  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.
