Force Balance at the Transition from Selective Withdrawal to Viscous Entrainment

We simulate the evolution of the steady-state interface in the selective withdrawal regime. Selective withdrawal ends when the upward pull exerted by the viscous flow in the withdrawing liquid layer overcomes the downward force due to surface tension. The lower-layer dynamics are unimportant. The dominant contribution to the surface-tension force comes from the large area where the interface is weakly deflected, instead of the small area where the surface is most distorted. A scaling estimate based on this idea yields results that agree quantitatively with both simulations and previous experiments.