Finding the point of no return: Dynamical systems theory applied to the moving contact-line instability

The wetting and dewetting of solid surfaces is ubiquitous in physical systems across a range of length scales, and it is well known that there are maximum speeds at which these processes are stable. Past this maximum, flow transitions occur, with films deposited on solids (dewetting) and the outer fluid entrained into the advancing one (wetting). These new flow states may be desirable, or not, and significant research effort has focused on understanding when and how they occur. Up until recently, numerical simulations captured these transitions by focussing on steady calculations. This review concentrates on advances made in the computation of the time-dependent problem, utilising dynamical systems theory. Facilitated via a linear stability analysis, unstable solutions act as 'edge states', which form the 'point of no return' for which perturbations from stable flow cease decaying and, significantly, show the system can become unstable before the maximum speed is achieved.