Non-Newtonian stress, collisional dissipation and heat flux in the shear flow of inelastic disks: a reduction via Grad's moment method

摘要

The non-Newtonian stress tensor, collisional dissipation rate and heat flux in the plane shear flow of smooth inelastic disks are analysed from the Grad-level moment equations using the anisotropic Gaussian as a reference. For steady uniform shear flow, the balance equation for the second moment of velocity fluctuations is solved semi-analytically, yielding closed-form expressions for the shear viscosity μ, pressure p, first normal stress difference N_1 and dissipation rate D as functions of (i) density or area fraction v, (ii) restitution coefficient e, (iii) dimensionless shear rate R, (iv) temperature anisotropy η (the difference between the principal eigenvalues of the second-moment tensor) and (v) angle between the principal directions of the shear tensor and the second-moment tensor. The last two parameters are zero at the Navier-Stokes order, recovering the known exact transport coefficients from the present analysis in the limit η ?→ 0, and are therefore measures of the non-Newtonian rheology of the medium. An exact analytical solution for leading-order moment equations is given, which helped to determine the scaling relations of R, η and with inelasticity. We show that the terms at super-Burnett order must be retained for a quantitative prediction of transport coefficients, especially at moderate to large densities for small values of the restitution coefficient (e-1). Particle simulation data for a sheared inelastic hard-disk system are compared with theoretical results, with good agreement for p, μ and N_1 over a range of densities spanning from the dilute to close to the freezing point. In contrast, the predictions from a constitutive model at Navier-Stokes order are found to deviate significantly from both the simulation and the moment theory even at moderate values of the restitution coefficient (e~0.9). Lastly, a generalized Fourier law for the granular heat flux, which vanishes identically in the uniform shear state, is derived for a dilute granular gas by analysing the non-uniform shear flow via an expansion around the anisotropic Gaussian state. We show that the gradient of the deviatoric part of the kinetic stress drives a heat current and the thermal conductivity is characterized by an anisotropic second-rank tensor, for which explicit analytical expressions are given.