Mixing in heterogeneous fluids: An examination of fluid property variations

Abstract

In the context of stirred tanks, “mixing” refers to the purposeful and controlled flow designed to minimize heterogeneity, such as variations in solute or additive concentration. Industries like food and polymer processing often encounter situations where fluid properties are closely tied to additive concentration. However, conventional engineering models of mixing, herein referred to as “homogeneous models”, typically assume that the influence of heterogeneous fluid properties on mixing dynamics is negligible. In these models, flow development is considered independent of mixing, and the fluid’s rheological properties and density are assumed to be uniform. This manuscript’s primary objective is to emphasize the potential for substantial inaccuracies in predicting mixing outcomes when the effects heterogeneous fluid properties are disregarded. We investigate the homogenization of an additive in a fluid-filled cylindrical tank stirred by an axisymmetric disk, where both fluid rheology and density are contingent on the additive concentration. We introduce and compare two models for predicting mixing development. The first model (model problem $T$) incorporates variations in fluid properties dependent on the additive concentration, while the second model (model problem $M$) simplifies the fluid properties to their average values. Our approach to modeling mixing centers on a concentration field governed by advection–diffusion. We illustrate that the mapping between the parameter spaces of the two model problems is far from one-to-one. For any given point in the parameter space of model problem $M$, three distinct parameter groups (buoyancy, Atwood number, and viscosity ratio) exhibit unconstrained variations within the corresponding subset of the parameter space of model problem $T$. As a concrete example, we investigate the impact of buoyancy on the evolution of velocity and additive concentration in model problem $T$. Our analysis characterizes the influence of buoyancy on the mixing rate by examining the asymptotic behavior of the concentration field. We find that the standard deviation of the concentration asymptotically converges to an exponential decay, with the intercept and decay rate diminishing as a power-law function of buoyancy. This underscores the significant effect that even slight variations in buoyancy can have on the mixing process. Finally, our results conclusively demonstrate that the recirculation zones, areas where fluid velocity is notable, in model problems M and T do not align. In model problem $M$, the well-mixed region and the recirculation zones closely coincide, but this alignment is not observed in model problem $T$. Collectively, our study provides a counterexample that challenges the hypothesis suggesting that the development of the well-mixed region, and the mixing rate within homogeneous models accurately represent the characteristics of mixing in real-world heterogeneous fluids.