1. Variational writings of equilibrium equations

1.1. Mechanics

We start from the following differential writing:

\(\text{Div}\sigma +r{\text{F}}^{m}=0\) eq 1.1-1

We will see later that we are still adopting the \(\sigma =\sigma \text{'}+{\sigma }_{p}I\) decomposition, where \(\sigma \text{'}\) refers to the effective constraint.

It is therefore up to the module for integrating balance equations to do the sum: \(\sigma =\sigma \text{'}+{\sigma }_{p}I\).

We will then write a variational form of [éq 1.1-1] in time \({t}^{\text{+}}\).

\(\{\begin{array}{c}{\sigma }^{\text{+}}=\sigma {\text{'}}^{\text{+}}+{\sigma }_{p}^{\text{+}}I\\ {\int }_{\Omega }{\sigma }^{\text{+}}\mathrm{.}\epsilon (\text{v})={\int }_{\Omega }{r}^{\text{+}}{\text{F}}^{{m}^{\text{+}}}\mathrm{.}\text{v}+{\int }_{\partial \Omega }{\text{f}}^{{\mathit{ext}}^{\text{+}}}\mathrm{.}\text{v}\forall \text{v}\in {U}_{\mathit{ad}}\end{array}\) eq 1.1-2

1.2. Hydraulics

We start from the following differential writing:

\(\frac{\mathit{dm}}{\mathit{dt}}+\text{Div}(\text{M})=0\) eq 1.2-1

It is considered that there can be two components, and for each of them two phases.

More precisely, the variables \({m}_{\mathrm{1,}}{\text{M}}_{1}\) and \({m}_{\mathrm{2,}}{\text{M}}_{2}\) each relate to a conservative mass constituent.

As a matter of principle, we set out:

\(\begin{array}{c}{m}_{1}={m}_{1}^{1}+{m}_{1}^{2};{\text{M}}_{1}={\text{M}}_{1}^{1}+{\text{M}}_{1}^{2}\\ {m}_{2}={m}_{2}^{1}+{m}_{2}^{2};{\text{M}}_{2}={\text{M}}_{2}^{1}+{\text{M}}_{2}^{2}\end{array}\)

What we will write:

\(\begin{array}{c}{m}_{\mathit{constituant}}=\sum _{\mathit{nb}\mathit{phase}\mathit{du}\mathit{constituant}}{m}_{\mathit{constituant}}^{\mathit{phase}}\\ {\text{M}}_{\mathit{constituant}}=\sum _{\mathit{nb}\mathit{phase}\mathit{du}\mathit{constituant}}{\text{M}}_{\mathit{constituant}}^{\mathit{phase}}\end{array}\)

In applications, for example, we could have:

2 components: air and water

2 phases for water

1 phase for air

We would then have:	\({m}_{1}^{1}\mathit{et}{\text{M}}_{1}^{1}\): mass supply and liquid water flow
	\({m}_{1}^{2}\mathit{et}{\text{M}}_{1}^{2}\): mass supply and steam flow
	\({m}_{2}^{1}\mathit{et}{\text{M}}_{2}^{1}\): mass input and dry air flow
	\({m}_{2}^{2}\mathit{et}{\text{M}}_{2}^{2}\): non-existent

It is considered that there are two pressures. No hypothesis is made about what pressures \({p}_{1}\mathit{et}{p}_{2}\) mean, it will depend on the laws of behavior and how we choose to write them: we could for example choose:

\(\begin{array}{c}{p}_{1}=\mathit{pression}\mathit{capillaire}(p(\mathit{gaz})-p(\mathit{liquide}))\\ {p}_{2}=\mathit{pression}\mathit{de}\mathit{gaz}(\mathit{vapeur}+\mathit{gaz})\end{array}\)

We will then write a variational form of [éq 1.2-1].

\(-{\int }_{\Omega }d\frac{({m}_{1}^{1}+{m}_{1}^{2})}{\mathit{dt}}{\pi }_{1}+{\int }_{\Omega }({\text{M}}_{1}^{1}+{\text{M}}_{1}^{2})\mathrm{.}\nabla {\pi }_{1}={\int }_{\partial \Omega }({\text{M}}_{\mathrm{1ext}}^{1}+{\text{M}}_{\mathrm{1ext}}^{2})\mathrm{.}{\pi }_{1}\forall {\pi }_{1}\in {P}_{\mathrm{1ad}}\) eq 1.2-2

