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A uniformly valid analytic solution of two-dimensional viscous flow over a semi-infinite flat plate. (English) Zbl 0931.76017
Summary: We apply a new analytic technique, namely the homotopy analysis method, to give an explicit, analytic, uniformly valid solution of the equation governing the two-dimensional laminar viscous flow over a semi-infinite flat plate, $$f'''(\eta)+ \alpha f(\eta)f''(\eta)+ \beta[1- f^{\prime 2}(\eta)] =0$$, under the boundary conditions $$f(0)= f'(0)= 0$$, $$f'(+\infty)= 1$$. This analytic solution is uniformly valid in the whole region $$0\leq \eta<+\infty$$. For Blasius’ (1908) flow ($$\alpha= 1/2$$, $$\beta= 0$$), this solution converges to Howarth’s (1938) numerical result and gives analytic value $$f''(0)= 0.332057$$. For the Falkner-Skan (1931) flow ($$\alpha=1$$), it gives the same family of solutions as Hartree’s (1937) numerical results, and provides a related analytic formula for $$f''(0)$$ when $$2\geq \beta\geq 0$$. Additionally, this analytic solution allows to prove that for $$-0.1988\leq \beta<0$$, the Hartree’s (1937) family of solutions possesses the property that $$f'\to 1$$ exponentially as $$\eta\to +\infty$$.

##### MSC:
 76D10 Boundary-layer theory, separation and reattachment, higher-order effects 76M45 Asymptotic methods, singular perturbations applied to problems in fluid mechanics
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