Mixing by Cutting and Shuffling: Remarkably Persistent Structures

Speaker:

Richard Lueptow - Professor of Mechanical Engineering and Senior Associate Dean for Research, McCormick School of Engineering

Talk Abstract:

Engineers and scientists think of mixing in terms of diffusion, chaotic advection (stirring), or turbulence. However, mixing can also be accomplished by cutting and shuffling, like that used to mix a deck of cards or the colors of a Rubik's Cube. While other means of mixing have long been studied, mixing by cutting and shuffling is not well explored or understood. Unlike the stretching and folding characteristic of chaotic advection, cutting and shuffling maps do not stretch the material and exhibit no chaotic behavior in the usual sense—yet they can mix quite efficiently under certain conditions. In a 3D geometry, a physical model of cutting and shuffling is a spherical tumbler that is half-filled with a granular material undergoing a bi-axial rotation protocol—a rotation about one axis followed by a rotation about another axis for each iteration. X-ray visualization of the flowing granular material reveals non-mixing regions. Simulations of the granular system confirm the non-mixing regions. To further explore cutting and shuffling of a hemisphere, the problem can be mathematically formulated as a Piecewise Isometries (PWIs) transformation that cuts an object into a finite number of pieces and rearranges them into the object's original shape. Computationally recording the cut locations from the PWI on the hemispherical shell yields beautifully intricate complex patterns. However, the PWI transformation requires the assumption of a non-physical granular flow. Hence, it is remarkable that non-mixing regions identified using PWIs correspond to surprisingly persistent non-mixing regions and global barriers to mixing that occur in both experiments and continuum model simulations. This extraordinary merging of the mathematics of PWIs, traditional dynamical systems approaches, and physical applications is leading toward a novel paradigm for understanding and predicting mixing in physical systems based on cutting and shuffling. Funded by NSF Grant #CMMI-1435065.

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