The flow of fluid through a pipe of uniform (circular) cross-section is known as Hagen–Poiseuille flow. The equations governing the Hagen–Poiseuille flow can be derived directly from the Navier–Stokes equation in cylindrical coordinates by making the following set of assumptions:
The flow is steady ( <math> partial(...)/partial t = 0 </math> ).
The radial and swirl components of the fluid velocity are zero ( <math> u_r = u_theta = 0 </math> ).
The flow is axisymmetric ( <math> partial(...)/partial theta = 0 </math> ) and fully developed (<math> partial u_z/partial z = 0 </math> ).
Then the second of the three Navier–Stokes momentum equations and the continuity equation are identically satisfied. The first momentum equation reduces to <math> partial p/partial r = 0 </math>, i.e., the pressure <math> p </math> is a function of the axial coordinate <math> z </math> only. The third momentum equation reduces to:
<math> fracfracleft(r fracright)= frac frac</math> where <math>mu</math> is the dynamic viscocity of the fluid.