# twoFluidEqn¶

Two fluid equations written as total mass density, momentum density, total charge density total current density and ion and electron energy. The two-fluid equations can also be written as two separate sets of euler equations, however, this form has the advantage that numerical diffusion is applied to the total charge density so that quasi-neutrality is enforced numerically.

\notag \begin{align} \frac{\partial}{\partial t}\left( \begin{array}{c} \rho \\ \rho\,u_{x} \\ \rho\,u_{y} \\ \rho\,u_{z} \\ \rho_{c}\\ j_{x}\\ j_{y}\\ j_{z}\\ e_{i}\\ e_{e}\\ \end{array} \right) + \nabla\cdot P = 0 \end{align}

where $$P$$ is defined as

$\begin{split}\begin{pmatrix} \rho_{i}\,u_{x\,i}+\rho_{i}\,u_{x\,e} & \rho_{i}\,u_{y\,i}+\rho_{e}\,u_{y\,e} & \rho_{i}\,u_{z\,i}+\rho_{e}\,u_{z\,e} \\ \rho_{i}\,u_{x\,i}^{2}+P_{i}+ \rho_{e}\,u_{x\,e}^{2}+P_{e} & \rho_{i}\,u_{x\,i}\,u_{y\,i}+\rho_{e}\,u_{x\,e}\,u_{y\,e} & \rho_{i}\,u_{x\,i}\,u_{z\,i}+\rho_{e}\,u_{x\,e}\,u_{z\,e} \\ \rho_{i}\,u_{y\,i}\,u_{x\,i}+\rho_{e}\,u_{y\,e}\,u_{x\,e} & \rho_{i}\,u_{y\,i}\,u_{y\,i} + P_{i}+\rho_{e}\,u_{y\,e}\,u_{y\,e} + P_{e} & \rho_{i}\,u_{y\,i}\,u_{z\,i}+\rho_{e}\,u_{y\,e}\,u_{z\,e} \\ \rho_{i}\,u_{z\,i}\,u_{x\,i}+\rho_{e}\,u_{z\,e}\,u_{x\,e} & \rho_{i}\,u_{z\,i}\,u_{y\,i}+\rho_{e}\,u_{z\,e}\,u_{y\,e} & \rho_{i}\,u_{z\,i}\,u_{z\,i} + P_{i}+\rho_{e}\,u_{z\,e}\,u_{z\,e} + P_{e} \\ \rho_{i}\,u_{x\,i}+\rho_{i}\,u_{x\,e} & \rho_{i}\,u_{y\,i}+\rho_{e}\,u_{y\,e} & \rho_{i}\,u_{z\,i}+\rho_{e}\,u_{z\,e} \\ r_{i}(\rho_{i}\,u_{x\,i}^{2}+P_{i})+ r_{e}(\rho_{e}\,u_{x\,e}^{2}+P_{e}) & r_{i}\rho_{i}\,u_{x\,i}\,u_{y\,i}+r_{e}\rho_{e}\,u_{x\,e}\,u_{y\,e} & r_{i}\rho_{i}\,u_{x\,i}\,u_{z\,i}+r_{e}\rho_{e}\,u_{x\,e}\,u_{z\,e} \\ r_{i}\rho_{i}\,u_{y\,i}\,u_{x\,i}+r_{e}\rho_{e}\,u_{y\,e}\,u_{x\,e} & r_{i}(\rho_{i}\,u_{y\,i}\,u_{y\,i} + P_{i})+r_{e}(\rho_{e}\,u_{y\,e}\,u_{y\,e} + P_{e}) & r_{i}\rho_{i}\,u_{y\,i}\,u_{z\,i}+r_{e}\rho_{e}\,u_{y\,e}\,u_{z\,e} \\ r_{i}\rho_{i}\,u_{z\,i}\,u_{x\,i}+r_{e}\rho_{e}\,u_{z\,e}\,u_{x\,e} & r_{i}\rho_{i}\,u_{z\,i}\,u_{y\,i}+r_{e}\rho_{e}\,u_{z\,e}\,u_{y\,e} & r_{i}(\rho_{i}\,u_{z\,i}\,u_{z\,i} + P_{i})+r_{e}(\rho_{e}\,u_{z\,e}\,u_{z\,e} + P_{e}) \\ u_{x\,i}\left(e_{i}+P_{i}\right) & u_{y\,i}\left(e_{i}+P_{i}\right) & u_{z\,i}\left(e_{i}+P_{i}\right)\\ u_{x\,e}\left(e_{e}+P_{e}\right) & u_{y\,e}\left(e_{e}+P_{e}\right) & u_{z\,e}\left(e_{e}+P_{e}\right)\\ \end{pmatrix}\end{split}$

With $$r_{i}=q_{i}/m_{i}$$ and $$r_{e}=q_{e}/m_{e}$$ where $$q_{e}$$ is the electron charge, $$q_{i}$$ is the ion charge, $$m_{e}$$ is the electron mass and $$m_{i}$$ is the ion mass. In addition the variables $$(\rho_{\alpha},u_{x\,\alpha},u_{y\,\alpha},u_{x\,\alpha},e_{\alpha},P_{\alpha})$$ are the species mass density, species x velocity, species y velocity, species z velocity, species total energy density, and species pressure respectively. In this case $$\alpha$$ represents the species, either $$e$$ for electron or $$i$$ for ion.

## Parameters¶

ionGamma (float)
Specific heat ratio for the ions
electronGamma (float)
Specific heat ratio for the electrons. Defaults to 5/3
ionMass (float)
ion mass
electronMass (float)
electron mass
ionCharge (float)
ion charge
electronCharge
electron charge
basementPressure (float)
The minimum pressure allowed. Defaults to 0.
basementDensity (float)
The minimum density allowed for the ions. Defaults to 0. The electron basement density is determined by multiplying by the mass ratio, therefore $$basementDensityElectrons = (me/mi)basementDensity$$

## Parent Updater Data¶

in (string vector, required)
1st Input Variable
1. $$\rho$$ mass density
2. $$\rho\,u_{x}$$ x momentum density
3. $$\rho\,u_{y}$$ y momentum density
4. $$\rho\,u_{z}$$ z momentum density
5. $$\rho_{c}$$ total charge density
6. $$j_{x}$$ x current density
7. $$j_{y}$$ y current density
8. $$j_{z}$$ z current density
9. $$e_{i}$$ ion energy density
10. $$e_{e}$$ electron energy density

## Example¶

An example twoFluidEqn equation block is given below:

<Equation twoFluid>
kind = twoFluidEqn
ionGamma = GAS_GAMMA
electronGamma = GAS_GAMMA
ionMass = ION_MASS
electronMass = ELECTRON_MASS
ionCharge = ION_CHARGE
electronCharge = ELECTRON_CHARGE
basementDensity = BASEMENT_DENSITY
basementPressure = BASEMENT_PRESSURE
</Equation>