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Discharge of a granular silo as a visco-plastic flow
L. Staron∗ and P.-Y. Lagr´ee∗
Granular matter is a well-known example of thixotropic material, able to flow like
a viscous fluid or resist shear stress like a solid, and evolving from one state to the
other over a distance typically of a few grain diameters. The discharge of a silo is
one of the situations where the thixotropic properties of granular matter are best
illustrated: while a dilute flow creates a free fall arch in the vicinity of the aperture,
higher regions undergo a slower shear, and region closer to the corners remains static
(Figure 1).
Unlike the clepsydra, the discharge of a granular silo implies a constant rate, dictated
by the size of the aperture, but independent of the height of material stored. This
phenomenology - known as the Berveloo law - is often understood as resulting from
the friction forces mobilized at the walls of the silo, thereby decreasing the apparent
weight of the material, and screening the bottom area from the pressure now partly
sustained by the walls (the ”Janssen effect”). This explanation fails however in the
case of wide systems for which walls are distant from several times the height of
material stored.
In this contribution, we simulate the continuum counterpart of the granular silo by
implementing the visco-plastic µ(I)-rheology in a 2D Navier-Stokes solver (Gerris).
We observe a constant discharge rate irrespective of the initial filling height, and we
recover the Berverloo scaling relating discharge rate and aperture size. This result
points at the existence of a yield stress, rather than at the mobilization of friction
forces at walls, as controling the discharge of the granular silo. Moreover, comparison
with Contact Dynamics simulations show the reliability of the µ(I)-rheology when
modeling complex flow of granular matter.
∗ CNRS - Universit´
e Pierre et Marie Curie Paris 6, UMR 7190, Institut Jean Le Rond d’Alembert,
F-75005 Paris, France.
Figure 1: Velocity field in a numerical granular silo (left, Contact Dynamics) and its viscoplastic continuum counterpart (right, Gerris solver).