## 3 Analytical Solution for the Inviscid Model with Constant Source Term
### Choice of the Parameters and Initial Conditions  
In order to compare the analytical solution to the results presented in Figures 2 and 3 extracted from the experiments performed by [12], we have to define the friction angle \(\delta\), the slope angle \(\theta\), the thickness of the granular layer \(h\), the thickness of the initially static layer \(b^{0}\), and the initial velocity profile \(U^{0}(Z)\). Glass beads of diameter \(d=0.7\) mm were used, with repose and avalanche angles of about \(23^{\circ}\) and \(25^{\circ}\), respectively. Because wall effects are known to increase the effective friction for granular flows in channels such as those of [12], we use here a friction angle of \(\delta=26^{\circ}\)[27; 32; 42]. We perform different tests by varying the slope angle \(\theta\) from \(19^{\circ}\) to \(24^{\circ}\). Indeed, for slope angles \(\theta\) much lower than \(\delta\), the interface reaches the top of the layer very quickly, preventing us from observing a possible erosion of the static layer. We set \(b^{0}=5\) mm, which is the thickness of the initial static layer in the experiments, and \(h=0.02\) m, corresponding to the mean thickness at the position \(X=90\) cm where the measurements were performed, see Figure 3. Thus unless specified, we always take  
\[\delta=26^{\circ},\qquad b^{0}=0.005\,\text{m},\qquad h=0.02\,\text{m}. \tag{18}\]The objective here is only to compare the order of magnitude and the general trend of the analytical solution to the experimental results, since the experiments are more complicated than the uniform granular layer and the initial conditions defined in the model. In particular, the thickness of the granular layer in the experiments may vary by up to 20% during the flow and slightly depends on the inclination angle (see Figure 3). Furthermore, the initial velocity profiles and the maximum velocity also depend on the inclination angle, whereas we impose here the same velocity profile for all the tests. Velocity profiles in experimental granular flows have been extensively measured in different regimes, see e.g., [25]. For free surface flows over rigid inclined beds, the velocity profiles vary with inclination, thickness of the flow, and time. Essentially, the velocity profiles may vary from a linear profile for thin layers over small slope angles to Bagnold-like profiles for higher inclinations. The same general trend is observed for thicker flows (see Figure 5 of [25]). For surface flows over a pile of static grains, the velocity profiles roughly exhibit an upper linear part in the flowing layer and a lower exponential tail near the static/flowing interface (see Figure 6 of [25]). This is consistent with the measurements shown in Figure 3. Furthermore, experimental results suggest that the shear rate \(\partial_{Z}U\) is almost constant and equal to \(0.5\sqrt{\frac{S}{d}}\) (see e.g., Equation (11) of [25]). As a result, for a linear profile of slope \(\alpha_{1}\) (see case (a) below), we choose \(\alpha_{1}=70\) s\({}^{-1}\), which is consistent with the velocity profiles measured at \(t=0\) s (see Figure 3). In order to investigate the different possible profiles of the velocity, we choose three initial velocity profiles defined, for all \(Z\in[b^{0},h]\), as:  
1. linear profile \(U^{0}(Z)=\alpha_{1}(Z-b^{0})\), with \(\alpha_{1}=70\) s\({}^{-1}\),
2. exponential profile \(U^{0}(Z)=\alpha_{2}(e^{\beta Z}-e^{\beta b^{0}})\), with \(\alpha_{2}=0.1\) ms\({}^{-1}\) and \(\beta=130\) m\({}^{-1}\),
3. Bagnold profile \(U^{0}(Z)=\alpha_{3}((h-b^{0})^{\frac{3}{2}}-(h-Z)^{\frac{3}{2}})\), with \(\alpha_{3}=545\) m\({}^{-1/2}\)s\({}^{-1}\). In each case, the maximum velocity is \(U^{0}(h)\simeq 1\) ms\({}^{-1}\). For each profile, Equation (9) provides explicitly the time evolution of the static/flowing interface position as follows:  
1. \(b(t)=\frac{S}{\alpha_{1}}t+b^{0}\),
2. \(b(t)=\frac{1}{\beta}\log\left(\frac{S}{\alpha_{2}}t+e^{\beta b^{0}}\right)\),
3. \(b(t)=h-\left((h-b^{0})^{3/2}-\frac{S}{\alpha_{3}}t\right)^{2/3}\).