developed analytic technique, namely, homotopy analysis method. By giving a comparison with the existing results, it is shown that the obtained analytic solutions are highly accurate and are in good agreement with the results already present in literature. Also, the present analytic solution is uniformly valid for all values of the dimensionless second grade parameter α. The effects of α and the Prandtl number Pr on velocity and temperature profiles are discussed through graphs.

1.
Blasius
,
H.
, 1908, “
Grenzschichten in Flussigkeiten mit Kleiner Reibung
,”
Z. Math. Phys.
,
56
, pp.
1
17
.
2.
Heimenz
,
K.
, 1911, “
Die Grenzschicht an einem in den gleichformigen Flussigkeitsstrom eingetauchten geraden Kreiszylinder
,” Ph.D. thesis, Gottingen.
3.
Glauert
,
M. B.
, 1956, “
The Laminar Boundary Layer on Oscillating Plates and Cylinders
,”
J. Fluid Mech.
0022-1120,
1
, pp.
97
110
.
4.
Stuart
,
J. T.
, 1959, “
The Viscous Flow Near a Stagnation Point When the External Flow Has Uniform Vorticity
,”
J. Aerosp. Sci.
0095-9820,
26
, pp.
124
125
.
5.
Davey
,
A.
,1961, “
Boundary Layer Flow at a Saddle Point of Attachment
,”
J. Fluid Mech.
0022-1120,
10
, pp.
593
610
.
6.
Tamada
,
K. J.
, 1979, “
Two-Dimensional Stagnation Point Flow Impinging Obliquely on a Plane Wall
,”
J. Phys. Soc. Jpn.
0031-9015,
46
, pp.
310
311
.
7.
Dorepaal
,
J. M.
, 1986, “
An Exact Solution of the Navier-Stokes Equation Which Describes Non-Orthogonal Stagnation-Point Flow in two Dimensions
,”
J. Fluid Mech.
0022-1120,
163
, pp.
141
147
.
8.
Zumbrunnen
,
D. A.
, 1991, “
Convective Heat and Mass Transfer in The Stagnation Region of a Laminar Planar jet Impinging on a Moving Surface
,”
ASME J. Heat Transfer
0022-1481,
113
, pp.
563
570
.
9.
Wang
,
C. Y.
, 1985, “
The Unsteady Oblique Stagnation Point Flow
,”
Phys. Fluids
0031-9171,
28
, pp.
2046
2049
.
10.
Ariel
,
P. D.
, 1994, “
Three-Dimensional Stagnation Point Flow of a Viscoelastic Fluid
,”
Mech. Res. Commun.
0093-6413,
21
, pp.
389
396
.
11.
Ariel
,
P. D.
, 1994, “
Stagnation Point Flow With Suction: An Approximate Solution
,”
ASME J. Appl. Mech.
0021-8936,
61
, pp.
976
978
.
12.
Ariel
,
P. D.
, 2001, “
Axisymmetric Flow of a Second Grade Fluid Past a Stretching Sheet
,”
Int. J. Eng. Sci.
0020-7225,
39
, pp.
529
553
.
13.
Bian
,
X.
, and
Rangel
,
R. H.
, 1996, “
The Viscous Stagnation Flow Solidification Problem
,”
Int. J. Heat Mass Transfer
0017-9310,
39
, pp.
3581
3594
.
14.
Mahapatra
,
T. R.
, and
Gupta
,
A. S.
, 2002, “
Heat Transfer in Stagnation Point Flow Towards a Stretching Sheet
,”
Heat Mass Transfer
0947-7411,
38
, pp.
517
521
.
15.
Baris
,
S.
, 2003, “
Steady Three-Dimensional Flow of a Second Grade Fluid Towards a Stagnation Point at a Moving Flat Plate
,”
Turk. J. Eng. Environ. Sci.
1300-0160,
27
, pp.
21
29
.
16.
Fosdick
,
R. L.
, and
Rajagopal
,
K. R.
, 1978, “
Uniqueness and Drag for Fluids of Second Grade in Steady Motion
,”
Int. J. Non-Linear Mech.
0020-7462,
13
, pp.
131
137
.
17.
Dunn
,
J. E.
, and
Fosdick
,
R. L.
, 1974, “
Thermodynamics, Stability and Boundedness of Fluids of Complexity and Fluids of Second Grade
,”
Arch. Ration. Mech. Anal.
0003-9527,
56
, pp.
191
252
).
18.
Dunn
,
J. E.
, and
Rajagopal
,
K. R.
, 1995, “
Fluids of Different Type—Critical Review and Thermodynamic Analysis
,”
Int. J. Eng. Sci.
0020-7225,
33
, pp.
689
729
.
19.
Nayfeh
,
A. H.
, 2000,
Perturbation Methods
,
Wiley
,
New York
.
20.
Liao
,
S. J.
, 2003,
Beyond Perturbation: Introduction to Homotopy Analysis Method
,
Chapman and Hall∕CRC
,
Boca Raton, FL
.
21.
