.. _vc_poiseuille_channel_flow:

=======================
Poiseuille Channel Flow
=======================

Poiseuille periodic flow can be described as a flow between two stationary plates, caused by a pressure gradient :math:`\mathrm{d}p/\mathrm{d}x` parallel to the plates.
This is illustrated in the figure below.

.. figure:: /_static/img/solver/validation/cases/poiseuille_channel.svg
    :width: 60%
    :align: center

We're mainly interested in the velocity profile from :math:`y=0`` to :math:`y=1` (top plate) in the flow direction.
This profile is analytically defined by equation :math:numref:`vc_poiseuille_periodic_vel_profile`

.. math:: 
    \frac{u(y)}{u_{\mathrm{avg}}}=6\left(y - y^2\right)
    :label: vc_poiseuille_periodic_vel_profile

And :math:`u_{\mathrm{avg}}` is the average velocity, which 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{h^{2}}{12\rho\nu}
    :label: vc_poiseuille_channel_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 (periodic) <02.1_poiseuille_channel_periodic.ipynb>
   Poiseuille (periodic and regularized) <02.2_poiseuille_channel_regularized.ipynb>
   Poiseuille (velocity-Neumann) <02.3_poiseuille_channel_vel_neumann.ipynb>
   Poiseuille (multilevel) <02.4_poiseuille_channel_multilevel.ipynb>