Abstract
Mixing of fluids and mixing of solids are both relatively mature fields. In contrast, mixing in systems where flowing and non-flowing regions coexist remains largely unexplored and little understood. Here we report remarkably persistent mixing and non-mixing regions in a three-dimensional dynamical system where randomness is expected. A spherical shell half-filled with dry non-cohesive particles and periodically rotated about two horizontal axes generates complex structures that vary non-trivially with the rotation angles. They result from the interplay between fluid-like mixing by stretching-and-folding, and solids mixing by cutting-and-shuffling. In the experiments, larger non-mixing regions predicted by a cutting-and-shuffling model alone can persist for a range of protocols despite the presence of stretching-and-folding flows and particle-collision-driven diffusion. By uncovering the synergy of simultaneous fluid and solid mixing, we point the way to a more fundamental understanding of advection driven mixing in materials with coexisting flowing and non-flowing regions.
Highlights
Mixing of fluids and mixing of solids are both relatively mature fields
We employ the mathematics of piecewise isometries to predict the geometric skeleton of the mixing and non-mixing regions, demonstrating that for a range of tumbling protocols, the mixing structure is a fundamental consequence of the dynamics of cutting-andshuffling
The B-set of non-mixing regions is the half-period offset of the A-set of non-mixing regions and vice versa. Both piecewise isometry (PWI) and flowing layer (FL) models capture the gross features of the large scale persistent mixing and non-mixing structures observed in experiments for the two protocols we have examined so far
Summary
Mixing of fluids and mixing of solids are both relatively mature fields. In contrast, mixing in systems where flowing and non-flowing regions coexist remains largely unexplored and little understood. A spherical shell half-filled with dry non-cohesive particles and periodically rotated about two horizontal axes generates complex structures that vary non-trivially with the rotation angles They result from the interplay between fluid-like mixing by stretching-and-folding, and solids mixing by cutting-and-shuffling. To study the interaction of mixing by stretching-and-folding with mixing by cutting-and-shuffling, we consider a geometrically simple 3D model system with localized flow—a spherical tumbler half-filled with a dry granular material and rotated alternately about orthogonal horizontal axes. Experiments with this system demonstrate the existence and extraordinary persistence of nonmixing island structures due to cutting-and-shuffling for many different experimental protocols, even in the presence of diffusion and flowing regions. This model fully captures the mixing and non-mixing regions observed in the experiments and provides insight into the interaction between stretching-and-folding and the underlying structure of the dynamical system based on cutting-and-shuffling
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
