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

Abstract

Non-Newtonian transport properties of an inertial suspension of inelastic rough hard spheres under simple shear flow are determined by 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 v and angular ω velocity space. As usual, each Fokker–Planck operator is constituted by a drag force term (proportional to v and/or ω) plus a stochastic Langevin term defined in terms of the background temperature Tex. 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 in the case of smooth inelastic hard spheres, our results show that both the temperature and the non-Newtonian viscosity increase drastically with an increase in 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. The present work extends previous theoretical results (H. Hayakawa and S. Takada, “Kinetic theory of discontinuous rheological phase transition for a dilute inertial suspension,” Prog. Theor. Exp. Phys. 2019, 083J01 and R. G. González and V. Garzó, “Simple shear flow in granular suspensions: Inelastic Maxwell models and BGK-type kinetic model,” J. Stat. Mech. 2019, 013206) to rough spheres.