Hagen-Poiseuille Flow in a Circular duct

For a duct of circular cross section, the steady-state, fully developed axial velocity is described by 

Assuming that the pressure gradient is a specified constant and that the viscosity is also constant, the solution (see Eq. 4.8) is 

Since the radial domain in the duct ranges between 0 r rw, the constant Ci must be zero. Otherwise, the velocity would become unbounded at the centerline. The other constant is determined easily from a no-slip condition at the wall, rw. The solution is 

In this form, it is apparent that the pressure gradient dp/dz must be negative, which produces a positive velocity profile. It is convenient to represent this solution in terms of a mean velocity, which by definition is 

It is apparent that the mean velocity is exactly half the maximum velocity, which occurs at r — 0. In terms of the mean velocity, the velocity profile is stated as 

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