## 4 Numerical Solution to the Viscous Free Interface Model
### Results and Comparison with Experiments for Constant Viscosity  
We consider a constant and uniform source term of the form (14). The case of a linear initial velocity profile is simulated using different values of the viscosity \(\
u\) (taken constant) and slope angles \(\theta\). The results obtained with the two methods of Sections 4.1 and 4.2 are always identical, thus we shall not specify which one is used. As discussed in Section 4.3, the static/flowing interface always penetrates the initially static layer, in contrast to what has been observed for the inviscid model, within a length \(b^{0}-b^{\min}\) proportional to \(\
u\), according to (36). In other words, the flow excavates the static bed. However, for \(\
u<10^{-5}\) m\({}^{2}\)s\({}^{-1}\), the penetrating length is too small to be observed. For \(\
u=10^{-5}\) m\({}^{2}\)s\({}^{-1}\) (Figure 9(a)), \(b^{0}-b^{\min}=8\times 10^{-4}\) m for \(\theta=24^{\circ}\) (with \(t^{c}=5\times 10^{-2}\) s) and \(b^{0}-b^{\min}=2\times 10^{-4}\) m for \(\theta=19^{\circ}\) (with \(t^{c}=4\times 10^{-3}\) s). Thus, the static/flowing interface penetrates only slightly within the initially static layer. As the viscosity increases (Figure 9(b)), the static/flowing interface penetrates deeper into the initially static layer and even reaches the bottom for \(\
u=10^{-4}\) m\({}^{2}\)s\({}^{-1}\) at \(\theta=24^{\circ}\) (Figure 9(c)). The results that best reproduce the experimental observation for the penetration of \(b(t)\) within the initially static layer are obtained with \(\
u\simeq 5\times 10^{-5}\) m\({}^{2}\)s\({}^{-1}\). In good agreement with the experiments, the simulations with viscosity predict that the depth and duration \(t^{c}\) of penetration of the static/flowing interface within the initially static layer increase with the slope angle. Furthermore, the values of \(b(t)\) and \(t^{c}\) are in reasonable agreement with those observed experimentally (Figure 9(b) and Figure 17 of [12]). Qualitatively similar results are obtained using different initial velocity profiles (Figure 12). However, the shape of \(b(t)\) is affected by the choice of the initial velocity profile. For example, for an exponential initial velocity profile with \(\theta=24^{\circ}\) and \(\
u=5\times 10^{-5}\) m\({}^{2}\)s\({}^{-1}\), the static/flowing interface position \(b(t)\) stagnates at an almost constant position for the first 0.5s, contrary to the case of a linear initial velocity profile (Figures 9(b) and 9(b)). The convex shape of \(b(t)\) during the migration of the static/flowing interface up to the free surface obtained with the viscous model is very different from that observed and from that obtained with the inviscid model, which predicted a linear shape related to the linear initial velocity profile. With the viscous model, the time evolution of \(b(t)\) and the velocity profile \(U(t,Z)\) are not so clearly related to the shape of the initial velocity profile, as shown for example in Figure 12 for \(\theta=24^{\circ}\). The velocity profiles for the viscous model are also very different from those for the inviscid model. Whatever the shape of the initial velocity profile, the velocity profiles later exhibit an exponential-like tail near the static/flowing interface, similar to that observed experimentally. They also exhibit a convex shape near the free surface which can be observed, although not as marked, in some but not all experimental velocity profiles (e.g., Figure 4(c)). While the maximum velocity decreases too fast at \(t=0.5\) s and \(\theta=24^{\circ}\) compared to the experiments and to the inviscid model, the velocity is much closer to the experiments at \(t=1\) s than with the inviscid model (Figure 9(b)). For \(\theta=19^{\circ}\) and \(\theta=22^{\circ}\), the decrease in velocity at time \(t=0.7\) s is overestimated in the viscous model. Finally, we briefly look at the stopping time \(t^{\text{stop}}\) of the whole granular layer. The stopping time for the inviscid model and for the experiments is given in Table 1, for different slope angles.