Sluice gates are used to regulate water level (or flow rate) in open channels. The figure shows a gate that is adjusted so that the upstream depth is maintained at a depth h₁. The density of water is r, and the acceleration of gravity is g. The water flow under the gate may be considered incompressible and inviscid.

Suppose the downstream depth is measured as h₂, that is, the quantities h₁, h₂, r, and g are known.

(a) Assuming uniform velocities at the far upstream and downstream stations1 and 2 derive an expression for the horizontal force F, per unit breadth, required to hold the gate in place. Check your result by showing that it gives zero when h₁=h₂ and the hydrostatic result when h₂=0.