The granular silo as a continuum plastic flow: The hour-glass vs the clepsydra

The granular silo is one of the many interesting illustrations of the thixotropic property of granular matter: a rapid flow develops at the outlet, propagating upwards through a dense shear flow while material at the bottom corners of the container remains static. For large enough outlets, the discharge flow is continuous; however, by contrast with the clepsydra for which the flow velocity depends on the height of fluid left in the container, the discharge rate of granular silos is constant. Implementing a plastic rheology in a 2D Navier-Stokes solver (following the mu(I)-rheology or a constant friction), we simulate the continuum counterpart of the granular silo. Doing so, we obtain a constant flow rate during the discharge and recover the Beverloo scaling independently of the initial filling height of the silo. We show that lowering the value of the coefficient of friction leads to a transition toward a different behavior, similar to that of a viscous fluid, and where the filling height becomes active in the discharge process. The pressure field shows that large enough values of the coefficient of friction ($\simeq$ 0.3) allow for a low-pressure cavity to form above the outlet, and can thus explain the Beverloo scaling. In conclusion, the difference between the discharge of a hourglass and a clepsydra seems to reside in the existence or not of a plastic yield stress.