Lattice models for granular-like velocity fields: Hydrodynamic limit

TL;DR

This paper analyzes a 1D lattice model for granular fluids, deriving hydrodynamic equations with fluctuating currents, and explores various flow regimes and their stability, highlighting deviations from local equilibrium.

Contribution

It introduces a detailed stochastic lattice model for granular velocity fields and derives hydrodynamic equations with fluctuations, extending understanding of granular flow regimes.

Findings

Hydrodynamic equations with fluctuating currents are derived.

Homogeneous cooling state exhibits linear instability.

Local equilibrium assumptions are generally invalid.

Abstract

A recently introduced model describing -on a 1d lattice- the velocity field of a granular fluid is discussed in detail. The dynamics of the velocity field occurs through next-neighbours inelastic collisions which conserve momentum but dissipate energy. The dynamics can be described by a stochastic equation in full phase space, or through the corresponding Master Equation for the time evolution of the probability distribution. In the hydrodynamic limit, equations for the average velocity and temperature fields with fluctuating currents are derived, which are analogous to those of granular fluids when restricted to the shear modes. Therefore, the homogeneous cooling state, with its linear instability, and other relevant regimes such as the uniform shear flow and the Couette flow states are described. The evolution in time and space of the single particle probability distribution, in all those regimes, is also discussed, showing that the local equilibrium is not valid in general. The noise for the momentum and energy currents, which are correlated, are white and Gaussian. The same is true for the noise of the energy sink, which is usually negligible.