Steven W. Meier

A dynamical systems approach to mixing and segregation of granular materials in tumblers

The physics of granular matter is one of the big questions in science. Granular matter serves as a prototype of collective systems far from equilibrium and fundamental questions remain. At the same time, an understanding of granular matter has tremendous practical importance. Among practical problems, granular mixing and its interplay with segregation is arguably at the top of the list in terms of impact. Granular mixing in three-dimensional systems is complicated, as flow induces segregation by particle size or density. Several approaches and points of view for analysis are possible in principle, ranging from continuum to discrete. Flow and segregation in three-dimensional systems is seemingly complicated; however, to a reasonable approximation, all of the dynamics takes place in a thin flowing surface layer. This observation, coupled with key experimental results, leads to a simple, compact and extensible continuum-based dynamical systems framework applicable to time-periodic flow in quasi-two-dimensional tumblers and three-dimensional systems (such as spheres and cubes) rotated about one or more axes of rotation. The case of time-periodic systems, in its simplest version, can be viewed as a mapping of a domain into itself. The placement of periodic points can be 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 occurring in the flow.

Linked twist map formalism in two and three dimensions applied to mixing in tumbled granular flows

We study the mixing properties of two systems: (i) a half-filled quasi-two-dimensional circular drum whose rotation rate is switched between two values and which can be analysed in terms of the existing mathematical formalism of linked twist maps; and (ii) a half-filled three-dimensional spherical tumbler rotated about two orthogonal axes bisecting the equator and with a rotational protocol switching between two rates on each axis, a system which we call a three-dimensional linked twist map, and for which there is no existing mathematical formalism. The mathematics of the three-dimensional case is considerably more involved. Moreover, as opposed to the two-dimensional case where the mathematical foundations are firm, most of the necessary mathematical results for the case of three-dimensional linked twist maps remain to be developed though some analytical results, some expressible as theorems, are possible and are presented in this work. Companion experiments in two-dimensional and three-dimensional systems are presented to demonstrate the validity of the flow used to construct the maps. In the quasi-two-dimensional circular drum, bidisperse (size-varying or density-varying) mixtures segregate to form lobes of small or dense particles that coincide with the locations of islands in computational Poincare sections generated from the flow model. In the 3d spherical tumbler, patterns formed by tracer particles reveal the dynamics predicted by the flow model.

Surface velocity in three-dimensional granular tumblers

A fundamental characteristic of granular flows is that they are typically restricted to thin layers of rapid surface flow. Thus, a complete understanding of surface flows is key for an accurate representation of the dynamics of the entire flow. Experiments were conducted in three-dimensional tumblers: cylindrical tumblers of various diameters, a double-cone tumbler, and a spherical tumbler, the Froude number for the last two being a function of the local geometry and ranging from 2.6 × 10 -5 to 7.5 x 10 -4 . Surface velocity measurements for 1 mm and 2 mm glass particles were obtained using particle tracking velocimetry. Results indicate that the streamwise surface velocity at the midpoint of the flowing layer is a linear function of local flowing layer length, regardless of tumbler shape, particle size, rotation rate, and fill fraction. In addition, the axial velocity of particles at the free surface is negligible. These results are key for the development of three-dimensional models of granular flows.