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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 $0\le \eta<+\infty$. 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 $2\ge \beta\ge 0$. Additionally, this analytic solution allows to prove that for $-0.1988\le \beta<0$, the Hartree’s (1937) family of solutions possesses the property that $f'\to 1$ exponentially as $\eta\to +\infty$.

76D10Boundary-layer theory, separation and reattachment, etc. (incompressible viscous fluids)
76M45Asymptotic methods, singular perturbations (fluid mechanics)
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