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TECHNICAL PAPERS

# Series Solution of Three-Dimensional Unsteady Laminar Viscous Flow Due to a Stretching Surface in a Rotating Fluid

[+] Author and Article Information
Yue Tan

State Key Laboratory of Ocean Engineering, School of Naval Architecture, Ocean, and Civil Engineering, Shanghai Jiao Tong University, Shanghai 200030, Chinatanyue@sjtu.edu.cn

Shi-Jun Liao1

State Key Laboratory of Ocean Engineering, School of Naval Architecture, Ocean, and Civil Engineering, Shanghai Jiao Tong University, Shanghai 200030, Chinasjliao@sjtu.edu.cn

1

Corresponding author.

J. Appl. Mech 74(5), 1011-1018 (Jan 29, 2007) (8 pages) doi:10.1115/1.2723816 History: Received September 08, 2006; Revised January 29, 2007

## Abstract

An analytic technique, namely the homotopy analysis method, is applied to solve the Navier–Stokes equations governing unsteady viscous flows due to a suddenly stretching surface in a rotating fluid. Unlike perturbation methods, the current approach does not depend upon any small parameters at all. Besides contrary to all other analytic techniques, it provides us with a simple way to ensure the convergence of solution series. In contrast to perturbation approximations which have about 40% average errors for the considered problem, our series solutions agree well with numerical results in the whole time region $0⩽t<+∞$. Explicit analytic expressions of the skin friction coefficients are given, which agree well with numerical results in the whole time region $0⩽t<+∞$. This analytic approach can be applied to solve some complicated three-dimensional unsteady viscous flows governed by the Navier–Stokes equations.

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## Figures

Figure 1

The 15th-order HAM approximation of f″(0,0) and g′(0,0) in the case of λ=1∕2

Figure 2

Comparison of f′(0,η) with the exact solution: solid line exact solution; filled circle: 15th-order HAM approximation; open circle: 20th-order HAM approximation

Figure 3

Comparison of numerical solutions with HAM results of f″(ξ,0) and g′(ξ,0) in the case of λ=1∕2 by means of ℏ=−1∕2; solid line: numerical result of f″(ξ,0); open circle: 15th-order HAM approximation of f″(ξ,0); dash line: numerical result of g′(ξ,0); filled circle: 15th-order HAM approximation of g′(ξ,0)

Figure 4

Comparison of numerical results (4) of CfxRex with analytic approximations; symbols: numerical result; solid line: 15th-order HAM approximation: (a)ℏ=−0.5; (b)ℏ=−0.25; dash line: perturbation approximation given by Nazar (4); dash-dotted line: 15th-order HPM approximation (ℏ=−1)

Figure 5

Comparison of numerical results (4) of CfyRex with analytic approximations; symbols: numerical result; solid line: 15th-order HAM approximation (a)ℏ=−0.5; (b)ℏ=−0.25; dash line: perturbation approximation given by Nazar (4); dash-dotted line: 15th-order HPM approximation (ℏ=−1)

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