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Research Papers

# On the Nonsimilarity Boundary-Layer Flows of Second-Order Fluid Over a Stretching Sheet

[+] Author and Article Information
Xiangcheng You

State Key Laboratory of Ocean Engineering, School of Naval Architecture, Ocean and Civil Engineering, Shanghai Jiao Tong University, Shanghai 200240, Chinaxcyou@sjtu.edu.cn

Hang Xu

State Key Laboratory of Ocean Engineering, School of Naval Architecture, Ocean and Civil Engineering, Shanghai Jiao Tong University, Shanghai 200240, Chinahangxu@sjtu.edu.cn

Shijun Liao1

State Key Laboratory of Ocean Engineering, School of Naval Architecture, Ocean and Civil Engineering, Shanghai Jiao Tong University, Shanghai 200240, Chinasjliao@sjtu.edu.cn

1

Corresponding author.

J. Appl. Mech 77(2), 021002 (Dec 08, 2009) (8 pages) doi:10.1115/1.3173764 History: Received June 27, 2008; Revised April 15, 2009; Published December 08, 2009; Online December 08, 2009

## Abstract

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.

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## Figures

Figure 1

A sketch of the physical problem

Figure 2

The curves fηη(ξ,0)∼ℏ at the 15th-order of approximation in the case of λ=1/2. Solid line: ξ=1/4, dashed line: ξ=1/2, dash-dotted line: ξ=3/4, and dash-dot-dotted line: ξ=1.

Figure 3

The curves of fηη(ξ,0)∼ξ in the case of λ=1/2 by means of different values of ℏ. Solid line: 20th-order HAM result when ℏ=−1/5, symbols: 25th-order HAM result when ℏ=−1/5, dashed line: 20th-order HAM result when ℏ=−1/2, and dash-dotted line: 20th-order HAM result when ℏ=−3/5.

Figure 4

The curves of fη(ξ,η) in the case of λ=1/2 by means of ℏ=−1/5. Solid line: ξ=1/4, dashed line: ξ=1/2, dash-dotted line: ξ=3/4, dash-dot-dotted line: ξ=1, and symbols: numerical results.

Figure 5

Cf∗(x) in the case of λ=1/2 by means of ℏ=−1/5. Solid line: 20th-order HAM approximation, symbols: 20th-order HAM approximation, dashed line: Cf∗(x)=−0.881819/x, and dash-dotted line: Cf∗(x)=−6.53215/x.

Figure 6

δ∗(x) in the case of λ=1/2 by means of ℏ=−1/5. Solid line: 20th-order HAM approximation, symbols: 25th-order HAM approximation, dashed line: δ∗(x)=1.62805x, and dash-dotted line: δ∗(x)=1.22472.

Figure 7

The 20th-order HAM approximation of Cf∗(x) at different values of λ. Solid line: λ=0, dashed line: λ=1/2, dash-dotted line: λ=1, and dash-dot-dotted line: λ=2.

Figure 8

The 20th-order HAM approximation of δ∗(x) at different values of λ. Solid line: λ=0, dashed line: λ=1/2, dash-dotted line: λ=1, and dash-dot-dotted line: λ=2.

Figure 9

The Uw(ξ) at different values of α. Solid line: α=0, dashed line: α=1/2, dash-dotted line: α=1, dotted line: α=3/2, and dash-dot-dotted line: α=2.

Figure 10

The 20th-order HAM approximation of δ∗(x) at different values of α when λ=1/2. Solid line: α=1/4, dashed line: α=1/2, dash-dotted line: α=3/4, and dash-dot-dotted line: α=1.

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