Cutting, Shuffling and Diffusion as a Coarsening Process

Abstract

Cutting and shuffling (CS) rearranges fluid without stretching, but homogenisation requires diffusion (D); this interaction is not well understood. We show that CSD is a coarsening process, with scale $\bar{\ell }=L(n_c\mathrm{Pe}/2{)}^{-1/3}$ rather than the Batchelor scale, and that cutting acts as a fractional Laplacian in wavenumber, giving the exact mixing rate $\gamma =2^{1/3}|a_1^{\prime }|n_c^{2/3}\lambda {\mathrm{Pe}}^{-1/3}$ for $n_c$ cuts per event at rate $\lambda$, with $a_1^{\prime }=-1.0188\dots$ the first zero of ${\mathrm{Ai}}^{\prime }$. This holds in any dimension for random and deterministic protocols, extends to finite-time mixing, and places CSD in a known non-equilibrium universality class.

arXiv comment6 pages, 5 figures