  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.
