  In many industrial processes, such as pouring a liquid or coating a rotating
cylinder, air bubbles are entrapped inside the liquid. We propose a novel
mechanism for this phenomenon, based on the instability of cusp singularities
that generically form on free surfaces. The air being drawn into the narrow
space inside the cusp destroys its stationary shape when the walls of the cusp
come too close. Instead, a sheet emanates from the cusp's tip, through which
air is entrained. Our analytical theory of this instability is confirmed by
experimental observation and quantitative comparison with numerical simulations
of the flow equations.
