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

Abstract

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 † mkb@uchicago.edu ‡ ic64@cornell.edu ¶ wzhang@uchicago.edu 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.

Selective withdrawal of polymer solutions: Experiments

“…In his work, the suction is represented by a point sink. More recently, several experimental and numerical studies have dealt with well-controlled Newtonian liquid-liquid systems [1,[5][6][7] and gas-liquid systems [8][9][10]. For liquid-liquid systems, the flow behavior may be classified into three regimes: subcritical, critical and supercritical (Fig.…”

“…The critical state is the threshold for the uptake of the interface. The hump height in the subcritical regime and the critical flow rate have been measured and computed [1,5,7].…”

Force Balance at the Transition from Selective Withdrawal to Viscous Entrainment

“…We mimic the effect of a deep lower layer by requiring that a constant pressure jump, of size p 0 , be maintained across the internal fluid surface I b . A previous study [11] has tested this model extensively against the experimental situation and found that it reproduces the measured dynamics, with the model parameter p 0 and the pinning condition, having little effect on the outcome. Here, we focus on the results for Q c .…”

“…The bulk flow in both layers then satisfies the linear Stokes flow equations, which we solve using a Green's function formulation [18]. The numerical formulation follows that devised by Kleine Berkenbusch, Cohen, & Zhang, and the full description can be found in the earlier study [11]. The key steps are that we first obtain an expression for the velocity on the fluid interface as an integral over a closed surface formed by the fluid interface, I f and the back surface I b .…”

Experimental and numerical studies on pumping viscoplastic fluids

“…While many studies focused on Newtonian fluids (Xue and Yue 1998;Zhou and Graebel 1990;Robinson et al 2010), only a few papers studied non-Newtonian fluids withdrawal. However, they were restricted in the investigation of surface/interface deformation and force balance (Zhou and Feng 2010;Blanchette and Zhang 2009;Berkenbusch et al 2008;Jeong 2007). There have been no experimental studies to examine the flow field of a non-Newtonian flow withdrawing near an intake.…”