Hydrodynamic singularities and clustering in a freely cooling inelastic gas

TL;DR

This paper investigates the development of clustering and singularities in a freely cooling inelastic gas using hydrodynamic equations, revealing that the flow becomes inertia-driven and develops finite-time singularities in a simplified one-dimensional model.

Contribution

It demonstrates that flow by inertia is a universal intermediate asymptotic in the nonlinear clustering of inelastic gases, using a simplified one-dimensional hydrodynamic approach.

Findings

Shear stress becomes negligible at late stages of clustering.

Flow by inertia leads to finite-time singularities in density and velocity gradient.

Inertia-driven flow is a generic feature of unstable free cooling in dilute inelastic gases.

Abstract

We employ hydrodynamic equations to follow the clustering instability of a freely cooling dilute gas of inelastically colliding spheres into a well-developed nonlinear regime. We simplify the problem by dealing with a one-dimensional coarse-grained flow. We observe that at a late stage of the instability the shear stress becomes negligibly small, and the gas flows solely by inertia. As a result the flow formally develops a finite time singularity, as the velocity gradient and the gas density diverge at some location. We argue that flow by inertia represents a generic intermediate asymptotic of unstable free cooling of dilute inelastic gases.