Water, assumed incompressible, flows steadily through a very wide rectangular duct of depth w$.$ The entrance velocity is constant, u$=$ U $0,$ and the exit velocity approximates turbulent flow, u $=$ u max $(1-$ y $/$ h$)^1/7.$ h $=$ half height of duct, y $=$ distance up from the centerline. This expression is only good for the top half of the duct, so the total flow is $2$ X the integral of u dA$.$ Integrate from $0$ to h

Determine the ratio U $0/$ u max for this flow.