Continuous equations =================== Mechanics: conservation of momentum ---------------------------------------------------- We note :math:`\sigma` the Cauchy stress tensor and :math:`s` the second (symmetric) Piola-Kirchhoff tensor. We denote :math:`P` as the :math:`{x}_{0}={x}_{S}(0)\to {x}_{S}({x}_{0},t)` transformation gradient. :math:`P=\frac{\partial {x}_{S}({x}_{0},t)}{\partial {x}_{0}}` We have: :math:`s=\text{det}P\text{.}{P}^{-1}\text{.}\sigma \text{.}{P}^{-T}`. The mechanical balance equations are written in configuration :math:`{\Omega }_{0}`: :math:`{\text{Div}}_{0}(P\text{.}s)+r{F}^{m}=0` We noted :math:`{\text{Div}}_{0}` as the divergence operator with respect to the :math:`{x}_{0}` space variables in the :math:`{\Omega }_{0}` configuration. Insofar as we hypothesize small displacements and small deformations of the skeleton, this equation can be approximated by: :math:`\text{Div}\sigma +r{F}^{m}=0` **eq 3.1-1** We will see later that we are still adopting the :math:`\sigma =\sigma \text{'}+{\sigma }_{p}I` decomposition, where :math:`\sigma \text{'}` refers to the effective constraint. It is therefore up to the module for integrating balance equations to do the sum: :math:`\sigma =\sigma \text{'}+{\sigma }_{p}I`. Hydraulics: mass conservation -------------------------------------- The Eulerian writing for the conservation of fluid mass for the constituent :math:`c` is written: :math:`\frac{{d}^{\mathrm{fl}}}{\mathrm{dt}}\int {}_{\Omega }\text{}\sum _{p}{\rho }_{c}^{p}\phi {S}^{p}d\Omega =0` We can then apply [:ref:`éq 2.3-1 <éq 2.3-1>`] by taking: :math:`{a}_{s}=0` and :math:`{{a}^{m}}_{c}^{p}=1` and [:ref:`éq 2.3-3 <éq 2.3-3>`] will give: :math:`\sum _{p}\frac{{d}^{s}{\rho }_{c}^{p}\phi {S}^{p}}{\mathrm{dt}}+\sum _{p}\text{Div}({w}_{c}^{p})=0` Using the definition of mass inputs [:ref:`éq 2.2.2.3-3 <éq 2.2.2.3-3>`], the definition of Lagrangian flows [:ref:`éq2.2.2.3-2 <éq2.2.2.3-2>`] we find the Lagrangian form of fluid mass conservation: :math:`\{\begin{array}{}\dot{{m}_{1}}+{\text{Div}}_{0}({M}_{1})=0\\ \dot{{m}_{2}}+{\text{Div}}_{0}({M}_{2})=0\end{array}` **eq 3.2-1** Energy equation --------------------- For thermodynamic functions, we systematically adopt a decomposition of the [:ref:`éq2.3-1 <éq2.3-1>`] type. This corresponds to the fact that the different energies all have a part carried by the solid and a part carried by the fluids. The part carried by the solid is characterized by a volume density while the parts carried by the fluid are characterized by mass densities, as we showed in paragraph [:ref:`§2.3 <§2.3>`]. **Total internet energy:** :math:`E=\int {}_{\Omega }\text{}({e}_{s}+\sum _{p,c}{\rho }_{c}^{p}\phi {S}^{p}{{e}^{m}}_{c}^{p})d\Omega` **eq 3.3.1** **Total entropy:** :math:`S=\int {}_{\Omega }\text{}({s}_{s}+\sum _{p,c}{\rho }_{c}^{p}\phi {S}^{p}{{s}^{m}}_{c}^{p})d\Omega` **eq 3.3.2** **Total enthalpy:** :math:`H=\int {}_{\Omega }\text{}({h}_{s}+\sum _{p,c}{\rho }_{c}^{p}\phi {S}^{p}{{h}^{m}}_{c}^{p})d\Omega` **eq 3.3.3** **Free energy:** :math:`\{\begin{array}{}\Psi =E-TS\\ {\Psi }_{s}={e}_{s}-{\text{Ts}}_{s}\\ {{\Psi }^{m}}_{c}^{p}={{e}^{m}}_{c}^{p}-T{{s}^{m}}_{c}^{p}\end{array}` **eq 3.3.4** **Free enthalpy:** :math:`\{\begin{array}{}G=H-TS\\ {g}_{s}={h}_{s}-T{s}_{s}\\ {{g}^{m}}_{c}^{p}={{h}^{m}}_{c}^{p}-T{{s}^{m}}_{c}^{p}\end{array}` **eq 3.3.5** Finally, by noting :math:`\dot{Q}(\Omega )` the heat rate received by a volume :math:`\Omega`, we have by definition: :math:`\dot{Q}(\Omega )=\underset{\partial \Omega }{\int }q\text{.