Adomian
,
G.
, 1975, “
Nonlinear Stochastic Differential Equations
,”
J. Math. Anal. Appl.
0022-247X,
55
, pp.
414
452
.
22.
Karmishin
,
A. V.
,
Zhukov
,
A. T.
, and
Kolosov
,
V. G.
, 1990,
Methods of Dynamics Calculation and Testing for Thin-Walled Structures
,
Mashiniostronie
,
Moscow
, in Russian.
23.
Lyapunov
,
A. M.
, 1992,
General Problems on Stability of Motions
,
Taylor and Francis
,
London
, English translation.
24.
Liao
,
S. J.
, 2006, “
An Analytic Solution of Unsteady Boundary-Layer Flows Caused by an Impulsively Stretching Plate
,”
Commun. Nonlinear Sci. Numer. Simul.
1007-5704,
11
, pp.
326
339
.
25.
Xu
,
H.
, and
Liao
,
S. J.
, 2005, “
Series Solutions of Unsteady Magnetohydrodynamic Flows of Non-Newtonian Fluids Caused by an Impulsively Stretching Plate
,”
J. Non-Newtonian Fluid Mech.
0377-0257,
129
, pp.
46
55
.
26.
Ali
,
A.
, and
Mehmood
,
A.
, 2008, “
Homotopy Analysis of Unsteady Boundary-Layer Flow Adjacent to Permeable Stretching Surface in a Porous Medium
,”
Commun. Nonlinear Sci. Numer. Simul.
1007-5704,
13
(
2
), pp.
340
349
.
27.
Mehmood
,
A.
, and
Ali
,
A.
, 2006, “
Analytic Solution of Generalized Three-dimensional Flow and Heat Transfer Over a Stretching Plane Wall
,”
Int. Commun. Heat Mass Transfer
0735-1933,
33
, pp.
1243
1252
.
28.
Liao
,
S. J.
, 1999, “
A Uniformly Valid Analytical Solution of 2D Viscous Flow Past a Semi Infinite Flat Plate
,”
J. Fluid Mech.
0022-1120,
385
, pp.
101
128
.
29.
Liao
,
S. J.
, 1999, “
An Explicit, Totally Analytic Approximate Solution for Blasius Viscous Flow Problems
,”
Int. J. Non-Linear Mech.
0020-7462,
34
, pp.
759
778
.
30.
Liao
,
S. J.
, and
Pop
,
I.
, 2004, “
Explicit Analytic Solution for Similarity Boundary-Layer Equations
,”
Int. J. Heat Mass Transfer
0017-9310,
47
(
1
), pp.
75
85
.
31.
Liao
,
S. J.
, and
Campo
,
A.
, 2002, “
Analytic Solutions of the Temperature Distribution in Blasius Viscous Flow Problems
,”
J. Fluid Mech.
0022-1120,
453
, pp.
411
425
.
32.
Liao
,
S. J.
, and
Cheung
,
K. F.
, 2003, “
Homotopy Analysis of Nonlinear Progressive Waves in Deep Water
,”
J. Eng. Math.
0022-0833,
45
(
2
), pp.
105
116
.
33.
Xu
,
H.
, 2004, “
An Explicit Analytic Solution for Free Convection About a Vertical Flat Plate Embedded in a Porous Media by Means of Homotopy Analysis Method
,”
Appl. Math. Comput.
0096-3003,
158
, pp.
433
443
.
34.
Liao
,
S. J.
, 2004, “
On the Homotopy Analysis Method for Nonlinear Problems
,”
Appl. Math. Comput.
0096-3003,
147
, pp.
499
513
.
35.
Liao
,
S. J.
, 2003, “
On the Analytic Solution of Magnetohydrodynamic Flows of Non-Newtonian Fluid Over a Stretching Sheet
,”
J. Fluid Mech.
0022-1120,
488
, pp.
189
212
.
36.
Liao
,
S. J.
, 2005, “
A New Branch of Solutions of Boundary-layer Flows Over an Impermeable Stretched Plate
,”
Int. J. Heat Mass Transfer
0017-9310,
48
(
12
), pp.
2529
2539
.
37.
Yang
,
C.
, and
Liao
,
S. J.
, 2006, “
On the Explicit, Purely Analytic Solution of Von Kármán Swirling Viscous Flow
,”
Commun. Nonlinear Sci. Numer. Simul.
1007-5704,
11
, pp.
83
93
.
38.
Wang
,
C.
,
Zhu
,
J. M.
,
Liao
,
S. J.
, and
Pop
,
I.
, 2003, “
On the Explicit Analytic Solutions of Cheng-Chang Equations
,”
Int. J. Heat Mass Transfer
0017-9310,
46
(
10
), pp.
1855
1860
.
39.
Allan
,
F. M.
, and
Syam
,
M. I.
, 2005, “
On Analytic Solution of the Non-Homogeneous Blasius Problem
,”
J. Comput. Appl. Math.
0377-0427,
182
, pp.
355
365
.
You do not currently have access to this content.