`Finding the point of no return': Understanding how a free-surface in a moving contact-line problem becomes unstable and how to suppress it*

The moving-contact line between a fluid, liquid and a solid is a ubiquitous phenomenon, and determining the maximum speed at which a liquid can wet/dewet a solid is a practically important problem. Using continuum models the maximum speed of wetting/dewetting can be found by calculating steady solutions of the governing equations and locating the critical capillary number, Ca_crit above which no steady-state solution can be found. Below Ca_crit, both stable and unstable steady-state solutions exist and if some appropriate measure of these solutions is plotted against Ca, a fold bifurcation appears where the stable and unstable branches meet. In this talk we develop a computational framework to describe this phenomenon and, by applying ideas from dynamical systems theory to the highly-dimensional complex system, show that, rather than just being a consequence of the fold bifurcation, the unstable solutions are `edge states' and a have profound importance on the transient behaviour of the system. Significantly, the system can become unstable when Ca<sub>crit</sub> due to finite amplitude interfacial `wobbles' are more dangerous than `stretch' perturbations.