A nonlinear theory of non-stationary low Mach number channel flows of freely cooling nearly elastic granular gases

TL;DR

This paper develops a nonlinear hydrodynamic theory for non-stationary, low Mach number, freely cooling granular gases in channels, revealing how heat diffusion influences clustering and leads to novel inhomogeneous cooling states.

Contribution

It introduces a reduced nonlinear nonlocal reaction-diffusion equation for channel flows, capturing clustering instability and the effects of heat diffusion on density blowup and cooling states.

Findings

Exact solutions show finite-time density blowup without heat diffusion.

Heat diffusion arrests blowup, leading to stable inhomogeneous cooling states.

The dynamics resemble Ostwald ripening, with only one dense region surviving.

Abstract

We use hydrodynamics to investigate non-stationary channel flows of freely cooling dilute granular gases. We focus on the regime where the sound travel time through the channel is much shorter than the characteristic cooling time of the gas. As a result, the gas pressure rapidly becomes almost homogeneous, while the typical Mach number of the flow drops well below unity. Eliminating the acoustic modes, we reduce the hydrodynamic equations to a single nonlinear and nonlocal equation of a reaction-diffusion type in Lagrangian coordinates. This equation describes a broad class of channel flows and, in particular, can follow the development of the clustering instability from a weakly perturbed homogeneous cooling state to strongly nonlinear states. If the heat diffusion is neglected, the reduced equation is exactly soluble, and the solution develops a finite-time density blowup. The heat diffusion, however, becomes important near the attempted singularity. It arrests the density blowup and brings about novel inhomogeneous cooling states (ICSs) of the gas, where the pressure continues to decay with time, while the density profile becomes time-independent. Both the density profile of an ICS, and the characteristic relaxation time towards it are determined by a single dimensionless parameter that describes the relative role of the inelastic energy loss and heat diffusion. At large values of this parameter, the intermediate cooling dynamics proceeds as a competition between low-density regions of the gas. This competition resembles Ostwald ripening: only one hole survives at the end.