Granular mixing and its interplay with segregation are complicated, since flow induces segregation by particle size or density. A canonical system for the study of granular flow is a partially filled rotating tumbler. Because, to a reasonable approximation, the dynamics in a tumbler take place in a thin flowing surface layer, a simple, compact, and extensible continuum model can be used to study the flow. However, a suitable general theoretical framework for mixing (and segregation) as well as appropriate mathematical tools to implement the theory are lacking. Full understanding requires an integrated effort consisting of new theory inspired by abstract mathematical concepts and supporting computational and experimental results. The theoretical work proposed here will focus on the application of Piecewise Isometries, Linked Twist Maps, and Lagrangian Coherent Structures. The ultimate objective of the research is to develop a theoretical framework using a dynamical systems approach that will lead to new mathematical tools to predict flow, mixing, segregation, and pattern formation for granular matter in 2d and 3d tumbler geometries with steady and time-periodic forcing. The approach is based on the geometry and symmetries of the 3d flow generated by the forcing (mixing protocols). Complementary experiments and simulations will be used to confirm the applicability of these theoretical approaches to real granular mixing and segregation problems.
The physics of the flow of granular matter is one of the big questions in science. Granular matter is a prototype of a complex system with collective behavior far from equilibrium. Yet many fundamental questions remain. At the same time, an understanding of granular flow has tremendous practical importance in situations ranging from landslides to food processing. Flowing granular systems are strongly disordered and yet display competition between chaos (mixing) and order (segregation). But inroads to date have been modest. Here, we introduce new dynamical systems approaches for the study of granular matter that are grounded in higher mathematics. Dynamical systems tools offer mathematical frameworks that can be exploited in the study of granular flow. Scientific progress here can have an immediate impact on technology and practice. Furthermore, the new mathematical approaches considered here have potential for broad-ranging impact on many physical systems, perhaps creating an entirely new paradigm much like the science of chaotic advection did for mixing of fluids in the 1990s.
Granular materials (particles, powders, pellets, grains, beads, etc.) are part of everyday life, from our breakfast cereal to sand on the beach. At a large scale, granular materials are encountered in oil exploration, agriculture, construction, chemical production, energy, minerals and ores, pharmaceuticals, plastics production, and geophysical phenomena such as landslides, beach erosion, dune motion, and river bed transport. A persistent, costly problem in many situations is the tendency for particles of different sizes, densities, or shapes to separate from one another when they are in motion due to flow or vibration, a process called segregation. A simple example is the Brazil nut effect in a can of mixed nuts—upon vibration (like that which can occur during shipping) small peanuts fall between the larger Brazil nuts to the bottom of the can forcing the Brazil nuts to rise to the top of the can. The result is a can of unmixed nuts. While this is a nuisance for mixed nuts or raisin bran breakfast cereal, it results in serious economic losses in the pharmaceutical, polymer, fertilizer, mining, agricultural, and consumer products industries, which process millions of tons of granular materials each year. An example is the pharmaceutical industry, which relies on the mixing of active and inactive powders to achieve the right mixture—too much active ingredient can be lethal, while too little does no good. Understanding and predicting mixing and segregation of granular materials are crucial. We have developed an exciting new approach to this problem based on a computational technique that we first used to predict pattern formation due to granular segregation. To get more realistic results, we modified the technique to include the diffusion that results as individual particles collide with one another. Combining our computational technique with an equation describing diffusion in flowing fluids (modified slightly to account for granular segregation) has enabled us to develop a remarkably effective new theory for predicting the segregation of flowing granular materials. This theory can predict granular mixing and segregation in many granular flow geometries for particles of two different sizes. An example comparing the theory predictions with experiments for segregation in a rotating cylindrical tumbler, like that used in many industrial situation, shows an excellent match (see figure). The dark regions of the theory corresponding to large particles match the light regions of the experiments, which also correspond to large particles. Our new approach has already been implemented for modeling consumer product manufacturing systems by a Fortune 100 company. A colleague in industry said that our theory represents more significant progress on this problem "than any other research team in the last 50+ years" and another says this theory is "a step change in our understanding of this historic problem." We believe that the theory can provide the basis for predicting granular mixing and segregation in much more complicated systems of many particle sizes, densities, or shapes. In another part of the research we considered the competition between chaos (mixing) and order (segregation) in flowing granular materials. This competition suggests that dynamical systems tools offer a mathematical framework that can be exploited to understand granular mixing and segregation. Dynamical systems is an abstract framework that consists of the state of a system at any instant and a dynamic rule specifying the immediately following state. One of the simplest tools that can be applied is called Piecewise Isometries, essentially "cutting and shuffling," much like what takes place in shuffling a deck of cards or mixing the colors of a Rubik’s cube. A simple system in which to test ideas related to cutting and shuffling is a spherical container (a tumbler) that is half filled with a granular material (sand or glass beads) and can be rotated about two different horizontal axes. Cutting and shuffling is achieved by first rotating the tumbler briefly about one horizontal axis (shuffling), switching the axis of rotation to a perpendicular horizontal axis (cutting), and rotating again (shuffling). Repeating this many times results in mixing, not to mention beautiful patterns of the overlapping cuts, depending on the angles over which the tumbler is rotated (see figure). The new approaches developed in this research were motivated by considering a dynamical systems framework for granular mixing and segregation and have resulted in new mathematical tools to predict flow, mixing, segregation, and pattern formation. This has led to a new paradigm for mixing by cutting and shuffling as well as improvements in understanding granular systems that are of immense economic importance in industries ranging from agriculture to pharmaceuticals. In addition, the research drew together a diverse team of investigators, collaborators, and students (including undergraduates, first generation Americans, and under-represented minorities) who will become the nation’s next generation of scientists and engineers.
