A container filled with viscous liquid
empties through a hole in the bottom. As the
filling height decreases, the liquid-air
interface forms an increasingly sharp tip.
At some finite distance h between the
tip and the hole the curvature of
the tip diverges like 1/(h-h^*)³,
forming a finite cone angle.

As h is decreased beyond h^*,
the cone angle vanishes like 1/h², see:
S. Courrech du Pont, J. Eggers
Sink flow deforms the interface between a viscous liquid and air into a tip singularity ,
Phys. Rev. Lett. 96 , 034501 (2006)

movie by S. Courrech du Pont