This work is devoted to an analytical description of the dynamics of the static/flowing interface in thin dry granular flows.
Our starting point is the asymptotic model derived in Bouchut et al. (2016) from an incompressible model with viscoplastic rheology including a Drucker--Prager yield stress.
This asymptotic model is based on the thin-layer approximation
(the flow is thin in the direction normal to the topography compared to its along-slope extension),
but the equations are not depth-averaged. It involves a free surface at the top of the flow
and a free time-dependent static/flowing interface at the bottom.
In the present work, we simplify this asymptotic model by decoupling the space coordinates,
and keeping only the dependence on time and on the normal space coordinate.
This entails introducing a time- and space-dependent source term, assumed here to be given,
and which represents the opposite of the net force acting on the flow, including gravity, pressure gradient, and internal friction.
We prove several properties of the resulting simplified model.
The crucial advantage of this simplified model is that we can obtain explicit solutions in the inviscid case,
for different shapes of the source term.
These explicit inviscid solutions exhibit a rich behaviour and qualitatively reproduce some physical features observed in granular flows.
Our main finding is that the zero of the source term, defined as the value of the normal space coordinate for which the source term vanishes,
plays a monitoring role in the dynamics of the static/flowing interface.
Indeed, this zero separates two layers for which the net force is either a driving or a resistive force.
Depending on the time variation of this zero, different interface dynamics are obtained,
from the starting to the arrest of the flow, including a progressive starting, a progressive stopping
and a sudden start of part of the granular mass.
These scenarios are expected to appear in the general asymptotic model when the source term
is nonlinearly coupled to the velocity.
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