The radial and swirl components of the fluid velocity are zero ( ).
The flow is axisymmetric ( ) and fully developed ( ).
Then the angular equation in the momentum equations and the continuity equation are identically satisfied. The first momentum equation reduces to , i.e., the pressure is a function of the axial coordinate only. The third momentum equation reduces to:
where is the dynamic viscosity of the fluid.
The solution is
Since needs to be finite at , . The no slip boundary condition at the pipe wall requires that at (radius of the pipe), which yields
The maximum velocity occurs at the pipe centerline ():
The average velocity can be obtained by integrating over the pipe cross section:
The Hagen–Poiseuille equation relates the pressure drop across a circular pipe of length to the average flow velocity in the pipe and other parameters. Assuming that the pressure decreases linearly across the length of the pipe, we have (constant). Substituting this and the expression for into the expression for , and noting that the pipe diameter , we get:
Rearrangement of this gives the Hagen–Poiseuille equation: