## 5 Discussion and Conclusions
Furthermore, the stopping time of the analytical front for the granular dam-break is  
\[t_{f}=2\sqrt{\frac{h_{0}}{g\cos\theta}}\frac{1}{\tan\delta-\tan\theta}. \tag{48}\]Equations (47) and (48) for the front velocity and the stopping time of the front for a depth-averaged model of a granular dam-break are very similar to Equations (15) and (17) respectively, found here for the velocity of the flow and for the stopping time of the granular layer in the non-averaged case and without viscosity. Although the initial penetration of the static/flowing interface into the static layer (erosion) can be reproduced by taking into account the viscosity, this leads at the same time to an underestimation of the stopping time. The viscous model better reproduces the exponential-like tail of the velocity profile near the static/flowing interface than the inviscid model, but overestimates the convexity of the velocity profile near the free surface. Furthermore, the change in shape of the velocity profile observed in the experiments during the stopping phase is reproduced by the viscous model, as opposed to the inviscid model. However, the decrease in maximum velocity near the surface is too fast in the viscous model. All these results suggest that (i) viscosity plays an important role near the static/flowing interface at depth and in this region a reasonable estimate for the viscosity is \(\
u\simeq 5\times 10^{-5}\) m\({}^{2}\)s\({}^{-1}\) and (ii) viscous effects in the experiments seem to be much smaller near the free surface. These observations suggest a non-constant viscosity, as proposed in the so-called \(\mu(I)\) flow law, e.g., [25; 26; 27; 28]. As in [32], we have used the \(\mu(I)\) flow law to derive the value of the viscosity, which becomes variable and nonlinear with respect to the shear rate. This flow law reproduces approximately the value of the viscosity at the static/flowing interface (i.e., \(\
u\simeq 5\times 10^{-5}\) m\({}^{2}\)s\({}^{-1}\)) and at the free surface (vanishing viscosity). However, this model does not allow us to significantly improve the accuracy of the behavior of the static/flowing interface close to stopping. This is because it imposes a Bagnold behavior close to the free surface, which is in contradiction with experiments that predict a linear behavior. A more detailed numerical simulation of the granular collapse over sloping beds with the complete viscoplastic model and the \(\mu(I)\) rheology gives a dynamic viscosity of around 0.2 Pa\(\cdot\) s near the front, increasing up to about 0.4 Pa\(\cdot\)s behind the front (see Figure 11 of [33] and Figure 10 of [32]), leading to a kinematic viscosity of \(\
u=1.3\times 10^{-4}\) m\({}^{2}\)s\({}^{-1}\) and \(\
u=2.6\times 10^{-4}\) m\({}^{2}\)s\({}^{-1}\), respectively, for a density of \(\rho=1550\)kg/m\({}^{3}\). Note also that our treatment of side wall effects deserves further improvements, since we simply increased the friction coefficient as done in [27; 32; 42]. Taking into account wall effects changes the static/flowing position, as shown in particular in [33] for the granular collapse considered here. More precisely, the static/flowing interface is found to be less deep within the flow when wall effects are accounted for (see Figures 3 and 4 of [33]). This effect could possibly increase the value of the viscosity needed in the model to reproduce the penetration of the static/flowing interface observed in the experiments. This would give values of the viscosity closer to that given by the \(\mu(I)\) rheology near the front (Figure 11 of [33]). It would be of interest to take into account the \(X\)-variations of the source \(S\) from ([41], Equation (2.12)), thereby accounting for topography, propagation and non-hydrostatic effects. This would make it possible to study erosion in flows over complex topography in the laboratory (e.g., [45]) or at the natural scale (e.g., [46]).