Exact results for non-Newtonian transport properties in sheared granular suspensions: inelastic Maxwell models and BGK-type kinetic model

Abstract

The Boltzmann kinetic equation for dilute granular suspensions under simple (or uniform) shear flow (USF) is considered to determine the non-Newtonian transport properties of the system. In contrast to previous attempts based on a coarse-grained description, our suspension model accounts for the real collisions between grains and particles of the surrounding molecular gas. The latter is modeled as a bath (or thermostat) of elastic hard spheres at a given temperature. Two independent but complementary approaches are followed to reach exact expressions for the rheological properties. First, the Boltzmann equation for the so-called inelastic Maxwell models (IMM) is considered. The fact that the collision rate of IMM is independent of the relative velocity of the colliding spheres allows us to exactly compute the collisional moments of the Boltzmann operator without the knowledge of the distribution function. Thanks to this property the transport properties of the sheared granular suspension can be \emph{exactly} determined. As a second approach, a Bhatnagar--Gross--Krook (BGK)-type kinetic model adapted to granular suspensions is solved to compute the velocity moments and the velocity distribution function of the system. The theoretical results show in general a good agreement with the approximate analytical results derived for inelastic hard spheres (IHS) by means of Grad's moment method and with computer simulations performed in the Brownian limiting case ($m/m_g\to \infty$, where $m_g$ and $m$ are the masses of the particles of the molecular and granular gases, respectively). In addition, as expected the IMM and BGK results show that the temperature and non-Newtonian viscosity exhibit and $S$ shape in a plane of stress-strain rate (discontinuous shear thickening, DST). The DST effect becomes more pronounced as the mass ratio $m/m_g$ increases.