3. Variational formulation of the problem

Here we will limit ourselves to presenting the problem with only the conditions at the limits of imposed temperature [§2.1], of imposed normal flow [§2.3] or of exchange [§2.4]. The boundary conditions of wall exchange [§2.5] are treated in [§4] and those with linear relationships [§2.2] are easily reduced to that of [§2.1].

Let \(\Omega\) be an open of \({ℝ}^{3}\), of border \(\Gamma ={\Gamma }_{1}\cup {\Gamma }_{2}\cup {\Gamma }_{3}\).

The weak formulation of the heat equation is:

\(\underset{\Omega }{\int }\rho {C}_{p}\frac{\partial T}{\partial t}\text{.}vd\Omega +\underset{\Omega }{\int }\lambda \nabla T\text{.}\nabla vd\Omega -\underset{\Gamma }{\int }\lambda \frac{\partial T}{\partial n}\text{.}vd\Gamma =\underset{\Omega }{\int }s\text{.}vd\Omega\)

where \(v\) is a sufficiently regular function cancelling uniformly over \({\Gamma }_{1}\). With the following boundary conditions:

\(\{\begin{array}{ccc}T={T}_{1}(r,t)& & \text{sur}{\Gamma }_{1}\\ \lambda \frac{\partial T}{\partial n}=q(r,t)& & \text{sur}{\Gamma }_{2}\\ \lambda \frac{\partial T}{\partial n}=h(r,t)({T}_{\text{ext}}(r,t)-T)& & \text{sur}{\Gamma }_{3}\end{array}\)

The variational formulation of the problem is:

\(\underset{\Omega }{\int }\rho {C}_{p}\frac{\partial T}{\partial t}\text{.}vd\Omega +\underset{\Omega }{\int }\lambda \nabla T\text{.}\nabla vd\Omega +\underset{{\Gamma }_{3}}{\int }hT\text{.}vd\Gamma =\underset{\Omega }{\int }s\text{.}vd\Omega +\underset{{\Gamma }_{2}}{\int }q\text{.}vd\Gamma +\underset{{\Gamma }_{3}}{\int }h{T}_{\text{ext}}\text{.}vd\Gamma\)