Cahn-Hiliard theory for unstable granular fluids

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 ~mechanismlike spinodal decomposition, or ‘‘uplifting’’ in structural geology!, and selective noise reduction ~mechanismlike peneplanation in structural geology! leading to vortex patterns. 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. Analytic expressions are derived for spatial correlation functions and structure factors that agree well with molecular dynamics simulations of a fluid of inelastic hard disks.