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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, ${f}^{\text{'}\text{'}\text{'}}\left(\eta \right)+\alpha f\left(\eta \right){f}^{\text{'}\text{'}}\left(\eta \right)+\beta \left[1-{f}^{\text{'}2}\left(\eta \right)\right]=0$, under the boundary conditions $f\left(0\right)={f}^{\text{'}}\left(0\right)=0$, ${f}^{\text{'}}\left(+\infty \right)=1$. 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 ${f}^{\text{'}\text{'}}\left(0\right)=0·332057$. 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 ${f}^{\text{'}\text{'}}\left(0\right)$ 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 ${f}^{\text{'}}\to 1$ exponentially as $\eta \to +\infty$.
##### MSC:
 76D10 Boundary-layer theory, separation and reattachment, etc. (incompressible viscous fluids) 76M45 Asymptotic methods, singular perturbations (fluid mechanics)