## 3 Analytical Solution for the Inviscid Model with Constant Source Term
### Results and Comparison with Experiments  
We compare here our analytical results with the experimental data shown in Figures 2 and 3. The velocity profiles at the position \(X=90\) cm from the gate at different times for slope angles \(\theta=19^{\circ}\), \(22^{\circ}\), \(23^{\circ}\), and \(24^{\circ}\) were taken from the experiments of [9, 12, 13]. The position of the static/flowing interface has been calculated using a velocity threshold of 1 cm/s. The material has been assumed to be static when the particle velocity is lower than this threshold. In order to observe velocity profiles on one side of the flow (through the transparent channel wall) and monitor the evolution of the interface separating flowing and static grains, black beads have been used as tracers at a volume fraction of about 50%. Measurements have been made from successive frames during short time intervals of 0.01-0.04 s ([13]).  
The \(b(t)\) curves are plotted in Figure 5 and the \(U(Z)\) profiles in Figures 6 and 7. The evolution of the static/flowing interface position \(b(t)\) obtained from the analytical solution (Figure 5) reproduces to a certain extent the experimental observations. The shape of \(b(t)\) is directly related to the velocity profile, as demonstrated by Equation (9). For the analytical solution, depending on the initial velocity profile, the stopping time is in the range 2.75-3 s for \(\theta=24^{\circ}\), 1.35-1.5 s for \(\theta=22^{\circ}\), and 0.75-0.85 s for \(\theta=19^{\circ}\), whereas \(t^{\text{stop}}\simeq 3.4\) s, \(t^{\text{stop}}\simeq 1.4\) s, and \(t^{\text{stop}}\simeq 0.9\) s in the experiments, respectively.  
As a result, the stopping time is well reproduced by the analytical solution, even though its strong increase for \(\theta=24^{\circ}\) is underestimated in the analytical solution. On the other hand, the penetration of the static/flowing interface within the initially static bed is not reproduced by the analytical solution that instead predicts a static/flowing interface rising towards the free surface at all times.  
Figure 5: Static/flowing interface position \(b\) as a function of time \(t\) for the inviscid model, different slope angles and an initially static granular layer of thickness \(b^{0}=5\) mm, using (**a**) a linear, (**b**) an exponential, and (**c**) a Bagnold initial velocity profile. The experimental results from Figure 2 are shown for comparison.  
Figure 6: Velocity profiles \(U(Z)\) at different times for the inviscid model, with a linear initial velocity profile and an initially static granular layer of thickness \(b^{0}=5\) mm over an inclined plane of slope (**a**) \(\theta=19^{\circ}\) and (**b**) \(\theta=24^{\circ}\). The experimental results from Figure 3 are shown for comparison.  
The decrease of the velocity with time is relatively well reproduced up to about 0.8 s. With the analytical solution, the velocity profiles maintain the same shape whereas the maximum velocity decreases, as roughly observed in the experiments. The decrease of the maximum velocity in the experiments and with the analytical solution are very similar (Figure 7). At later times (\(t\geq 1\) s) and at \(\theta=22^{\circ}\) and \(\theta=24^{\circ}\), the experiments show a clear change in the velocity profile (see Figure 3c at \(t=1\) s and \(t=2\) s) that is not reproduced by the constant shape of the velocity profiles predicted by the analytical solution. Furthermore, the maximum velocity decreases much faster in the experiments. In the experiments, the velocity profiles seem to be closer to linear for smaller slopes (\(\theta=19^{\circ}\) and \(\theta=22^{\circ}\)) and more exponential for \(\theta=24^{\circ}\). Referring to Equation (16) and Figure 5, this may explain why \(b(t)\) measured experimentally has an exponential shape at \(\theta=24^{\circ}\), whereas it is closer to linear for smaller slope angles.