Mixing with Piecewise Isometries

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Mixing of granular materials is an important engineering process, and a very complex problem. In this thesis, I use granular mixing in a half-full biaxial spherical tumbler (BST), a spherical container that rotates sequentially about two orthogonal axes, to motivate the study of mixing with piecewise isometries (PWIs), a rich discrete dynamical system with potential applications in mixing design. The set of rotations for the BST constitutes an iteration, which corresponds to a PWI transformation under simplifying assumptions. Trajectories for singularities of a PWI in the limit of infinite iterations reveal an inherent structure defined as the exceptional set, and we show that its complement comprises non-mixing regions. The measures of different exceptional sets are compared to established measures of mixing, and conditions limiting the dynamics of the BST to that of a PWI are relaxed to understand the impact of a theoretical exceptional set in physical applications for future studies.

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  • 02/13/2018
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