Mixing and Segregation of Granular Materials: A Dynamical Systems ApproachPublic Deposited
The physics of granular materials is one of the big questions in science. Granular materials serve as a prototype of collective systems far from equilibrium and fundamental questions remain. At the same time, an understanding of granular materials has tremendous practical importance. Among practical problems, granular mixing and its interplay with segregation often produces unexpected results, and analytical tools are in the early stages of development. As flow intended to create mixing induces segregation by particle size or density, analysis becomes complicated; however, all of the dynamics occur in a thin flowing surface layer. This observation, coupled with two key experimental results, is used to develop a continuum-based theoretical framework applicable to time-periodic flow in quasi-two-dimensional tumblers (circular and polygonal cross-sections) and three-dimensional systems (spheres and cubes) rotated about one or more axes of rotation. The first experimental observation is that the streamwise velocity at the midlength of the rapidly flowing surface layer in quasi-two-dimensional tumblers varies linearly with depth with a maximum at the free surface. The second experimental result, which is described in this dissertation, is that the streamwise velocity at the midlength of the free surface in three-dimensional tumblers is proportional to the local flowing layer length. The case of time-periodic systems, in its simplest version, is viewed as a mapping of a domain into itself and is studied using a dynamical systems approach. Central to this case is the character of periodic points. The placement of periodic points is investigated using symmetry concepts. The character of the periodic points and associated manifolds provides a skeleton for the flow and a template for segregation processes in the flow. Finally, a constitutive model treating the different particle types as interpenetrating continua is coupled to the continuum model. Patterns produced by this interpenetrating continua model capture experimental segregation patterns. An assumption in this approach is that the flow affects segregation, but segregation does not affect flow. However, when this assumption is not valid, a radial streak segregation pattern may form in quasi-two-dimensional tumblers. This radial streak segregation pattern coarsens to as few as one streak over several hundred revolutions.