Extended kinetic theory applied to pressure-controlled shear flows of frictionless spheres between rigid, bumpy planes

We numerically investigate, through discrete element simulations, the steady flow of identical, frictionless spheres sheared between two parallel, bumpy planes in the absence of gravity and under a fixed normal load. We measure the spatial distributions of solid volume fraction, mean velocity, intensity of agitation and stresses, and confirm previous results on the validity of the equation of state and the viscosity predicted by the kinetic theory of inelastic granular gases. We also directly measure the spatial distributions of the diffusivity and the rate of collisional dissipation of the fluctuation kinetic energy, and successfully test the associated constitutive relations of the extended kinetic theory, i.e., a kinetic theory which includes the role of velocity correlations. We then phrase and numerically integrate a system of differential equations governing the flow, with suitably modified boundary conditions. We show a remarkable qualitative and quantitative agreement with the results of the discrete simulations. In particular, we study the effect of (i) the coefficient of collisional restitution, (ii) the imposed load and (iii) the bumpiness of the planes on the profiles of the hydrodynamic fields, the ratio of shear stress-to-pressure and the gap between the bumpy planes. Finally, we predict the critical value of the imposed load above which crystallization occurs, based on the value of the solid volume fraction near the boundaries obtained from the numerical solution of the kinetic theory. This notably reproduces what we observe in the discrete simulations.