Favre averaged Navier-Stokes equations
The instantaneous continuity equation (1), momentum equation (2) and energy equation (3) for a compressible fluid can be written as:
For a Newtonian fluid, assuming Stokes Law for mono-atomic gases, the viscous stress is given by:
Where the trace-less viscous strain-rate is defined by:
The heat-flux, , is given by Fourier's law:
Where the laminar Prandtl number is defined by:
To close these equations it is also necessary to specify an equation of state. Assuming a calorically perfect gas the following relations are valid:
Where , , and are constant.
The total energy is defined by:
Equations (1)-(9), supplemented with gas data for , , and perhaps , form a closed set of partial differential equations, and need only be complemented with boundary conditions.