Tip singularity on a liquid-gas interface

In our experiment, an interface between a viscous liquid and air is deformed by a sink flow of constant flow rate to form a sharp tip. The interface shape is recorded using a microscope with 1 $\rm{\mu m}$ resolution. The curvature at the tip is controlled by the distance $h$ between the tip and the sink. As a critical distance $h^{\star}$ is approached, the curvature diverges like $1/(h-h^{\star})^3$ and the tip becomes cone-shaped. As the distance to the sink is decreased further, the opening angle of the cone vanishes like $h^2$. No evidence for air entrainment was found, except when the tip was inside the orifice.