Non-Newtonian hydrodynamics for a dilute granular suspension under uniform shear flow

摘要

We study in this work a steady shearing laminar flow with null heat flux(usually called "uniform shear flow") in a gas-solid suspension at low density.The solid particles are modeled as a gas of smooth hard spheres with inelasticcollisions while the influence of the surrounding interstitial fluid on thedynamics of grains is modeled by means of a volume drag force, in the contextof a rheological model for suspensions. The model is solved by means of threedifferent but complementary routes, two of them being theoretical (Grad'smoment method applied to the corresponding Boltzmann equation and an exactsolution of a kinetic model adapted to granular suspensions) and the otherbeing computational (Monte Carlo simulations of the Boltzmann equation). Unlikein previous studies on granular sheared suspensions, we include in our Grad'ssolution nonlinear terms in the stress tensor in the collisional momentassociated with the momentum transfer. This theoretical enhancement allows usfor the detection and evaluation of the normal stress differences in the planenormal to the laminar flow. In addition, the exact solution of the kineticmodel gives the explicit form of the velocity moments of the velocitydistribution function. Comparison between our theoretical and numerical resultsshows in general a good agreement for the non-Newtonian rheological properties,the kurtosis (fourth velocity moment of the distribution function) and thevelocity distribution of the kinetic model for quite strong inelasticity andnot too large values of the (scaled) friction coefficient characterizing theviscous drag force. This shows the accuracy of our analytical results thatallows us to describe in detail the flow dynamics of the granular suspensionwith zero heat flux throughout the paper.