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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 larger dynamic range of the numerical results 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 robust trend in the steady-state interface evolution. For large deflections, the hump height is proportional to the logarithm of the curvature at the hump tip; thus small changes in hump height correspond to large changes in the value of the hump curvature.

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