Observation of cusps during the levelling of free surfaces in viscous flows

We experimentally study the formation of cusps at the free surfaces of visc ous fluids in three simple cases that portray possible natural or industria l procceses. Two cases concern levelling driven by gravity: the case (A) of a sinusoidal surface and the case (B) of a single groove. In the third cas e (C), an initially sinusoidal surface evolves under the action of a fast e nough lateral compression that the effects of gravity are negligible. Case (A) shows a critical aspect ratio above which the cusps form. Case (B) allo ws a more detailed study of the evolution of the cuspidal structure, which does not change in shape but reduces its size according to a simple power l aw dependence in time. In case (C), cusps form even for small initial aspec t ratios.