An exact solution of the inelastic Boltzmann equation for the Couette flow with uniform heat flux

TL;DR

This paper derives an exact analytical solution for the inelastic Boltzmann equation describing steady Couette flow in granular gases with uniform heat flux, revealing novel rheological and thermal properties.

Contribution

It presents a new class of exact solutions for the inelastic Boltzmann equation with uniform heat flux, linking shear rate and thermal gradient to inelasticity.

Findings

Reduced moments are polynomials in thermal gradient with coefficients depending on inelasticity.

Rheological properties are independent of thermal gradient and match simple shear flow.

Heat flux follows a generalized Fourier's law with an additional component.

Abstract

In the steady Couette flow of a granular gas the sign of the heat flux gradient is governed by the competition between viscous heating and inelastic cooling. We show from the Boltzmann equation for inelastic Maxwell particles that a special class of states exists where the viscous heating and the inelastic cooling exactly compensate each other at every point, resulting in a uniform heat flux. In this state the (reduced) shear rate is enslaved to the coefficient of restitution $\alpha$, so that the only free parameter is the (reduced) thermal gradient $\epsilon$. It turns out that the reduced moments of order $k$ are polynomials of degree $k-2$ in $\epsilon$, with coefficients that are nonlinear functions of $\alpha$. In particular, the rheological properties ($k=2$) are independent of $\epsilon$ and coincide exactly with those of the simple shear flow. The heat flux ($k=3$) is linear in the thermal gradient (generalized Fourier's law), but with an effective thermal conductivity differing from the Navier--Stokes one. In addition, a heat flux component parallel to the flow velocity and normal to the thermal gradient exists. The theoretical predictions are validated by comparison with direct Monte Carlo simulations for the same model.