A kind of approximation solution technique which does not depend upon small parameters. II: An application in fluid mechanics.(English)Zbl 1031.76542

Summary: In this paper, the non-linear approximate technique called Homotopy Analysis Method proposed by Liao is further improved by introducing a non-zero parameter into the traditional way of constructing a homotopy. The 2D viscous laminar flow over an infinite flat-plain governed by the non-linear differential equation $$f'''(\nu)+f(\nu)f'(\nu)/2 = 0$$ with boundary conditions $$f(0) = f'(0) = 0,f'(+\infty) = 1$$ is used as an example to describe its basic ideas. As a result, a family of approximations is obtained for the above-mentioned problem, which is much more general than the power series given by Blasius [Z. Math. Phys. 36, 1 (1908)] and can converge even in the whole region $$\nu\in[0, +\infty)$$. Moreover, the Blasius’ solution is only a special case of ours. We also obtain the second-derivative of $$f(\nu)$$ at $$\nu= 0$$, i.e. $$f''(0) = 0.33206$$, which is exactly the same as the numerical result given by Howarth [Proc. R. Soc. Lond. A 164, 547 (1938)].

MSC:

 76M25 Other numerical methods (fluid mechanics) (MSC2010) 76D10 Boundary-layer theory, separation and reattachment, higher-order effects
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