Fluid acceleration down inclined channel – unsteady?

[humphreybogart]

1. Can a fluid which is accelerating down an inclined channel with constant acceleration still satisfy the continuity equation for a constant-density fluid?

Since the acceleration will be down the slope, in the streamwise direction, then surely ∇⋅U > 0?

2. Assuming that continuity is satisfied above, then is the flow unsteady?

I can't seem to agree with my textbook that it is, since I imagine if you used an accelerating reference frame (with acceleration equal to that of the accelerating fluid), then the flow would be steady. Am I misunderstanding something?

 

[Orodruin]

If the cross sectional area of the flow decreases.

This flow would not be steady in the accelerated frame because different parts of the fluid move at different velocities and so fluid would flow away from you if you follow a stream line.