.. _vc_couette_flow:

============
Couette Flow
============

Couette flow can be defined as a flow between two plates, initially at rest, in which the top plate starts moving with a constant velocity :math:`U` while the bottom plate is kept with a null velocity. 
This configuration is illustrated below:

.. figure:: /_static/img/solver/validation/cases/couette.svg
    :width: 70%
    :align: center

This flow consists of a unidirectional flow with a velocity field that varies only at :math:`y`-direction. 
An analytical solution for it can be found in :footcite:t:`Pozrikidis2011-do`, and through this solution the velocity can be written as:

.. math::
    u\left(y,t\right)=U\frac{y}{h}-2\frac{U}{\pi}\sum_{n=1}^{\infty}\frac{1}{n}e^{-n^{2}\pi^{2}\frac{\nu t}{h^{2}}}\mathrm{sin}\left[n\pi\left(1-\frac{y}{h}\right)\right]
    :label: vc_startupcouette

After a sufficient time, the flow reaches a steady state Couette flow with a linear velocity profile:

.. math::
    u\left(y\right)=U\frac{y}{h}
    :label: vc_couette

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

   Startup Couette case<01_startup_couette.ipynb>

.. footbibliography::