}nd\Gamma +\underset{\Omega }{\int }\Thetad \Omega` **eq 3.3.6** Finally, we recall that the enthalpy of fluids is calculated by the formula: :math:`h=e+\frac{p}{\rho }` **eq 3.3.7** The first principle ~~~~~~~~~~~~~~~~~~~~~ With the definitions given above, it is written: :math:`-\sum _{p,c}\text{Div}\left({{h}^{m}}_{c}^{p}{\mathrm{M}}_{c}^{p}\right)+\mathrm{\sigma }\mathrm{:}\dot{\mathrm{\epsilon }}+\sum _{p,c}{\mathrm{M}}_{c}^{p}\text{.}{\mathrm{F}}^{m}+\mathrm{\Theta }-\text{Div}q=0` **eq 3.3.1-1** This writing corresponds to equation (22) in chapter III -2-3 of [:ref:`bib1 `], in which we neglected inertia terms. For homogeneous media, it corresponds to equation (31) in paragraph IV-3-2 of [:ref:`bib3 `]. The second principle ~~~~~~~~~~~~~~~~~~~~~~ Its fairly well known form is: :math:`\dot{s}+\sum _{p,c}\text{Div}({{s}^{m}}_{c}^{p}{M}_{c}^{p})+\text{Div}(\frac{q}{T})-\frac{\Theta }{T}\ge 0` **eq 3.3.2-1** Using the classical thermodynamic considerations [:ref:`bib1 `] linked to the introduction of free enthalpy [:ref:`éq 3.3.5 <éq 3.3.5>`], we show that we must necessarily have: :math:`\sigma -\frac{\partial \Psi }{\partial \varepsilon }=0` **eq 3.3.2-2** :math:`{{g}^{m}}_{c}^{p}-\frac{\partial \Psi }{\partial {m}_{c}^{p}}=0` **eq 3.3.2-3** :math:`s+\frac{\partial \Psi }{\partial T}=0` **eq 3.3.2-4** Energy equation ~~~~~~~~~~~~~~~~~~~~~~~ Quite often, it is considered that, since the transformations are reversible, the second principle ultimately provides equality. In addition, in [:ref:`éq 3.3.2-1 <éq 3.3.2-1>`] the unknown temperature :math:`T` is replaced by a constant value called the reference temperature. It is finally a linearization of [:ref:`éq 3.3.2-1 <éq 3.3.2-1>`] justified if the temperature variations are "small". Note that the transport term :math:`\sum _{p,c}\text{Div}({{s}^{m}}_{c}^{p}{M}_{c}^{p})` complicates the treatment of nonlinearity due to the presence of temperature in the denominator of the other terms in [:ref:`éq 3.3.2-1 <éq 3.3.2-1>`]. We work in enthalpy in order to overcome this difficulty. We start from the equation of the first principle [:ref:`éq 3.3.1-1 <éq 3.3.1-1>`] into which we inject the equations [:ref:`éq 3.3.2-2 <éq 3.3.2-2>`], [], [:ref:`éq 3.3.2-3 <éq 3.3.2-3>`], [:ref:`éq 3.3.2-4 <éq 3.3.2-4>`], and the definition of free enthalpy [:ref:`éq 3.3-5 <éq 3.3-5>`] into which we obtain: :math:`T\dot{s}+\sum _{p,c}({{h}^{m}}_{c}^{p}\dot{{m}_{c}^{p}}-T{{s}^{m}}_{c}^{p}\dot{{m}_{c}^{p}})=-\sum _{p,c}\text{Div}({{h}^{m}}_{c}^{p}{M}_{c}^{p})+\sum _{p,c}{M}_{c}^{p}\text{.}{F}^{m}+\Theta -\text{Div}q` **eq 3.3.3-1** We then ask: :math:`\deltaq \text{'}=T\deltas -T\sum _{p,c}{s}^{{m}_{c}^{p}}\delta {m}_{c}^{p}` **eq 3.3.3-2** The quantity :math:`Q\text{'}` has the dimension of one energy per unit volume. It represents the heat received by the system in a transformation for which there is no heat input by entering a fluid having an enthalpy. Although :math:`\deltaq \text{'}` is not an exact differential, we take this quantity as a state variable. Finally, the energy equation adopted has the following form: :math:`\sum _{p,c}{{h}^{m}}_{c}^{p}\dot{{m}_{c}^{p}}+\dot{Q\text{'}}+\sum _{p,c}\text{Div}({{h}^{m}}_{c}^{p}{M}_{c}^{p})+\text{Div}q-\sum _{p,c}{M}_{c}^{p}\text{.}{F}^{m}=\Theta` **eq 3.3.3-3**