ABSTRACT
We locate the free surface where the velocity profile exhibits a sharp change in slope. We place the origin of our coordinate system (x, s, z) there and on the centerline of the channel. Above the free surface, grains are thrown upwards in ballistic trajectories to a dilute region experiencing rare collisions. In this ballistic layer where s < 0, ux/
√ gd varies little and ν decreases
away from the free surface with d2ν/ds2 > 0.
