The problem of the free streamline solutions of the Falkner–Skan equation is revisited in this paper. Until now, such solutions were found for negative values of the pressure gradient parameter $\beta$ only. All of them are associated with slip velocities $-1<f^{{}^{\prime }}\left(0\right)<\infty$ and emerge from the trivial solution $f=\eta$ of the problem. The present paper shows, however, that in the positive range $1<\beta <\infty$, just below the interval $-1<f^{{}^{\prime }}\left(0\right)<\infty$, a further branch of free streamline solutions of slip velocities $-2\le f^{{}^{\prime }}\left(0\right)<-1$ exists. These new solutions emerge from an exact solution of the Falkner–Skan equation which describes the flow in a converging channel with moving boundaries at the saddle–node bifurcation point $\left(f^{{}^{\prime }}\left(0\right)=-2,f^{{}^{\prime \prime }}\left(0\right)=0\right)$. For the large-$\beta$ asymptotics of this solution branch a new algorithm is presented. The occurrence of further free streamline solutions in the range $\beta <0$, as well as the existence of free streamlines of vanishing slip velocities, $f^{{}^{\prime \prime }}\left(0\right)=f^{{}^{\prime }}\left(0\right)=0$, both for positive and negative values of $\beta$ is also addressed in the paper. The flow inside a cone is also considered shortly and the occurrence of free streamline solutions is pointed out also in this case.

本文重新讨论了Falkner-Skan方程的自由流线解的问题。直到现在，对于压力梯度参数的负值仍发现了这样的解决方案$\beta$只要。所有这些都与滑移速度有关$-\mathrm{1个}<F^{{}^{\prime }}\left(0\right)<\infty$ 并从平凡的解决方案中脱颖而出 $F=\eta$问题。但是，本文件表明，在积极范围内$\mathrm{1个}<\beta <\infty$，正好在间隔以下 $-\mathrm{1个}<F^{{}^{\prime }}\left(0\right)<\infty$，是滑移速度的免费精简解决方案的进一步分支 $-\mathrm{2个}\le F^{{}^{\prime }}\left(0\right)<-\mathrm{1个}$存在。这些新的解决方案源于Falkner-Skan方程的精确解，该方程描述了在汇合通道中的流动，并且在鞍-节点分叉点处有移动边界$\left(F^{{}^{\prime }}\left(0\right)=-\mathrm{2个}，F^{{}^{\prime \prime }}\left(0\right)=0\right)$。对于大型$\beta$该解决方案的渐近性提出了一种新的算法。范围内出现其他免费精简解决方案$\beta <0$以及滑行速度消失的自由流线的存在， $F^{{}^{\prime \prime }}\left(0\right)=F^{{}^{\prime }}\left(0\right)=0$，无论是正值还是负值 $\beta$该文件中也有介绍。在短时间内还考虑了圆锥体内的流动，并且在这种情况下也指出了自由流线解决方案的出现。