> ## Documentation Index
> Fetch the complete documentation index at: https://compflowlab.mintlify.site/llms.txt
> Use this file to discover all available pages before exploring further.

# Governing Equation

CompFlowLab solves the coupled conservation equations for mass, momentum, total energy, and species mass fractions:

```math theme={null}
\frac{\partial q}{\partial t} + \frac{\partial f}{\partial x} - \frac{\partial f_v}{\partial x} = H
```

where $q$ is the vector of conserved variables defined as

```math theme={null}
q = \begin{pmatrix} \rho \\ \rho u \\ \rho h^0 - p \\ \rho Y_l \end{pmatrix}
```

Here, $\rho$ is the density, $u$ is the velocity in the x-direction, and $Y_l$ is the mass fraction of the $l^{th}$ species. The total specific enthalpy $h^0$ is defined as

```math theme={null}
h^0 = h + \frac{1}{2} u^2 = \sum_{l}^N h_l Y_l + \frac{1}{2} u^2
```

# Fluxes

The fluxes are decomposed into inviscid $f$ and viscous $f_v$ components:

```math theme={null}
f = \begin{pmatrix} \rho u \\ \rho u^2 + p \\ \rho u h^0 \\ \rho u Y_l \end{pmatrix}, \quad f_v = \begin{pmatrix} 0 \\ \tau_{xx} \\ u \tau_{xx} - q_x \\ \rho D_l \frac{\partial Y_l} { \partial x} \end{pmatrix}
```

Here, $D_l$ denotes the effective diffusion coefficient of the $l^th$ species into the mixture. In practice, this approximates multicomponent diffusion as the binary diffusion of each species into a mixture.

# Heat Flux

The heat flux in the x-direction, $q_x$, is defined as

```math theme={null}
q_x = -K \frac{\partial T}{\partial x} + \sum_{l=1}^N \rho D_l \frac{\partial Y_l}{\partial x} h_l + q_{source}
```

where:

* $K$ is the thermal conductivity,
* $T$ is the temperature,
* $Y_l$ are the species mass fractions,
* $h_l$ are the species specific enthalpies,
* $q_{source}$ represents volumetric heat sources (e.g., radiation or external heating). \
  \
  These three terms correspond to **conduction**, **species diffusion**, and **volumetric heat generation**, respectively.\\

# Shear Stress

The shear stress $\tau_{xx}$ is defined in terms of the molecular viscosity and velocity gradient:\\

```math theme={null}
\tau_{xx} = \frac{4}{3} \mu \frac{\partial u}{\partial x}
```

where $\mu$ is the molecular viscosity.

# Source term

The source term $H$ accounts for chemical reactions and includes one entry per species equation representing the production or destruction rate of the $l^th$ species, $\omega_l$:

```math theme={null}
H = \begin{pmatrix} 0 \\ 0 \\ 0 \\ \omega_l \end{pmatrix}
```


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