Depth-averaged model for entraining granular flow in gradually change width channel

Erosion at the base of a debris flow controls how large the flow become and how far it can travel. Due to the complexity of the geological structure, the width of the valley is highly varied which profoundly affect the base erosion of debris flow. To understand the relationship between the entrainment rate and the valley width, we develop a new depth-averaged model to describe dry granular flow in a gradually changed width channel. The model achieves closure with three balance equations, mass, momentum, and kinetic energy. We also assume a linearized mu(I) rheology and Coulomb friction along the sidewalls. By introducing the width varying terms into the equations, the theory predicts the granular entrainment in the channel. Furthermore, by doing a scaling analysis, we find two dimensionless numbers can characterize the flow behavior. To validate the theory, we conduct experiments with a 40-cm diameter vertical rotating drum and 2mm- diameter grains. The parabolic plates are placed in the drum to create a convex/concave channel. The grain motion is monitored using a high-speed camera and Particle Tracking Velocimetry. The empirical results are remarkably consistent with our theoretical predictions.