Boundary conditions, loading and initial condition
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Refer to [:external:ref:`R5.02.01 <R5.02.01>`] for thermal boundary conditions and loads leading to linear temperature equations as well as for the initial condition.




Nonlinear normal flow
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These are Neumann-type conditions, defining the flow entering the domain.


.. _RefEquation 2.1-1:

:math:`-q(x,t)\mathrm{.}n=g(x,T)` on the border :math:`\mathrm{\Gamma }` eq 2.1-1

where :math:`g(x,T)` is a function of the temperature and possibly of the space variable :math:`x` and/or of the time variable :math:`t` and/or time and :math:`n` designates the external normal at the border :math:`\Gamma`, :math:`q` is the heat flow vector (directed according to decreasing temperatures).

This expression makes it possible to introduce, for example, exchange-type conditions with a convective exchange coefficient dependent on temperature:


.. _RefEquation 2.1-2:

:math:`-q(x,t)\mathrm{.}n=g(x,T)=h(x,T)({T}_{\text{ext}}(x,t)-T)` eq 2.1-2




Nonlinear normal flow - infinity radiation-like condition
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A particular case of the previous boundary conditions is the infinity radiation of gray bodies which results in a special case of function :math:`g(x,T)`:

:math:`-q(x,t)\mathrm{.}n=\sigma \epsilon \left[(T(x)+\text{273}\text{.}\text{15}{)}^{4}-({T}_{\infty }+\text{273}\text{.}\text{15}{)}^{4}\right]` eq 2.2-1

The characteristics to be defined when defining this load are emissivity, the Stefan-Boltzmann constant :math:`\sigma =\mathrm{5,73}{.10}^{-8}\mathrm{usi}` and the temperature to infinity.

:math:`T(r)` and :math:`{T}_{\infty }` are then expressed in degrees Celsius. :math:`–273.15°C` is the temperature of absolute zero.