Cutting and shuffling a hemisphere: Nonorthogonal axes

Abstract

We examine the dynamics of cutting-and-shuffling a hemispherical shell driven by alternate rotation about two horizontal axes using the framework of piecewise isometry (PWI) theory. Previous restrictions on how the domain is cut-and-shuffled are relaxed to allow for non-orthogonal rotation axes, adding a new degree of freedom to the PWI. A new computational method for efficiently executing the cutting-and-shuffling using parallel processing allows for extensive parameter sweeps and investigations of mixing protocols that produce a low degree of mixing. Non-orthogonal rotation axes break some of the symmetries that produce poor mixing with orthogonal axes and increase the overall degree of mixing on average. Arnold tongues arising from rational ratios of rotation angles and their intersections, as in the orthogonal axes case, are responsible for many protocols with low degrees of mixing in the non-orthogonal-axes parameter space. Arnold tongue intersections along a fundamental symmetry plane of the system reveal a new and unexpected class of protocols whose dynamics are periodic, with exceptional sets forming polygonal tilings of the hemispherical shell.

Pattern formation in a fully three-dimensional segregating granular flow

“…2, the tumbler is rotated by angle θ z about the z-axis and then by angle θ x about the x-axis, where the z-axis and x-axis both lie in the horizontal plane. In the figure and in most previous studies, the axes are orthogonal, but in general, the angle γ between them can have any value [16,17]. This biaxial rotation protocol is specified by the triple (θ z , θ x , γ) and typically is repeated many times.…”

“…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).…”

Potentialities and Limitations of Machine Learning to Solve Mixing Problems: A Case Study

“…In one dimension, cutting-and-shuffling is described mathematically by interval exchange transforms (IETs) [42][43][44][45][46][47][48][49][50], a subset of broader cut-and-shuffle mixing actions called piecewise isometries (PWIs) [41,[51][52][53]. An IET cuts a one dimensional domain into intervals that are then reordered according to a specific permutation.…”