Non-Newtonian rheology in inertial suspensions of inelastic rough hard spheres under simple shear flow

TL;DR

This paper investigates the non-Newtonian behavior of inertial suspensions of inelastic rough spheres under shear flow, revealing how shear rate and roughness influence viscosity, temperature, and stability.

Contribution

It introduces a kinetic model combining Boltzmann and Fokker-Planck equations to analyze the rheology of inelastic rough spheres, including rotational effects and stability analysis.

Findings

Viscosity and temperature increase with shear rate, showing discontinuous shear thickening.

Roughness slightly reduces viscosity jump but affects rotational temperature.

Identifies parameter regions where steady shear flow becomes linearly unstable.

Abstract

Non-Newtonian transport properties of an inertial suspension of inelastic rough hard spheres under simple shear flow are determined from the Boltzmann kinetic equation. The influence of the interstitial gas on rough hard spheres is modeled via a Fokker-Planck generalized equation for rotating spheres accounting for the coupling of both the translational and rotational degrees of freedom of grains with the background viscous gas. The generalized Fokker-Planck term is the sum of two ordinary Fokker-Planck differential operators in linear $\mathbf{v}$ and angular $\boldsymbol{\omega}$ velocity space. As usual, each Fokker-Planck operator is constituted by a drag force term (proportional to $\mathbf{v}$ and/or $\boldsymbol{\omega}$) plus a stochastic Langevin term defined in terms of the background temperature $T_\text{ex}$. The Boltzmann equation is solved by two different but complementary approaches: (i) by means of Grad's moment method, and (ii) by using a Bhatnagar-Gross-Krook (BGK)-type kinetic model adapted to inelastic rough hard spheres. As occurs in the case of \emph{smooth} inelastic hard spheres, our results show that both the temperature and the non-Newtonian viscosity increase drastically with increasing the shear rate (discontinuous shear thickening effect) while the fourth-degree velocity moments also exhibit an $S$-shape. In particular, while high levels of roughness may slightly attenuate the jump of the viscosity in comparison to the smooth case, the opposite happens for the rotational temperature. As an application of these results, a linear stability analysis of the steady simple shear flow solution is also carried out showing that there are regions of the parameter space where the steady solution becomes linearly unstable.