\(-{\int }_{\Omega }d\frac{({m}_{2}^{1}+{m}_{2}^{2})}{\mathit{dt}}{\pi }_{2}+{\int }_{\Omega }({\text{M}}_{2}^{1}+{\text{M}}_{2}^{2})\mathrm{.}\nabla {\pi }_{2}={\int }_{\partial \Omega }({\text{M}}_{\mathrm{2ext}}^{1}+{\text{M}}_{\mathrm{2ext}}^{2})\mathrm{.}{\pi }_{2}\forall {\pi }_{2}\in {P}_{\mathrm{1ad}}\) eq 1.2-3

After discretization by a theta method:

\(\begin{array}{c}-{\int }_{\Omega }({m}_{1}^{1\text{+}}+{m}_{1}^{2\text{+}}){\pi }_{1}+\theta \Delta t{\int }_{\Omega }({\text{M}}_{1}^{1\text{+}}+{\text{M}}_{1}^{2\text{+}})\mathrm{.}\nabla {\pi }_{1}=\\ -{\int }_{\Omega }({m}_{1}^{1\text{-}}+{m}_{1}^{2\text{-}}){\pi }_{1}-(1-\theta )\Delta t{\int }_{\Omega }({\text{M}}_{1}^{1\text{-}}+{\text{M}}_{1}^{2\text{-}})\mathrm{.}\nabla {\pi }_{1}\\ +\Delta t{\int }_{\partial \Omega }({\text{M}}_{1\mathit{ext}}^{1\theta }+{\text{M}}_{\mathrm{1ext}}^{2\theta })\mathrm{.}{\pi }_{1}\forall {\pi }_{1}\in {P}_{\mathrm{1ad}}\end{array}\) eq 1.2-4

\(\begin{array}{c}-{\int }_{\Omega }({m}_{2}^{1\text{+}}+{m}_{2}^{2\text{+}}){\pi }_{2}+\theta \Delta t{\int }_{\Omega }({\text{M}}_{2}^{1\text{+}}+{\text{M}}_{2}^{2\text{+}})\mathrm{.}\nabla {\pi }_{2}=\\ -{\int }_{\Omega }({m}_{2}^{1\text{-}}+{m}_{2}^{2\text{-}}){\pi }_{2}-(1-\theta )\Delta t{\int }_{\Omega }({\text{M}}_{2}^{1\text{-}}+{\text{M}}_{2}^{2\text{-}})\mathrm{.}\nabla {\pi }_{2}\\ +\Delta t{\int }_{\partial \Omega }({\text{M}}_{2\mathit{ext}}^{1\theta }+{\text{M}}_{\mathrm{2ext}}^{2\theta })\mathrm{.}{\pi }_{2}\forall {\pi }_{2}\in {P}_{\mathrm{2ad}}\end{array}\) eq 1.2-5

Note:

In the context of saturated permanent HM modeling, the term \(\frac{{\mathit{dm}}_{1}^{1}}{\mathit{dt}}\) disappears from the writing of the conservation of fluid mass. The latter is simply written:

\(\text{Div}({\text{M}}_{1}^{1})=0\)

The corresponding variational form is written as:

\({\int }_{\Omega }{\text{M}}_{1}^{1}\mathrm{.}\nabla {\pi }_{1}={\int }_{\partial \Omega }{\text{M}}_{\mathrm{1ext}}^{1}\mathrm{.}{\pi }_{1}\forall {\pi }_{1}\in {P}_{\mathrm{1ad}}\)

1.3. Thermal

We introduce the enthalpies of each phase of each constituent: \({h}_{cm}^{p}\)

We note: \({\mathit{np}}_{c}\) the number of phases of the component c.

We adopt the rule for summing mute indices:

\({h}_{cm}^{p}{\text{M}}_{c}^{p}=\sum _{i=1}^{{\mathit{np}}_{c}}{h}_{\mathit{cm}}^{i}{\text{M}}_{c}^{i}\) \({h}_{cm}^{p}\frac{{\mathit{dm}}_{c}^{p}}{\mathit{dt}}=\sum _{i=1}^{{\mathit{np}}_{c}}{h}_{\mathit{cm}}^{i}\frac{{\mathit{dm}}_{c}^{i}}{\mathit{dt}}\)

The thermal (or energy) equation is written as:

\(\frac{\mathit{dQ}\text{'}}{\mathit{dt}}+{h}_{cm}^{p}\frac{{\mathit{dm}}_{c}^{p}}{\mathit{dt}}+\text{Div}({h}_{cm}^{p}{\text{M}}_{c}^{p}+\text{q})=R+{\text{M}}_{c}^{p}\mathrm{.}{\text{F}}^{m}\) eq 1.3-1

We will then write a variational form of [éq 1.3-1] without injecting the hydraulic equilibrium equation into it:

