Liquid interfaces in viscous straining flows: Numerical studies of the   selective withdrawal transition
  This paper presents a numerical analysis of the transition from selective withdrawal to viscous entrainment. In our model problem, an interface between two immiscible layers of equal viscosity is deformed by an axisymmetric withdrawal flow, which is driven by a point sink located some distance above the interface in the upper layer. We find that steady-state hump solutions, corresponding to selective withdrawal of liquid from the upper layer, cease to exist above a threshold withdrawal flux, and that this transition corresponds to a saddle-node bifurcation for the hump solutions. Numerical results on the shape evolution of the steady-state interface are compared against previous experimental measurements. We find good agreement where the data overlap. However, the numerical results' larger dynamic range allows us to show that the large increase in the curvature of the hump tip near transition is not consistent with an approach towards a power-law cusp shape, an interpretation previously suggested from inspection of the experimental measurements alone. Instead the large increase in the curvature at the hump tip reflects a logarithmic coupling between the overall height of the hump and the curvature at the tip of the hump. 
Marko Kleine Berkenbusch, Itai Cohen, Wendy W. Zhang