Liquid interfaces in viscous straining flows: numerical studies of the selective withdrawal transition

In selective withdrawal, the interface between two liquid layers is deformed by an imposed withdrawal flow. A shape transition occurs at a threshold flow rate that changes the topology of the interface from a steady-state hump to an entrained spout. Near the transition a very sharp hump tip occurs, with the minimum tip radius far smaller than the characteristic lateral length-scale. Previous measurements (Cohen & Nagel (2002)) suggest the sharp hump is created via an approach towards a singular steady-state shape which is cut off at a small lengthscale. To help unfold the mechanism underlying the shape transition and to determine the origin of the cutoff lengthscale, we construct a numerical model of selective withdrawal. Its results are consistent with experimental measurements, but it has a larger dynamic range near the transition. The increased range enables us to identify the shape transition as a saddle-node bifurcation. The transition does not involve an approach towards a singular shape. At threshold, the hump radius is comparable to the hump height. We also analyze selective withdrawal under a variety of withdrawal conditions and find that the different results collapse when the lengthscales are appropriately rescaled.