The flow of fluid through a pipe of uniform (circular) cross-section is known as Hagen-Poiseuille flow. The Hagen-Poiseuille flow is an exact solution of the Navier-Stokes equations
in fluid mechanics
. The equations governing the Hagen-Poiseuille flow can be derived from the Navier-Stokes equation in cylindrical coordinates
by making the following set of assumptions:
- The flow is steady ( ).
- The radial and swirl components of the fluid velocity are zero ( ).
- The flow is axisymmetric ( ) and fully developed ( ).
Then the second of the three Navier-Stokes 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:
- The solution is
needs to be finite at
. The no slip boundary condition
at the pipe wall requires that
(radius of the pipe), which yields
Thus we have finally the following parabolic velocity profile:
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: