An alternative expression can be written by using the relationship between wall shear stress and pressure gradient (Eqs. 6 and 10) is given by,
                                            (11)
This velocity profile is parabolic in radial coordinate system. The velocity is maximum at the centerline of the pipe and becomes zero at the pipe wall. Integrating Eq. (11), the volume flow rate can be obtained as,
                                            (12)
Now, the average velocity in terms of volume flow rate is given by,
                                            (13)
so that for this flow,
                                          (14)
                                          (15)