In this paper, the nonsimilarity boundary-layer flows of second-order fluid over a flat sheet with arbitrary stretching velocity are studied. The boundary-layer equations describing the steady laminar flow of an incompressible viscoelastic fluid past a semi-infinite stretching flat sheet are transformed into a partial differential equation with variable coefficients. An analytic technique for highly nonlinear problems, namely, the homotopy analysis method, is applied to give convergent analytical approximations, which agree well with the numerical results given by the Keller box method. Furthermore, the effects of physical parameters on some important physical quantities, such as the local skin-friction coefficient and the boundary-layer thickness, are investigated in detail. Mathematically, this analytic approach is rather general in principle and can be applied to solve different types of nonlinear partial differential equations with variable coefficients in physics.

1.
Beard
,
D. W.
, and
Walters
,
K.
, 1964, “
Elastico-Viscous Boundary Layer Flows
,”
Proc. Cambridge Philos. Soc.
0068-6735,
60
, pp.
667
674
.
2.
Rajagopal
,
K. R.
,
Na
,
T. A.
, and
Gupta
,
A. S.
, 1984, “
Flow of a Viscoelastic Fluid Over a Stretching Sheet
,”
Rheol. Acta
0035-4511,
23
, pp.
213
215
.
3.
Vajravelu
,
K.
, and
Rollins
,
D.
, 1991, “
Heat Transfer in a Viscoelastic Fluid Over a Stretching Sheet
,”
J. Math. Anal. Appl.
0022-247X,
158
, pp.
241
255
.
4.
Vajravelu
,
K.
, and
Roper
,
T.
, 1999, “
Flow and Heat Transfer in a Second Grade Fluid Over a Stretching Sheet
,”
Int. J. Non-Linear Mech.
0020-7462,
34
, pp.
1031
1036
.
5.
Sarma
,
M. S.
, and
Rao
,
B. B.
, 1998, “
Heat Transfer in a Viscoelastic Fluid Over a Stretching Sheet
,”
J. Math. Anal. Appl.
0022-247X,
222
, pp.
268
275
.
6.
Dandapat
,
B. S.
, and
Gupta
,
A. S.
, 1989, “
Flow and Heat Transfer in a Viscoelastic Fluid Over a Stretching Sheet
,”
Int. J. Non-Linear Mech.
0020-7462,
24
, pp.
215
219
.
7.
Pontrelli
,
G.
, 1995, “
Flow of a Fluid of Second Grade Over a Stretching Sheet
,”
Int. J. Non-Linear Mech.
0020-7462,
30
, pp.
287
293
.
8.
Cortell
,
R.
, 2006, “
A Note on Flow and Heat Transfer of a Viscoelastic Fluid Over a Stretching Sheet
,”
Int. J. Non-Linear Mech.
0020-7462,
41
, pp.
78
85
.
9.
Liao
,
S. J.
, 1992, “
The Proposed Homotopy Analysis Technique for the Solution of Nonlinear Problems
,” Ph.D. thesis, Shanghai Jiao Tong University, Shanghai, China.
10.
Liao
,
S. J.
, 2003,
Beyond Perturbation: Introduction to the Homotopy Analysis Method
,
Chapman and Hall
,
London
/
CRC
,
Boca Raton, FL
.
11.
Liao
,
S. J.
, 2004, “
On the Homotopy Analysis Method for Nonlinear Problems
,”
Appl. Math. Comput.
0096-3003,
147
, pp.
499
513
.
12.
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
, pp.
2529
2539
.
13.
Xu
,
H.
,
Liao
,
S. J.
, and
Pop
,
I.
, 2006, “
Series Solutions of Unsteady Boundary Layer of Non-Newtonian Fluids Near a Forward Stagnation Point
,”
J. Non-Newtonian Fluid Mech.
0377-0257,
139
, pp.
31
34
.
14.
Liao
,
S. J.
, and
Tan
,
Y.
, 2007, “
A General Approach to Obtain Series Solutions of Nonlinear Differential Equations
,”
Stud. Appl. Math.
