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.

Figure 1

(a) Selective withdrawal and viscous entrainment regimes (superposed images). Liquid is withdrawn at volume flux $Q$ through a tube suspended above the interface separating two viscous, immiscible liquids. Below $Q_c$, the interface shape is a steady-state hump (black). Above $Q_c$, a spout forms (grey). The lower-layer liquid has viscosity ${\mu }_0$ and density ${\rho }_0$; the upper layer has viscosity $\mu$ and density $\rho$. The interface has surface tension $\sigma$. Photos courtesy of Cohen & Nagel [4]. (b) Schematic of the numerical model. The suspended tube in the experiment is modeled by a point sink at height $S$. The closed surface used in the boundary integral simulation is the combination of $I_f$, the portion of the interface within the pinning radius $a$, and $I_b$, a back-surface mimicking the effect of a deep lower layer.

Figure 2

Evolution of steady hump solutions. (a) Hump height $H/a$ vs withdrawal flux $Q/\widehat{Q}$. The asterisks indicate the end of selective withdrawal. (b) Close up of the steady hump shape at $Q_c$. The interface in the numerical model extends to the pinning radius $r/a=1$.

Figure 3

(a) Rescaled threshold $Q_c(\mu /\sigma )/S^2$ as a function of $S/a$, for three values of ${\mu }_0/\mu$. The asterisk corresponds to the rescaled threshold withdrawal flux for an interface deformed by a uniform flow. (b) Rescaled measurements for ${\mu }_0/\mu =0.001$ (solid circle), 0.0013 (cross), 0.006 (open circle), 0.021 (upper triangle), 0.3 (square), 1.26 (diamond), and 1.7 (lower triangle). All data from [4]. The dashed line indicates numerical results for ${\mu }_0/\mu =1$, $S\gg a$ by Lister [3].