Bagnold velocity profile for steady-state dense granular chute flow with base slip

The three-halves Bagnold profile for granular flow down an incline, assuming no-slip at the base, is a generally accepted velocity profile that applies in many instances. In the intermediate dense regime, the material behaviour resembles a Bingham fluid and the widely accepted friction model - μ(I) rheology originally defined within the realms of visco-plasticity applies. Here for steady-state dense granular flow, we derive a generalization of the Bagnold profile which applies to the above- mentioned inter-particle friction approach, for which the Navier slip boundary condition applies at the base, and the Bagnold profile is included as a special case. We extend the Bagnold profile for a Navier slip base and we provide an expression for the slip length ℓ which demonstrates the dependence on the fundamental physical constants appearing in the μ(I) framework. The given velocity profile provides a simple formula that captures the major flow characteristics of dense granular chute flow.