Boundary-layer development beyond a critical value

Two initial-value problems are considered involving a parameter \(\alpha \), the corresponding steady states have a critical value at \(\alpha =\alpha _c<0\) with steady state solutions possible only if \(\alpha \ge \alpha _c\). The aim is to compare how the solution to these two problems evolves in time. For the first problem we find that a solution exists for all time with a two-region structure having a self-similar relatively thin front region moving away from the wall and an inner inviscid region, increasing exponentially in thickness at a rate proportional to \(\hbox {e}^{|\alpha |\,t}\). A finite-time singularity develops in the second problem, the nature of this singularity being derived in terms of the time difference \(\tau =t_s-t\), where the singularity occurs at time \(t_s\), showing that the velocity becomes singular at a rate proportional to \(\tau ^{-1}\) and the thickness of the boundary-layer increasing at a rate proportional to \(\tau ^{-1/2}\).