4. Variational writing of equilibrium equations

4.1. Mechanics

We note \({U}_{ad}\) the set of kinematically admissible displacement fields, that is to say the elements of \({({H}^{1}(\Omega ))}^{3}\) verifying the boundary conditions while moving on the part of \(\partial \Omega\) supporting such conditions [bib3].

The variational form of [éq 3.1-1] is:

\(\{\begin{array}{c}\mathrm{\sigma }=\mathrm{\sigma }\text{'}+{\sigma }_{p}\mathrm{I}\\ \int {}_{\mathrm{\Omega }}\text{}\mathrm{\sigma }\text{.}\mathrm{\epsilon }\left(\mathrm{v}\right)=\int {}_{\mathrm{\Omega }}\text{}r{\mathrm{F}}^{m}\text{.}\mathrm{v}d\mathrm{\Omega }+\int {}_{\partial \mathrm{\Omega }}\text{}{\mathrm{f}}^{\mathit{ext}}\text{.}\mathrm{v}d\mathrm{\Gamma }\text{}\forall \mathrm{v}\in {U}_{\mathit{ad}}\end{array}\) eq 4.1-1

4.2. Hydraulics

We denote \({P}_{1\mathrm{ad}}\) (resp. \({P}_{2\mathrm{ad}}\)) the set of admissible pressure fields \({\pi }_{1}\) (resp. \({\pi }_{2}\)), that is to say the elements of \({H}^{1}(\Omega )\) verifying the pressure boundary conditions \({P}_{1}\) (resp. \({P}_{2}\)) on the part of \(\partial \Omega\) supporting such conditions [bib3]. The variational form of [éq 3.2-1] is:

\(\begin{array}{}-{\int }_{\Omega }(\dot{{m}_{1}^{1}}+\dot{{m}_{1}^{2}})\text{.}{\pi }_{1}+{\int }_{\Omega }({M}_{1}^{1}+{M}_{1}^{2})\text{.}\nabla {\pi }_{1}d\Omega =\text{}\\ {\int }_{\partial \Omega }({M}_{1\mathrm{ext}}^{1}+{M}_{1\mathrm{ext}}^{2})\text{.}{\pi }_{1}d\Gamma \forall {\pi }_{1}\in {P}_{1\mathrm{ad}}\\ -{\int }_{\Omega }(\dot{{m}_{2}^{1}}+\dot{{m}_{2}^{2}})\text{.}{\pi }_{2}+{\int }_{\Omega }({M}_{2}^{1}+{M}_{2}^{2})\text{.}\nabla {\pi }_{2}d\Omega =\text{}\\ {\int }_{\partial \Omega }({M}_{2\mathrm{ext}}^{1}+{M}_{2\mathrm{ext}}^{2})\text{.}{\pi }_{2}d\Gamma \forall {\pi }_{2}\in {P}_{2\mathrm{ad}}\end{array}\}\) eq 4.2-1

which shows scalar hydraulic flows \({M}_{i\mathrm{ext}}^{j}\) at the edges.

4.3. Thermal

We note \({T}_{\mathrm{ad}}\) the set of admissible temperature fields \(\tau\), that is to say the elements of \({H}^{1}(\Omega )\) verifying the temperature boundary conditions on the part of \(\partial \Omega\) supporting such conditions. [bib3]. The variational form of [éq 3.3.3-3] is:

\(\begin{array}{}{\int }_{\Omega }\dot{Q}\text{'}\text{.}\tau d\Omega +\sum _{p,c}{\int }_{\Omega }{{h}^{m}}_{c}^{p}\dot{{m}_{c}^{p}}\text{.}\tau d\Omega -{\int }_{\Omega }(\sum _{p,c}{{h}^{m}}_{c}^{p}{M}_{c}^{p}+q)\text{.}\nabla \tau d\Omega =\text{}\\ {\int }_{\Omega }(\Theta +\sum _{p,c}{M}_{c}^{p}\text{.}{F}^{m})\text{.}\tau d\Omega -{\int }_{\partial \Omega }(\sum _{p,c}{{h}^{m}}_{c}^{p}{M}_{c\mathrm{ext}}^{p}+{q}_{\mathrm{ext}})\text{.}\tau d\Gamma \\ \forall \tau \in {T}_{\mathrm{ad}}\end{array}\) eq 4.3-1

Note that, unlike other presentations, and in particular [bib8] we did not inject the mass conservation equations, and we integrated the transport term \(\sum _{p,c}\text{Div}({{h}^{m}}_{c}^{p}{M}_{c}^{p})\) in part.

This last point has the advantage of not causing higher-order derivatives to appear, and, on the contrary, of naturally causing boundary conditions relating to heat input linked to hydraulic flows to appear: \(\sum _{p,c}{\int }_{\partial \Omega }{{h}^{m}}_{c}^{p}{M}_{c\mathrm{ext}}^{p}\text{.}\tau d\Gamma\).

It may in fact be considered that the heat flow conditions directly define:

\(\stackrel{~}{{q}_{\mathit{ext}}}={{h}^{m}}_{c}^{p}{M}_{c\mathit{ext}}^{p}+{q}_{\mathit{ext}}\)