Past exam of the theoretical mechanics course of the University of Oxford / 2011 / A / 2 / b / ii

Use autonomous third-order reduction using the dependent variable. On a monotone branch where $f$ can serve as the independent variable, the chain rule givesTherefore the reduced second-order equation isThe wall and matching conditions becomeFor the usual positive outer flow this is the limit $f\to +\infty$. The reduced equation is singular at $G=0$: if $f^{\prime \prime }(0)=b\ne 0$, then $f\sim b{\eta }^2/2$ and $G\sim b\eta$, so $G$ has square-root behavior as a function of $f$ near the wall. A finite value for $G_f(0)$ should not be imposed as an extra boundary condition.