# Persistent structures in a 3D dynamical system with solid and fluid regions

## TL;DR

This study reveals the coexistence of persistent mixing and non-mixing regions in a 3D dynamical system with solid and fluid regions, highlighting the interplay between different mixing mechanisms and their mathematical modeling.

## Contribution

It uncovers how solid and fluid mixing mechanisms interact in a 3D system, demonstrating the persistence of larger islands despite complex flows, advancing understanding of mixed-material advection.

## Key findings

- Larger islands predicted by cutting-and-shuffling persist despite fluid mixing.
- Smaller islands predicted by the model are not observed in practice.
- The interplay between stretching-and-folding and cutting-and-shuffling explains complex structures.

## Abstract

Remarkably persistent mixing and non-mixing regions (islands) are observed to coexist in a three-dimensional dynamical system where randomness is expected. The track of an x-ray opaque particle in a spherical shell half-filled with dry non-cohesive particles and periodically rotated about two axes reveals interspersed structures that are spatially complex and vary non-trivially with the rotation angles. The geometric skeleton of the structures forms from the subtle interplay between fluid-like mixing by stretching-and-folding, and solids mixing by cutting-and-shuffling, which is described by the mathematics of piecewise isometries. In the physical system, larger islands predicted by the cutting-and-shuffling model alone can persist despite the presence of stretching-and-folding flows and particle-collision-driven diffusion, while predicted smaller islands are not observed. 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 both solid and flowing regions.