Problem overview
========================

The heat equation has strong nonlinearities under certain conditions. This is the case when the material undergoes phase changes: these are accompanied by a sudden variation in characteristic quantities (heat capacity, enthalpy). This nonlinearity is all the more accentuated when one treats the convection-diffusion problem, where the term transport dependent on the enthalpy function appears. The aim of this modeling is to deal with this last problem in a steady state (stationary case).

In all cases, it is assumed that the speed field is known a priory. The case of a mobile solid is quite frequent in practice. In particular, it relates to welding applications or surface treatment that involve a heat source moving in a given direction and at a given speed. The thermal problem is then studied in a frame of reference linked to the source.

The partial derivative problem results from the global heat balance equation for any domain :math:`\Omega` which is written as:


.. _RefEquation 1-1:

:math:`\begin{array}{ccc}\frac{d}{\text{dt}}\underset{\Omega }{\mathrm{\int }}\mathit{\rho \beta d}\Omega \mathrm{=}& \underset{\Omega }{\mathrm{\int }}\mathit{Qd}\Omega \mathrm{-}& \underset{\mathrm{\partial }\Omega }{\mathrm{\int }}q\text{.}nd\Gamma \\ \text{accumulation}& \text{création}+& \text{entrée-sortie}\end{array}` eq 1-1

In this equation, :math:`O` represents a related domain, within the system under study, which is followed in its movement, :math:`\beta` represents the specific enthalpy of the material and :math:`\rho` designates its density. :math:`Q` is a volume heat source, :math:`q` is the heat flow across border :math:`\partial \Omega` (:math:`n` being the outside normal), and :math:`d/\text{dt}` is the **particle derivative**.

The first term in [:ref:`éq 1-1 <éq 1-1>`] is written (see for example [:ref:`bib1 <bib1>`]):


.. _RefEquation 1-2:

:math:`\frac{d}{\text{dt}}\underset{\Omega }{\mathrm{\int }}\mathit{\rho \beta d}\Omega \mathrm{=}\underset{\Omega }{\mathrm{\int }}(\frac{\mathrm{\partial }(\rho \beta )}{\mathrm{\partial }t}+\text{div}(\rho \beta V))d\Omega` eq 1-2

or, taking into account mass conservation :math:`(\frac{\mathrm{\partial }\rho }{\mathrm{\partial }t}+\text{div}(\rho V)\mathrm{=}0)` [:ref:`bib1 <bib1>`]:


.. _RefEquation 1-3:

:math:`\frac{d}{\text{dt}}\underset{\Omega }{\mathrm{\int }}\rho \beta d\Omega \mathrm{=}\underset{\Omega }{\mathrm{\int }}(\rho \frac{\mathrm{\partial }\beta }{\mathrm{\partial }t}+\rho V\text{.}\text{grad}\beta )d\Omega` eq 1-3

where :math:`V` is the movement speed vector for the :math:`\Omega` domain. :math:`V` is entered under the simple CONVECTION keyword from commands AFFE_CHAR_THER and AFFE_CHAR_THER_F.

The second term of the second member of [:ref:`éq 1-1 <éq 1-1>`] is written, taking into account the divergence theorem and Fourier's law :math:`(q\mathrm{=}\mathrm{-}k(T)\text{grad}T)`:


.. _RefEquation 1-4:

:math:`\underset{\mathrm{\partial }\Omega }{\mathrm{\int }}q\text{.}nd\Gamma \mathrm{=}\underset{\Omega }{\mathrm{\int }}\text{div}qd\Omega \mathrm{=}\mathrm{-}\underset{\Omega }{\mathrm{\int }}\text{div}(k(T)\text{grad}T)d\Omega` eq 1-4

where :math:`T` is the temperature and :math:`k(T)` is the thermal conductivity of the material, a function of temperature.

The equation [:ref:`éq 1-1 <éq 1-1>`] must be satisfied for any :math:`\Omega` domain, so it comes:


.. _RefEquation 1-5:

:math:`\rho \frac{\mathrm{\partial }\beta }{\mathrm{\partial }t}+\rho V\text{.}\text{grad}\beta \mathrm{-}\text{div}(k(T)\text{grad}T)\mathrm{=}Q` in :math:`\Omega` eq 1-5

**Note:**

*Note that the classical case with,* :math:`k(T)=k` * *(constant) and* :math:`V=0` *, and where the specific enthalpy is a linear function of temperature,* :math:`\beta (T)\mathrm{=}\text{CT}` *returns the well-known classical equation:*

:math:`\rho C\frac{\mathrm{\partial }T}{\mathrm{\partial }t}\mathrm{-}\mathit{kDT}\mathrm{=}Q` *in* :math:`\Omega`

*where* :math:`\Delta` *is Laplacian and* :math:`C` *(constant) represents specific heat.*

The partial derivative problem treated by the command THER_NON_LINE_MO [:external:ref:`U4.33.04 <U4.33.04>`], consists in solving the equation [:ref:`éq 1-5 <éq 1-5>`] in the stationary case (directly in the permanent state) with boundary conditions on the border :math:`\partial \Omega`.

This problem is formally written in the following form:


.. _RefEquation 1-6:

:math:`\begin{array}{cc}V\text{.}\text{grad}u(T)-\text{div}(k(T)\text{grad}T)=Q,& \text{dans}\Omega ,\\ +\text{conditions aux limites}& \text{sur}\partial \Omega \end{array}` eq 1-6

where we adopted the notation, valid for the whole sequence, :math:`u(T)\mathrm{=}\rho \beta (T)` where :math:`\rho` is constant, defining the volume enthalpy.