Hisao Hayakawa

The rheology of a dilute binary mixture of inertial suspension under simple shear flow is analyzed in the context of the Boltzmann kinetic equation. The effect of the surrounding viscous gas on the solid particles is accounted for by means of a deterministic viscous drag force plus a stochastic Langevin-like term defined in terms of the environmental temperature Tenv. Grad’s moment method is employed to determine the temperature ratio and the pressure tensor in terms of the coefficients of restitution, concentration, the masses and diameters of the components of the mixture, and the environmental temperature. Analytical results are compared against event-driven Langevin simulations for mixtures of hard spheres with the same mass density m1/m2 = (σ(1)/σ(2))3, mi and σ(1) being the mass and diameter, respectively, of the species i. It is confirmed that the theoretical predictions agree with simulations of various size ratios σ(1)/σ(2) and for elastic and inelastic collisions in a wide range of parameter space. It is remarkable that the temperature ratio T1/T2 and the viscosity ratio η1/η2 (ηi being the partial contribution of the species i to the total shear viscosity η = η1 + η2) discontinuously change at a certain shear rate as the size ratio increases; this feature (which is expected to occur in the thermodynamic limit) cannot be completely captured by simulations due to the small system size. In addition, a Bhatnagar–Gross–Krook (BGK)-type kinetic model adapted to mixtures of inelastic hard spheres is exactly solved when Tenv is much smaller than the kinetic temperature T. A comparison between the velocity distribution functions obtained from Grad’s method, the BGK model, and simulations is carried out.

We propose a simple model comprising three particles to study the nonlinear mechanical response of jammed frictional granular materials under oscillatory shear.