The above problem is known as Hagen-Poiseuille flow.

We start with the steady, incompressible Navier-Stokes equations in a cylindrical coordinate system. In such a system, the governing equations are:

These equations may be derived by a control volume approach.

For a fully developed flow, the derivatives with respect to z- vanish, except for the pressure gradient term dp/dz. The continuity yields v equal to zero. Setting v to be zero in the v- momentum equation gives us ?p/?r =0, that is, pressure does not vary with r, at a given z- location.
 
 

In the u- momentum equation, many terms drop out, giving

Integrating this equation once, we get

For u to be bounded as r goes to zero, we require C to be zero.

Integrating again, and applying the boundary condition u=0 at r=R, we get