Deriving Half-Max Velocity in Pipe Flow

[jonnygeordie]

Could someone help me with this. In pipe flow show that the average velocity is half of the maximum velocity. I have it written in my notes that this is true, but not how to derive it.

 

[Hao]

You can get the answer if you solve Navier-Stokes equation in cylindrical coordinates with the following assumptions:

1) Flow is incompressible.
2) Pipe pressure decreases linearly.
3) Flow rate is finite.
4) Flow rate is static, hence, the fluid only flows in one particular direction (you can guess which one based on the symmetry of a cylindrical pipe).
5) Flow rate at the sides of the pipe = 0.