On the free streamline solutions of the Falkner–Skan equation

Abstract

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 β only. All of them are associated with slip velocities −1<f′0<∞ and emerge from the trivial solution f=η of the problem. The present paper shows, however, that in the positive range 1<β<∞, just below the interval −1<f′0<∞, a further branch of free streamline solutions of slip velocities −2≤f′0<−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 f′0=−2,f′′0=0. For the large-β asymptotics of this solution branch a new algorithm is presented. The occurrence of further free streamline solutions in the range β<0, as well as the existence of free streamlines of vanishing slip velocities, f′′0=f′0=0, both for positive and negative values of β 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.