Navier-Stokes hydrodynamics of thermal collapse in a freely cooling granular gas

TL;DR

This paper uses Navier-Stokes hydrodynamics to analyze the long-term clustering and thermal collapse in a freely cooling dilute granular gas, revealing tendencies toward finite-time density blowups and cluster formation.

Contribution

It demonstrates that granular hydrodynamics predicts thermal collapse and clustering in dilute gases, providing analytical and numerical evidence for finite-time blowups in higher dimensions.

Findings

Homogeneous cooling state becomes unstable via sub-critical bifurcation in circular containers.

Thermal collapse occurs unarrested by heat diffusion in certain geometries.

Hydrodynamic simulations confirm analytical scaling laws near collapse.

Abstract

We employ Navier-Stokes granular hydrodynamics to investigate the long-time behavior of clustering instability in a freely cooling dilute granular gas in two dimensions. We find that, in circular containers, the homogeneous cooling state (HCS) of the gas loses its stability via a sub-critical pitchfork bifurcation. There are no time-independent solutions for the gas density in the supercritical region, and we present analytical and numerical evidence that the gas develops thermal collapse unarrested by heat diffusion. To get more insight, we switch to a simpler geometry of a narrow-sector-shaped container. Here the HCS loses its stability via a transcritical bifurcation. For some initial conditions a time-independent inhomogeneous density profile sets in, qualitatively similar to that previously found in a narrow-channel geometry. For other initial conditions, however, the dilute gas develops thermal collapse unarrested by heat diffusion. We determine the dynamic scalings of the flow close to collapse analytically and verify them in hydrodynamic simulations. The results of this work imply that, in dimension higher than one, Navier-Stokes hydrodynamics of a dilute granular gas is prone to finite-time density blowups. This provides a natural explanation to the formation of densely packed clusters of particles in a variety of initially dilute granular flows.