A note on free streamline solutions to the Falkner–Skan equation

Abstract

The Falkner–Skan equation, defined by the parameter \(\beta \), is considered subject to a free streamline (zero wall shear) boundary condition. Solutions are found only in \(\beta <0\), the solution becoming singular as \(\beta \rightarrow 0\). Several sets of solutions are seen in \(\beta <0\), each emerging from the trivial solution \(f \equiv \eta \) at \(\beta =-\frac{1}{2} -k, \,k=0,1,2,\ldots \). The first of these sets of solutions has \(f'(0)\) monotone with \(\beta \), the solution terminating as \(\beta \rightarrow 0\) and becoming singular as \(\beta \rightarrow -1\). The other sets of solutions each have a saddle-node bifurcation giving two solution branches, becoming singular and terminating as \(\beta \rightarrow -1\). The asymptotic limits of \(\beta \rightarrow 0\) and \(\beta \rightarrow -1\) are discussed.