## 5 Discussion and Conclusions  
We have considered the 1D (in the direction normal to the flow) non-averaged model with static/flowing dynamics that was initially proposed in [41]. This model is based on the description of a granular material by a yield stress viscoplastic rheology. Here we have assumed that the source term in this model is constant, which corresponds to simple shear solutions to the viscoplastic model. Thus, all the quantities are assumed to be independent of the down-slope variable and in particular the flow thickness is constant. We have shown that this free interface model can be solved with reliable and relatively simple numerical methods. We have compared model solutions for both the inviscid and viscous models to observations from experiments on granular flow over an inclined static layer of grains. The analytical solution for the inviscid model and the numerical results for the viscous model reproduce quantitatively some essential features of the change with time of the velocity, of the static/flowing interface position, and of the stopping time of the granular mass, even though the flow thickness in the experiments is not perfectly uniform and the initial velocity profile changes with slope angle and flow thickness as discussed below. The analytical solution for the inviscid model shows that the evolution of the static/flowing interface position is proportional to the source term and inversely proportional to the shear rate (Equation (9)). For the viscous model (with constant viscosity), the analysis shows that the evolution of the interface is related to the viscosity, the source term, and the first- and third-order derivative of the source term and the velocity, respectively (Equation (8)). Owing to the appearance of this third-order derivative, the dynamics of the static/flowing interface cannot be reduced to a simple differential equation in terms of depth-averaged quantities. While the shape of the initial velocity profile is preserved at all times for the inviscid model according to Equation (15), the viscous model predicts that an exponential-like tail near the static/flowing transition and a convex shape near the free surface develop. The viscous contribution enables the static/flowing interface to initially penetrate within the static layer (which is eroded), as observed in the experiments, and contrary to the inviscid model. The viscoplastic model used here has the great advantage of involving only two parameters, i.e., the friction coefficient \(\mu_{s}\) and the viscosity \(\
u\) (or the coefficient \(\mu_{2}\) for the \(\mu(I)\) rheology), whereas the so-called partial fluidization theory, involving an order parameter to describe the transition between static and flowing material, also reproduces the erosion of the static bed and the velocity profiles predicted here with the viscous model [7], but at the cost of additional empirical equations for the time-change of a state parameter. One of the important results of the analysis lies in the explicit expressions obtained from the analytical solution, especially for the time evolution of the static/flowing interface. The dynamics are controlled by the source term \(S\) that is constant when the pressure is assumed to be hydrostatic and when the slope and thickness are assumed to be constant. In such a case, \(S(t,Z)=S=g\cos\theta(\tan\delta-\tan\theta)\), with \(\mu_{s}=\tan\delta\). Comparable results have been obtained from the analytical solution of thin-layer depth-averaged equations for a granular dam-break (i.e., with non-constant \(h\)) [43,44]. This analytical solution predicts a granular mass front velocity that decreases linearly with time  
\[U_{f}=\max\left(2\sqrt{kgh_{0}\cos\theta}-\left(g\cos\theta(\tan\delta-\tan \theta)\,t,0\right)\,,\right. \tag{47}\]  
where \(k\simeq 0.5\) (see [9]).