Poiseuille Pipe Flow

Poiseuille pipe flow can be described as a pressure driven axissimetric flow that occurs in a circular cross-section pipe, as illustrated below.

This results in an unidirectional flow to which an analytical solution of the velocity profile with \(0 \leq r \leq 1\) is given by (1)

(1)\[\frac{u(r)}{u_{\mathrm{avg}}}=2\left(1 - r^2\right)\]

In which \(u_{\mathrm{avg}}\) is the average velocity that can be obtained numerically, calculating its value using the simulation data, or using the analytical equation:

(2)\[u_{\mathrm{avg}}=\left(-\frac{\mathrm{d}p}{\mathrm{d}x}\right)\frac{\left(0.5d\right)^{2}}{8\rho\nu}\]

where the pressure gradient can be set through pressure inlet/outlet boundary conditions, or adding a external force to the bulk \(F_{x}=-\mathrm{d}p/\mathrm{d}x\).