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

• Published in: Journal of fluid mechanics 2014-10, Vol.757, p.251-296
• Main Authors: Saha, Saikat, Alam, Meheboob
• Format: Article
• Language: English
• Subjects: Anisotropy; Coefficients; Cross-disciplinary physics: materials science; rheology; Density; Exact sciences and technology; Flow velocity; Fluctuations; Fluid mechanics; Freezing; Freezing point; Granular solids; Heat; Heat transfer; Material form; Mathematical analysis; Mathematical models; Navier-Stokes equations; Physics; Rheology; Shear flow; Tensors; Thermal conductivity
• ISSN: 0022-1120; 1469-7645
• DOI: 10.1017/jfm.2014.489

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 $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}\mu $ , pressure $p$ , first normal stress difference ${\mathcal{N}}_1$ and dissipation rate ${\mathcal{D}}$ as functions of (i) density or area fraction $\nu $ , (ii) restitution coefficient $e$ , (iii) dimensionless shear rate $R$ , (iv) temperature anisotropy $\eta $ (the difference between the principal eigenvalues of the second-moment tensor) and (v) angle $\phi $ 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 $\eta ,\phi \to 0$ , 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 $R$ , $\eta $ and $\phi $ 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 ( $e \ll 1$ ). Particle simulation data for a sheared inelastic hard-disk system are compared with theoretical results, with good agreement for $p$ , $\mu $ and ${\mathcal{N}}_1$ 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 ( $e\sim 0.9$ ). 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