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>From Streamline Jumping to Strange Eigenmodes and Three-Dimensional Chaos: A Tour of the Mathematical Aspects of Granular Mixing in Rotating Tumblers.
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From Streamline Jumping to Strange Eigenmodes and Three-Dimensional Chaos: A Tour of the Mathematical Aspects of Granular Mixing in Rotating Tumblers.
Under the assumption of simple shear within a thin surface layer (the flowing layer) above a fixed bed of granular material, a kinematic continuum model of granular flow in a tumbler of convex cross-section rotating about a single axis is derived from first principles. For a half-full circular container, an exact closed-form solution is found. A numerical simulation methodology is developed for transport and mixing in non-circular geometries. Whereas previous studies focused on benchmarking this model, the goal of this dissertation is to investigate a number of its salient mathematical aspects.;In the Lagrangian frame, the limit of a vanishing flowing layer is considered, showing that particle trajectories become discontinuous, specifically the composition of isometries. In this limit, the "symptoms" of chaotic advection (streamline crossing, stretching and folding) are absent, leading to the identification of a new mixing mechanism: streamline jumping. Comparisons to experiments verify that these vanishing-flowing-layer dynamics form the "skeleton" of granular mixing.;Whereas streamline crossing leads to stretching and folding of material, streamline jumping leads to "cutting and shuffling." Simple examples are constructed to contrast stretching and folding from cutting and shuffling, specifically showing the latter system is not chaotic in the usual sense yet leads to mixing.;The Eulerian picture of mixing is also considered. A comparative analysis of eigenmodes of the advection-diffusion operator associated with a granular flow and the corresponding Poincare section and finite-time Lyapunov exponent field shows that experimental mixing and segregation patterns are composed of eigenmodes, whose structure is determined by coherent structures created by chaotic advection. To do so, a novel modification of the mapping method for scalar transport is developed to incorporate the effects of diffusion.;Finally, mixing in a three-dimensional (3D) spherical tumbler is studied. The location of period-one points of the flow is found analytically. Particle trajectories are shown to be restricted to two-dimensional surfaces under symmetric conditions but a 3D volume otherwise. Parametric studies identify pathological behaviors and optimal protocols. The vanishing-flowing-layer limit is developed as well, showing that cutting and shuffling leads to the growth of intermaterial area without stretching and folding.
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