The Knudsen temperature jump and the Navier-Stokes hydrodynamics of granular gases driven by thermal walls

TL;DR

This paper enhances the hydrodynamic modeling of granular gases driven by thermal walls by incorporating the Knudsen temperature jump boundary condition, validated through molecular dynamics simulations, leading to improved accuracy in predicting temperature profiles.

Contribution

It introduces a boundary condition correction for the Navier-Stokes equations accounting for the Knudsen temperature jump, with the numerical pre-factor determined via MD simulations.

Findings

Hydrodynamic predictions match MD simulations well.

The modified boundary condition improves temperature profile accuracy.

The approach applies to both elastic and inelastic granular gases.

Abstract

Thermal wall is a convenient idealization of a rapidly vibrating plate used for vibrofluidization of granular materials. The objective of this work is to incorporate the Knudsen temperature jump at thermal wall in the Navier-Stokes hydrodynamic modeling of dilute granular gases of monodisperse particles that collide nearly elastically. The Knudsen temperature jump manifests itself as an additional term, proportional to the temperature gradient, in the boundary condition for the temperature. Up to a numerical pre-factor of order unity, this term is known from kinetic theory of elastic gases. We determine the previously unknown numerical pre-factor by measuring, in a series of molecular dynamics (MD) simulations, steady-state temperature profiles of a gas of elastically colliding hard disks, confined between two thermal walls kept at different temperatures, and comparing the results with the predictions of a hydrodynamic calculation employing the modified boundary condition. The modified boundary condition is then applied, without any adjustable parameters, to a hydrodynamic calculation of the temperature profile of a gas of inelastic hard disks driven by a thermal wall. We find the hydrodynamic prediction to be in very good agreement with MD simulations of the same system. The results of this work pave the way to a more accurate hydrodynamic modeling of driven granular gases.