Yield-Stress Fluid Mixing: Localization Mechanisms and Regime Transitions

TL;DR

The paper addresses how yield-stress fluids affect mixing under laminar, two-dimensional stirring by a circular impeller. By simulating a viscoplastic Bingham fluid with a circular stirrer in an infinite domain and analyzing dye dispersion, the authors identify three localization mechanisms—finite vortex advection, vortex entrapment, and complete suppression of vortex shedding—that partition mixing into three regimes SE, ST, and NS. A regime map in the $(Re,Bn)$ plane and the concept of effective Reynolds numbers reveal that regime transitions resemble bluff-body flow transitions, offering a mechanistic framework for predicting mixing behavior in stirred tanks. The work provides practical guidance on how yield stress controls mixing efficiency and localization, with implications for design and operation of processes involving yield-stress fluids.

Abstract

We explore the mechanisms and regimes of mixing in yield-stress fluids by simulating the stirring of an infinite, two-dimensional domain filled with a Bingham fluid. A cylindrical stirrer moves along a circular path at constant speed to stir the fluid, with an initially quiescent domain marked by a passive dye in the lower half, facilitating the analysis of dye interface evolution and mixing dynamics. We first examine the mixing process in Newtonian fluids, identifying three key mechanisms: interface stretching and folding around the stirrer's path, diffusion across streamlines, and dye advection and interface stretching due to vortex shedding. Introducing yield stress into the system leads to notable localization effects in mixing, manifesting through three mechanisms: advection of vortices within a finite distance of the stirrer, vortex entrapment near the stirrer, and complete suppression of vortex shedding at high yield stresses. Based on these mechanisms, we classify three distinct mixing regimes in yield-stress fluids: (i) Regime SE, where shed vortices escape the central region, (ii) Regime ST, where shed vortices remain trapped near the stirrer, and (iii) Regime NS, where no vortex shedding occurs. These regimes are quantitatively distinguished through spectral analysis of energy oscillations, revealing transitions and the critical Bingham and Reynolds numbers. The transitions are captured through effective Reynolds numbers, supporting a hypothesis that mixing regime transitions in yield-stress fluids share fundamental characteristics with bluff-body flow dynamics. The findings provide a mechanistic framework for understanding and predicting mixing behaviors in yield-stress fluids, suggesting that the localization mechanisms and mixing regimes observed here are archetypal for stirred-tank applications.

Figures (21)

• Figure 1: Schematic of the domain geometry and initial conditions. The solid white line indicates the stirrer's path. The red and blue colors indicate the dyed and dye-free regions. Note that the figure is not to scale.
• Figure 2: Time evolution of (a) normalized variance of dye concentration for different mesh sizes, (b) relative error in dye concentration variance, $E_R(\sigma^2_{6}) = \frac{\sigma^2_{6} - \sigma^2_{6,\ 1.3\times10^5}}{\sigma^2_{6,\ 1.3\times10^5}}$, (c) kinetic energy for different mesh sizes, and (d) relative error in kinetic energy, $E_R({KE}) = \frac{{KE} - {KE}_{1.3\times10^5}}{{KE}_{1.3\times10^5}}$, for $Re = 150$ and $Bn = 2$.
• Figure 3: (a–f) Snapshots of the dye concentration field in a Newtonian fluid, $Bn = 0$, $Re = 50$, at times ${T} = 1, 3, 10, 15, 30, \text{and}~50$. The white circle marks the stirrer's path, while grey lines represent the streamlines. The radius of the field of view is provided in the caption. (g) Time evolution of the normalized variance, with circular markers indicating the time instances of the snapshots in (a–f).
• Figure 4: (a–f) Snapshots of the vorticity field in a Newtonian fluid, $Bn = 0$, $Re = 50$, at times ${T} = 1, 3, 10, 15, 30, \text{and}~50$. The white circle marks the stirrer's path, while grey lines represent the streamlines. The radius of the field of view is provided in the caption.
• Figure 5: Time evolution of the vortex centers in a Newtonian fluid, $Bn = 0$, $Re = 50$, over different time intervals, as indicated by the colorbar. Lighter shades correspond to earlier times within each subfigure. The white solid line represents the stirrer's path, while the black dashed lines denote the subdomain boundary.