Mixing by cutting and shuffling

Abstract

Dynamical systems theory has proven to be a successful approach to understanding mixing, with stretching and folding being the hallmark of chaotic mixing. Here we consider the mixing of a granular material in the context of a different mixing mechanism -cutting and shuffling-as a complementary viewpoint to that of traditional chaotic dynamics. Cutting and shuffling has a theoretical foundation in a relatively new area of mathematics called piecewise isometries (PWIs) with properties that are fundamentally different from the stretching and folding mechanism of chaotic advection. To demonstrate the effect of the cutting and shuffling combined with stretching and folding, we consider the mixing of granular materials of two different colors in a half-filled spherical tumbler that is rotated alternately about orthogonal axes. Mixing experiments using 1 mm particles in a 14 cm diameter tumbler are compared to PWI maps. The experiments are readily related to the PWI theory using continuum model simulations. By comparing experimental, simulation, and theoretical results, we demonstrate that mixing in a three-dimensional granular system can be viewed as mixing by traditional chaotic dynamics (stretching and folding) built on an underlying framework, or skeleton, of mixing due to cutting and shuffling. We further demonstrate that pure cutting and shuffling can generate a well-mixed system, depending on the angles through which the tumbler is rotated. We also explore the generation of interfacial area between the two colors of material resulting from both stretching in the flowing layer and cutting due to switching the axis of rotation.

Cutting and shuffling a hemisphere: Nonorthogonal axes

“…Typically, the stable set has structure at all scales; for this protocol it forms intricate, but incomplete, circle packing of the HS [6,43]. Cells are the maximally open neighborhoods around periodic points of the domain that contain points with the same periodic itinerary [i.e., a symbolic representation of an orbit by way of the atom labels (1,2,3,4) in Fig. 2(a)] [7,51].…”

“…For example, 4 }, i.e. q (j) = Q (1) j Q (4) . We will show that when α = β, there is a relatively simple closed form expression for q (j) , for all j ≥ 1.…”

“…Simplifying the product q (j) = Q (1) j Q (4) , and letting x = cos(γ/2) sin(α/2) = c γ s α , it can be shown that q (j)…”

Pattern formation in a fully three-dimensional segregating granular flow

“…These non-mixing regions also can be predicted by the more abstract mathematical theory of piecewise isometries (PWI) [19,20], where discontinuities can generate complex dynamical behaviors as seen in various applications [21,22]. The PWI map, which applies to the limiting case of an infinitely thin flowing layer at the free surface [23][24][25], captures the skeleton of the underlying flow generated by the fundamental framework of cuttingand-shuffling, a mechanism for mixing discrete materials [15][16][17][25][26][27]. (b) Bottom view of a segregation experiment in a half-full spherical tumbler with 15% large (d = 4 mm) blue particles and 85% small (d = 1.5 mm) red particles by volume for the same protocol as in (a).…”

Fluid Flow and Distributive Mixing Analysis in the Cavity Transfer Mixer

“…In this way, it is possible to run a number of simulations and investigate the mixing dynamics in different CTM configurations, the results of which confirm how the fluid shuffling and redistribution among the cavities are the major mixing drivers. Actually shuffling is generally recognized as one of the main mechanisms of material mixing and it is usually combined with cutting to mix granular systems, while stretching and folding are considered typical of fluid systems. However, recent works investigate cases in which combined stretching‐and‐folding and cutting‐and‐shuffling actions take place.…”