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Quantifying Mixing using Magnetic Resonance Imaging
Published on: January 25, 2012
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Predicting mixing via resonances: Application to spherical piecewise isometries
Lachlan D Smith1, Paul P Park2, Paul B Umbanhowar3
1Department of Chemical and Biological Engineering, Northwestern University, Evanston, Illinois 60208, USA.
Physical Review. E
|July 16, 2017
Summary
We developed an analytic method to identify nonmixing regions in spherical piecewise isometries (PWIs). This method reveals complex mixing efficacy distributions and analytically determines resonances caused by mode-locking phenomena.
Area of Science:
- Physics
- Applied Mathematics
- Complex Systems
Background:
- Spherical piecewise isometries (PWIs) are mathematical models used to study dynamical systems.
- Understanding mixing efficacy is crucial in fields like granular flow and statistical mechanics.
- Previous methods for analyzing mixing in such systems were limited.
Purpose of the Study:
- To develop an analytic method for identifying nonmixing regions in orientation-preserving spherical PWIs.
- To apply this method to quantify the mixing efficacy of PWIs derived from granular flow in a biaxial tumbler.
- To analytically determine local minima in mixing efficacy (resonances) and their underlying causes.
Main Methods:
- Development of an analytic technique to compute areas of nonmixing regions.
- Application of the method to a specific class of spherical PWIs originating from granular flow simulations.
- Analysis of the distribution of mixing efficacy across the parameter space.
Main Results:
- An analytic method for finding nonmixing regions in spherical PWIs was successfully developed.
- The study revealed a complex, non-uniform distribution of mixing efficacy in the investigated PWI class.
- Local minima in mixing efficacy, termed resonances, were identified and analytically predicted.
- These resonances were attributed to the interaction of two mode-locking-like phenomena.
Conclusions:
- The developed analytic method provides a powerful tool for understanding mixing in spherical PWIs.
- Mixing efficacy in granular flow systems modeled by PWIs is not uniformly distributed and exhibits predictable resonances.
- The findings offer insights into controlling and optimizing mixing processes in relevant physical systems.
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