## Introduction

Fluid mixing is an important process in industries, nature, and everyday life. Today it is well known that simple, non-turbulent velocity fields can create complex distributions of an advected field by exponential stretching and folding of material lines and surfaces. This phenomenon is termed “chaotic advection,”1 and it has been studied by many researchers.2–7 A majority of previous research has been restricted to Newtonian fluids, often in model flows under idealized conditions. Most flows of industrial interest, by contrast, are three-dimensional (3-D) and complex. Further, real high viscosity fluids often contain polymers or solids (such as fermentation broths, pastes, and colloids), and they are hardly ever Newtonian. Non-Newtonian fluids are common in industry and even in the human body (such as blood, saliva, and synovial fluid). They play a significant role in the materials, biochemical, polymer, and pharmaceutical industries. Although there has been recent progress, much remains to be learned about the fundamental phenomena controlling flow and mixing of non-Newtonian fluids in 3-D flows. The basic flow phenomena are at best partially understood, and as a result design and scale-up of non-Newtonian flows in mixing and reaction operations is difficult.

Shear-thinning viscosity and yield stress (viscoplastic) behavior are very common properties of non-Newtonian fluids. Such materials are frequently encountered in industrial problems (such as pastes and suspensions). Although the concept of yield stress is often challenged (see Ref. 8 for a review), its physical approximation has proved very useful in a wide range of applications, including mixing. For example, Wichterle and Wein9 showed that, in a stirred tank, a yield stress fluid is mobile around the impeller where shear stresses are high, whereas the same fluid is stagnant away from the impeller where shear stresses are low. Mobile regions are often called “caverns.” Several studies attempted to correlate cavern size to the amount of energy introduced to the system via torque measurements using empirical models.10–14 Such models improved over the years by assuming different cavern shapes (spherical, cylindrical, and toroidal), accounting for different forces (tangential and axial15), assuming different flow regimes inside the cavern,16 and using different rheological models. Although these models are able to predict the cavern size under a range of operating conditions, they are highly specific to the geometry and fluid considered. They provide limited insight into the general mixing mechanism of shear-thinning yield stress fluids.

Most flows of industrial relevance are 3-D. Numerical simulation of such flows is far from trivial due, in part, to complex geometry. Recent advances on numerical algorithms and discretization methods have allowed the study of chaotic mixing in 3-D flows with reasonable success.17–21 One advantage of numerical simulations is that, once a validated solution is obtained, they can provide a wealth of information that would be difficult to obtain experimentally. For shear-thinning yield stress fluids, numerical simulations must also be combined with a proper rheological model. Although mixing of yield stress fluids in industrially relevant devices is of interest, the majority of the previous numerical studies have been restricted to 2-D flows.22–25 In such flows stagnant regions and plugs with rigid rotation have been encountered, which indicates that mixing of yield stress fluids is far from trivial. The investigation of mixing of yield stress fluids in 2-D flows has been thorough. It has laid the foundation for the advances in numerical methods and rheological models that can be used to investigate 3-D flows. Numerical investigations into the mixing of yield stress fluids in 3-D systems13, 26–29 have been conducted with different levels of success. In some cases good agreement between experimental and simulated velocity fields is achieved. However, as with previous experimental investigations, most studies focused on empirical relationships that are highly specific. The Lagrangian properties of the flow are seldom analyzed.

In this article, we investigate the effects of shear-thinning viscosity and yield stress (viscoplastic) behavior on mixing in a 3-D flow using experiments and numerical simulations. A 3-D flow is generated in a stirred tank equipped with Rushton impellers. Mixing is investigated in experiments by means of velocity measurements and tracer visualizations. Symmetry-breaking methods are used to investigate mixing “performance.” Numerical simulations of shear-thinning viscoplastic fluids are performed using CFD and a viscosity model, which is based on rheological measurements. Stretching statistics and scale behavior are investigated using simulations.