Predicting the Transition from Selective Withdrawal to Entrainment in Two-Fluid Systems: Role of Shear-Thinning Fluids and Transitional Regimes.

Selective withdrawal, a process where fluid is extracted from one layer without disturbing adjacent layers, is crucial in applications such as petroleum extraction, emulsion processing, and environmental fluid mechanics. While prior research has made strides in understanding selective withdrawal in Newtonian fluids, the behavior of non-Newtonian fluids in such systems remains largely unexplored. This study bridges that gap by experimentally and theoretically investigating selective withdrawal where the lower fluid exhibits shear-thinning behavior. Soybean oil and Polydimethylsiloxane (PDMS silicone oil) are used as representative upper fluids. The research focuses on determining the critical submergence height for entrainment and understanding the dynamics at the fluid-fluid interface. Through detailed statistical analysis using the Games-Howell post-hoc test, significant differences in submergence height were identified as a function of flow rate and fluid properties. A notable finding was the delineation of two regimes: an inertia-dominated regime characterized by high Weber numbers $(We)$ and low Ohnesorge numbers $(Oh)$, and a transitional regime where viscous effects become prominent, though inertial forces remain significant. The experimental data collapsed onto a scaling curve, with the submergence height $(S)$ in the inertia-dominated regime scaling as $S \propto We^{1/5}$ and in the transitional regime scaling as $S \propto Re^{-1/3}$. This collapse confirmed the theoretical scaling analysis and demonstrated that as Reynolds number increases in the inertia-dominated regime, the submergence height $(S)$ approaches a constant. Although the introduction of shear-thinning fluids significantly reduced the submergence height at similar flow rates, an observation that was statistically significant, the overall collapse of the data indicates that the non-Newtonian effects of the lower layer remain subdominant in this experimental regime. The apparent viscosity of the shear-thinning fluids does not sufficiently alter the dominant inertial and viscous forces driving the system. A critical contribution of this work is the discovery of a generalized scaling law for the submergence height in our experimental regime, which makes an attempt to capture both the inertial and viscous effects. In addition, a novel observation was made regarding rag layers, particularly in soybean oil systems, where the stable emulsion particles at the interface, causing intermittent entrainment at higher submergence heights. This behavior underscores the dynamic complexity of rag layers in altering interface dynamics. Further studies are needed to refine the scaling laws, particularly for phenomena beyond the transitional regime, and to explore the dynamic behavior of rag layers at fluid interfaces under varying shear conditions.