The boundary conditions of the 2D Inclined Channel Model

The second model that we study is sketched in Fig. fig:b_sketch. The ``set-up'' is a two dimensional channel of depth $L_y$ and of length $L_x$, vertically confined by a bottom and a top inelastic wall, with periodic boundary conditions in the direction parallel to the flow. The channel is tilted up by an angle $\phi$ with respect to the horizontal so that gravity has both components $g_x=g\sin{\phi}$ and $g_y=g\cos{\phi}$.

Figure 4.2: A sketch of the second model where the only energy source is gravity, with components in both directions

This model mimics the experiment performed by Azanza et al. [6], where a stationary flow in a two dimensional inclined channel was observed at a point far from the source of the granular material. The assumption of periodic boundary conditions in the direction of the flow is consistent with the observed stationarity, due to the balance between the gravity drift and the damping effect of inelastic collisions (for a discussion of the possible regimes that can be shown by one particle in presence of this balance, see [153]).