The hydrodynamic limit of viscoelastic granular gases

TL;DR

This paper rigorously derives an incompressible Navier-Stokes-Fourier system with dynamic forcing from the Boltzmann equation for viscoelastic granular gases, revealing new inelastic-hydrodynamic effects and confirming Haff's law.

Contribution

It provides the first rigorous derivation of a hydrodynamic limit for viscoelastic granular gases with velocity-dependent restitution coefficient.

Findings

Derivation of a new incompressible Navier-Stokes-Fourier system with time-dependent forcing.

Explicit self-similar rescaling capturing nontrivial inelastic effects.

Determination of the granular temperature dissipation rate (Haff's law).

Abstract

We obtain the first rigorous derivation of an incompressible Navier-Stokes-Fourier system with self-consistent and time-dependent forcing terms from the inelastic hard-spheres Boltzmann equation associated to the relevant case of viscoelastic granular gases. The model's inelasticity is measured by the so-called restitution coefficient which, for viscoelastic particles, depends on the relative velocities of particles. Through a suitable self-similar change of variables, a balanced dynamic between energy inflow and outflow naturally emerges in the model which permits its analysis. In contrast, such balanced dynamic does not emerge naturally in the constant restitution case and has to be imposed in our previous contribution (Alonso, Lods, Tristani, M\'emoires SMF). The exact self-similar rescaling, which allows to capture nontrivial inelastic-hydrodynamic effects, presents itself explicitly in terms of the Knudsen number and the restitution coefficient. The consequence of such scaling is a non-autonomous rescaled Boltzmann equation whose solutions converge, in a specific weak sense, towards the aforementioned hydrodynamic limit. The incompressible Navier-Stokes-Fourier system obtained by this process appears to be new in this context. As a byproduct of the analysis, we determine the exact dissipation rate of the granular temperature known as \emph{Haff's law}.