## 4 Numerical Solution to the Viscous Free Interface Model
### Variable Viscosity
We choose \(\mu_{2}=\tan(28^{\circ})\), which leads to \(\
u(Z=b^{0})\simeq 5\times 10^{-5}\) m\({}^{2}\)s\({}^{-1}\), corresponding to the order of magnitude of \(\
u\) taken in Section 4.4. We consider the case of a linear initial velocity profile, with slope angles \(\theta=19^{\circ},22^{\circ},24^{\circ}\). The numerical results for the static/flowing interface position \(b(t)\) and velocity profiles \(U(Z)\) are plotted in Figures 14 and 15. In comparison with numerical results of Figures 10b and 11a,b obtained with the constant viscosity model, the shapes are similar, in particular for the initial erosion of the static bed. The behavior of the interface position \(b(t)\) reaching the free surface close to the stopping time is here closer to the prediction of the inviscid model (see Figure 5a). However, this behavior does not recover (since \(\
u\) vanishes at the free surface) the expected behavior, which is roughly linear, of the experimental results of Figure 2. The comparison with the experimental results is better with the \(\mu(I)\) rheology than with a constant viscosity. For clarity, the static/flowing interfaces and the velocity profiles corresponding to the different cases are plotted in Figure 16 for \(\theta=22^{\circ}\). \[I=\frac{2\|D\|d}{\sqrt{p_{dyn}/\rho_{s}}}, \tag{38}\]0 Static/flowing interface position \(b\) as a function of time \(t\), for variable viscosity associated with the \(\mu(I)\) law, with linear initial velocity profile and slope angles \(\theta=19^{\circ}\), \(\theta=22^{\circ}\), \(\theta=24^{\circ}\). The experimental results from Figure 2 are shown for comparison. It is important to explain the behavior of the solution close to the free surface. As said above, there is no need to impose any boundary condition at the free surface, because the pressure vanishes there. Nevertheless, the behavior close to the free surface of the solution to the parabolic problem (3) is deduced from the property that \(\partial_{Z}(v\partial_{Z}U)\) is bounded, which gives \(v\partial_{Z}U\sim h-Z\). For the model with constant viscosity, this leads to \(\partial_{Z}U\sim h-Z\) and \(U\) has a (rather) parabolic profile close to the free surface. For the model with \(\mu(I)\) rheology, Equation (41) implies that \(I\) tends to a finite value as \(Z\to h\), and thus with Equation (40), we obtain \(\partial_{Z}U\sim\sqrt{h-Z}\), and \(U\) has a (rather) Bagnold profile close to the free surface. We conclude that in any case, the Neumann condition \(\partial_{Z}U=0\) is recovered at the free surface, even if not enforced explicitly. For the model with the \(\mu(I)\) rheology, \(\partial_{Z}U\) tends to zero more slowly than for the model with constant viscosity, leading to a behavior closer to the prediction of the inviscid model. Nevertheless, the property \(\partial_{Z}U=0\) at the free surface, which seems to be a consequence of the incompressible viscoplastic model, makes it impossible to obtain a good representation of the experimental stopping phase. Note that the \(\mu(I)\) rheology was introduced precisely to represent the Bagnold profile (without a static phase) \(U(Z)=c(h^{3/2}-(h-Z)^{3/2})\), where \(c\) is determined by the relation \(\mu(I)=\tan\theta\), which gives  
\[I=\frac{2\|D\|d}{\sqrt{p_{dyn}/\rho_{s}}}, \tag{38}\]1  
\[I=\frac{2\|D\|d}{\sqrt{p_{dyn}/\rho_{s}}}, \tag{38}\]2 Static/flowing interface position \(b\) as a function of time \(t\) (left) and velocity profiles \(U(Z)\) at time \(t=0.5\) s (right), with linear initial velocity profile and slope angle \(\theta=22^{\circ}\), for respectively our model without viscosity, with constant viscosity \(\
u=5\times 10^{-5}\) m\({}^{2}\)s\({}^{-1}\), variable viscosity associated with the \(\mu(I)\) law, and experimental measurements. \[I=\frac{2\|D\|d}{\sqrt{p_{dyn}/\rho_{s}}}, \tag{38}\]3 Velocity profiles \(U(Z)\) at different times, for variable viscosity associated with the \(\mu(I)\) law, with linear initial velocity profile and slope angles \(\theta=19^{\circ}\) and \(\theta=24^{\circ}\).