This is an incomplete file brought over from the original Wiki and needs to be heavily edited and updated.
A general set of mass conservation equations can be written as \f[ \frac{\partial \rho_i}{\partial t} + \nabla\cdot(\rho_i\vec{v}) + \nabla\cdot\vec{J}_i = \dot{\omega}_i \quad \forall; i\in\mathscr{M}, \f] where the index \f$i;\in\mathscr{M}\f$ refers to the \f$i^\text{th}\f$ mass component being modeled.
\f$\vec{J}i\f$ is the diffusion mass flux of component \f$i\f$. When chemical nonequilibrium is assumed, components may be individual species and/or energy states. In this case \f$J_i = \rho_i \vec{V}i ;\forall;i\in\mathscr{S}\f$ and \f$\mathscr{M} = \mathscr{S}\f$ is the set of species/energy state indices being considered. For chemical equilibrium, \f$\vec{J}i = \sum{k\in\mathscr{S}} \nu{ik} M{w,i}/M_{w,k} \rho_k \vec{V}_k\f$ if elemental diffusion is allowed, or \f$\vec{J}_i = 0\f$ otherwise.
The production rates of component \f$i\f$, \f$\dot{\omega}i\f$ represent any process which may produce or destroy a mass component such as elementary chemical reactions. In general, both mass diffusion fluxes and production rates are linearly dependent following \f$\sum{i\in\mathscr{M}}\vec{J}i = \vec{0}\f$ and \f$\sum{i\in\mathscr{M}}\dot{\omega}_i = 0\f$.
Momentum conservation is written as \f[ \frac{\partial\rho\vec{v}}{\partial t} + \nabla\cdot(\rho\vec{v}\otimes\vec{v})
- \nabla\cdot(p\matrix{I}) - \nabla\cdot\matrix{\tau} = \vec{0}, \f] where \f$\matrix{\tau}\f$ is the second order shear stress tensor \f[ \matrix{\tau} = \eta \left[ \nabla \vec{v} + (\nabla \vec{v})^T - \frac{2}{3} (\nabla\cdot \vec{v}) \matrix{I} \right]. \f] The total density is defined simply as \f$\rho = \sum_{i\in\mathscr{M}} \rho_i\f$. Pressure is given by Dalton's Law as \f$p = \sum_{k\in\mathscr{S}} p_k = \sum_{k\in\mathscr{H}} n_k k_B T_h + n_e k_B T_e\f$, where the heavy particle translation temperature \f$T_h\f$ may or may not equal the electron temperature \f$T_e\f$ depending on the model in use.
The internal energy of the system may be considered split into \f$n^\mathscr{E}\f$ different energy modes which follow Boltzmann distributions according to an individual temperature prescribed to each mode. Total energy conservation is then written accordingly as \f[ \frac{\partial \rho E}{\partial t} + \nabla\cdot(\rho\vec{v}H) = \nabla\cdot(\matrix{\tau}\cdot\vec{v}) -\nabla\cdot\vec{q}, \f] where the total energy and enthalpy densities are the summation of each internal energy mode and the bulk kinetic energy density. \f[ \rho E = \sum_{m\in\mathscr{E}} \rho e^m + \frac{1}{2}\rho\vec{v}\cdot\vec{v} \f] \f[ \rho H = \sum_{m\in\mathscr{E}} \rho h^m + \frac{1}{2}\rho\vec{v}\cdot\vec{v} \f] \f$\mathscr{E} = {1,\dots,n^\mathscr{E}}\f$ is the set of energy mode indices. Likewise, the total heat flux vector is the summation of heat fluxes due to each energy mode. \f[ \vec{q} = \sum_{m\in\mathscr{E}} \vec{q}^m \f] \f[ \vec{q}^m = \sum_{k\in\mathscr{S}}\rho_k h_k^m\vec{V}_k-\lambda^m\nabla T^m \f]
\f$n^\mathscr{E}-1\f$ additional energy conservation equations are necessary to close the system. They may be written as \f[ \frac{\partial\rho e^m}{\partial t} + \nabla\cdot(\rho\vec{v}e^m) = -\nabla\cdot\vec{q}^m + \Omega^m - \delta_{m\mathscr{I_e}}; p_e\nabla\cdot\vec{v} \quad \forall; m \in {2,\dots,n^\mathscr{E}} \f] where \f$\delta\f$ is the Kronecker delta function and \f$\mathscr{I_e}\f$ is the index of the energy mode which includes the free electron energy contribution. \f$\Omega^m\f$ is the energy transfered into the energy mode \f$m\f$ (and out of all other modes).
| Symbol | Description | Function |
|---|---|---|
| \f$n^\mathscr{E}\f$ | Number of energy modes | (not available) |
| \f$n^\mathscr{M}\f$ | Number of mass components | (not available) |
| \f$\vec{J}_i\f$ | Mass diffusion fluxes | (not available) |
| \f$\dot{\omega}_i\f$ | Mass production rates | (not available) |
| \f$T^m\f$ | Temperature for each mode | Mutation::Thermodynamics::StateModel::getTemperatures() |
| \f$T_h\f$ | Heavy particle translation temperature | (not available) |
| \f$T_e\f$ | Electron translation temperature | (not available) |
| \f$\lambda^m\f$ | Thermal conductivity of each energy mode | (not available) |
| \f$e^m\f$ | Energy per mass of each mode | (not available) |
| \f$h^m\f$ | Enthalpy per mass of each mode | (not available) |
| \f$\Omega^m\f$ | Energy transfer source terms | (not available) |
| \f$\mathscr{I_e}\f$ | Index of free electron energy mode | (not available) |