0022-2526,
119
, pp.
297
355
.
15.
Liao
,
S. J.
, 2009, “
Notes on the Homotopy Analysis Method: Some Definitions and Theorems
,”
Commun. Nonlinear Sci. Numer. Simul.
1007-5704,
14
, pp.
983
997
.
16.
Liao
,
S. J.
, 2009, “
A General Approach to Get Series Solution of Non-Similarity Boundary Layer Flows
,”
Commun. Nonlinear Sci. Numer. Simul.
1007-5704,
14
, pp.
2144
2159
.
17.
Abbasbandy
,
S.
, 2006, “
The Application of the Homotopy Analysis Method to Nonlinear Equations Arising in Heat Transfer
,”
Phys. Lett. A
0375-9601,
360
, pp.
109
113
.
18.
Sajid
,
M.
,
Hayat
,
T.
, and
Asghar
,
S.
, 2006, “
On the Analytic Solution of the Steady Flow of a Forth Grade Fluid
,”
Phys. Lett. A
0375-9601,
355
, pp.
18
26
.
19.
Zhu
,
S. P.
, 2006, “
An Exact and Explicit Solution for the Valuation of American Put Options
,”
Quant. Finance
1469-7688,
6
, pp.
229
242
.
20.
Zhu
,
S. P.
, 2006, “
A Closed-Form Analytical Solution for the Valuation of Convertible Bonds With Constant Dividend Yield
,”
ANZIAM J.
1445-8735,
47
, pp.
477
494
.
21.
Yabushita
,
K.
,
Yamashita
,
M.
, and
Tsuboi
,
K.
, 2007, “
An Analytic Solution of Projectile Motion With the Quadratic Resistance Law Using the Homotopy Analysis Method
,”
J. Phys. A
0305-4470,
40
, pp.
8403
8416
.
22.
Wu
,
Y.
, and
Cheung
,
K. F.
, 2008, “
Explicit Solution to the Exact Riemann Problems and Application in Nonlinear Shallow Water Equations
,”
Int. J. Numer. Methods Fluids
0271-2091,
57
, pp.
1649
1668
.
23.
Rajagopal
,
K. R.
, 1984, “
On the Creeping Flow of the Second Order Fluid
,”
J. Non-Newtonian Fluid Mech.
0377-0257,
15
, pp.
239
246
.
24.
Hayat
,
T.
, and
Sajid
,
M.
, 2007, “
Analytic Solution for Axisymmetric Flow and Heat Transfer of a Second Grade Fluid Past a Stretching Sheet
,”
Int. J. Heat Mass Transfer
0017-9310,
50
, pp.
75
84
.
25.
Dunn
,
J. E.
, and
Rajagopal
,
K. R.
, 1995, “
Fluids of Differential Type-Critical Review and Thermodynamic Analysis
,”
Int. J. Eng. Sci.
0020-7225,
33
, pp.
689
729
.
26.
Cebeci
,
T.
, and
Bradshaw
,
P.
, 1984,
Physical and Computational Aspects of Convective Heat Transfer
,
Springer-Verlag
,
New York
.
27.
Adomian
,
G.
, 1976, “
Nonlinear Stochastic Differential Equations
,”
J. Math. Anal. Appl.
0022-247X,
55
, pp.
441
452
.
28.
Karmishin
,
A. V.
,
Zhukov
,
A. T.
, and
Kolosov
,
V. G.
, 1990,
Methods of Dynamics Calculation and Testing for Thin-Walled Structures
,
Mashionstroyenie
,
Moscow
, in Russian.
29.
Awrejcewicz
,
J.
,
Andrianov
,
I. V.
, and
Manevitch
,
L. I.
, 1998,
Asymptotic Approaches in Nonlinear Dynamics
,
Springer-Verlag
,
Berlin
.
30.
Lyapunov
,
A. M.
, 1992,
General Problem on Stability of Motion
,
Taylor & Francis
,
London
.
You do not currently have access to this content.