Cahn-Hilliard Theory for Unstable Granular Flows

TL;DR

This paper develops a Cahn-Hilliard-type theory to describe the early-stage spatial correlations and pattern formations in freely cooling granular fluids, aligning well with molecular dynamics simulations.

Contribution

It introduces a novel theoretical framework for granular flow fluctuations, capturing density clustering and vortex structures with quantitative predictions.

Findings

Density clustering exhibits a diffusive growth of length scale.

Velocity correlations decay algebraically as r^{-d}.

The theory matches molecular dynamics simulation results.

Abstract

A Cahn-Hilliard-type theory for hydrodynamic fluctuations is proposed that gives a quantitative description of the slowly evolving spatial correlations and structures in density and flow fields in the early stages of evolution of freely cooling granular fluids. Two mechanisms for pattern selection and structure formation are identified: unstable modes leading to density clustering (compare spinodal decomposition), and selective noise reduction (compare peneplanation in structural geology) leading to vortex structures. As time increases, the structure factor for the density field develops a maximum, which shifts to smaller wave numbers. This corresponds to an approximately diffusively growing length scale for density clusters. The spatial velocity correlations exhibit algebraic decay $\sim r^{-d}$ on intermediate length scales. The theoretical predictions for spatial correlation functions and structure factors agree well with molecular dynamics simulations of a system of inelastic hard disks.