.. _vc_poiseuille_pipe_flow:

====================
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.

.. figure:: /_static/img/solver/validation/cases/poiseuille_pipe.svg
    :width: 80%
    :align: center

This results in an unidirectional flow to which an analytical solution of the velocity profile with :math:`0 \leq r \leq 1` is given by :math:numref:`vc_poiseuille_pipe_vel_profile`

.. math::
    \frac{u(r)}{u_{\mathrm{avg}}}=2\left(1 - r^2\right)
    :label: vc_poiseuille_pipe_vel_profile

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

.. math::
    u_{\mathrm{avg}}=\left(-\frac{\mathrm{d}p}{\mathrm{d}x}\right)\frac{\left(0.5d\right)^{2}}{8\rho\nu}
    :label: vc_poiseuille_pipe_avg_velocity

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

.. toctree::
   :maxdepth: -1
   :hidden:

   Poiseuille Pipe (periodic) <03.1_poiseuille_pipe_periodic.ipynb>
   Poiseuille Pipe (velocity-Neumann) <03.2_poiseuille_pipe_vel_neumann.ipynb>