\(\begin{array}{c}{\int }_{\mathrm{\Omega }}\frac{\mathit{dQ}\text{'}}{\mathit{dt}}\mathrm{\tau }+{\int }_{\mathrm{\Omega }}{h}_{c\phantom{\rule{2em}{0ex}}m}^{p}\frac{{\mathit{dm}}_{c}^{p}}{\mathit{dt}}\mathrm{\tau }-{\int }_{\mathrm{\Omega }}({h}_{c\phantom{\rule{2em}{0ex}}m}^{p}{\text{M}}_{c}^{p}+\text{q})\mathrm{.}\nabla \mathrm{\tau }={\int }_{\mathrm{\Omega }}(R+{\text{M}}_{c}^{p}\mathrm{.}\text{F})\mathrm{\tau }-{\int }_{\partial \mathrm{\Omega }}({h}_{c\phantom{\rule{2em}{0ex}}m}^{p}{\text{M}}_{c\mathit{ext}}^{p}+{\text{q}}_{\mathit{ext}})\mathrm{.}\mathrm{\tau }\\ \phantom{\rule{18em}{0ex}}\forall \mathrm{\tau }\in {T}_{\mathit{ad}}\hfill \end{array}\) eq1.3-2

The discretization of [éq 1.3-2] by theta method leads to:

\(\begin{array}{c}{\int }_{\mathrm{\Omega }}(Q{\text{'}}^{\text{+}}-Q{\text{'}}^{\text{-}})\mathrm{\tau }-\mathrm{\theta }\mathrm{\Delta }t{\int }_{\mathrm{\Omega }}(({h}_{c\phantom{\rule{2em}{0ex}}m}^{p\text{+}}{\text{M}}_{c}^{p\text{+}}+{\text{q}}^{\text{+}}))\nabla \mathrm{\tau }(1-\mathrm{\theta })\mathrm{\Delta }t{\int }_{\mathrm{\Omega }}(({h}_{c\phantom{\rule{2em}{0ex}}m}^{p\text{-}}{\text{M}}_{c}^{p\text{-}}+{\text{q}}^{\text{-}}))\nabla \mathrm{\tau }+\dots \\ +\mathrm{\theta }{\int }_{\mathrm{\Omega }}{h}_{c\phantom{\rule{2em}{0ex}}m}^{p\text{+}}({m}_{c\phantom{\rule{2em}{0ex}}m}^{p\text{+}}-{m}_{c\phantom{\rule{2em}{0ex}}m}^{p\text{-}})\mathrm{\tau }+(1-\mathrm{\theta }){\int }_{\mathrm{\Omega }}{h}_{c\phantom{\rule{2em}{0ex}}m}^{p\text{-}}({m}_{c\phantom{\rule{2em}{0ex}}m}^{p\text{+}}-{m}_{c\phantom{\rule{2em}{0ex}}m}^{p\text{-}})\mathrm{\tau }\phantom{\rule{4em}{0ex}}=\phantom{\rule{0ex}{0ex}}\\ \mathrm{\theta }\mathrm{\Delta }t{\int }_{\mathrm{\Omega }}{\text{M}}_{c}^{p\text{+}}\mathrm{.}{\text{F}}^{m}\mathrm{\tau }+(1-\mathrm{\theta })\mathrm{\Delta }t{\int }_{\mathrm{\Omega }}{\text{M}}_{c}^{p\text{-}}\mathrm{.}{\text{F}}^{m}\mathrm{\tau }+\mathrm{\Delta }t{\int }_{\mathrm{\Omega }}{R}^{\mathrm{\theta }}\mathrm{\tau }-\mathrm{\Delta }t{\int }_{\mathrm{\Omega }}({h}_{c\phantom{\rule{2em}{0ex}}m}^{p}{\text{M}}_{\mathit{ext}}^{p\mathrm{\theta }}+{\text{q}}_{\mathit{ext}}^{\mathrm{\theta }})\mathrm{.}\mathrm{\tau }\\ \phantom{\rule{20em}{0ex}}\forall \mathrm{\tau }\in {T}_{\mathit{ad}}\hfill \end{array}\) eq 1.3-3

We notice in equation [éq 1.3-3] a term for heat input by the flow of fluid at the edge of the domain: \({\int }_{\partial \Omega }({h}_{cm}^{p}{\text{M}}_{c\mathit{ext}}^{p\theta }+{\text{q}}_{\mathit{ext}}^{\theta })\mathrm{.}\tau\).

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

\(\stackrel{̃}{\text{q}}{}_{\mathit{ext}}^{\theta }={h}_{cm}^{p}{\text{M}}_{c\mathit{ext}}^{p\theta }+{\text{q}}_{\mathit{ext}}^{\theta }\)