Reference problem
=====================


Geometry
---------


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    :height: 2.3654in


.. _RefImage_1000000000000A1F0000058B33E289E31DB72F88.png:

We consider the plane thermal problem of a square cavity (with a side equal to 1) where heat propagates:


* by convection (i.e. the particles constituting the middle of the cavity move at a speed :math:`u` assumed to be constant here); the speed :math:`u` is supposed to make an angle of :math:`67.5°` with the axis :math:`x`,


* by conduction.


Material properties
-----------------------

We take :math:`\rho {C}_{P}=1.` :math:`\lambda ={10}^{-6}`

Hence a diffusivity :math:`\alpha =\frac{\lambda }{\rho {C}_{P}}={10}^{-6}`

and as we take :math:`\parallel u\parallel =1`, we have the Peclet number :math:`{p}_{e}=\frac{\parallel u\parallel \mathrm{.}L}{\alpha }={10}^{6}` (:math:`L` is the characteristic length, here :math:`L=1.`).


Boundary conditions and loads
-------------------------------------

On segments :math:`\mathrm{AB}` and :math:`\mathrm{BC}`, a temperature :math:`T=1.` is imposed

On segment :math:`\mathrm{AE}`, we impose a temperature :math:`T=0.`

On the other 2 sides, we have the condition by default, namely, we have zero flow.