Calculate time derivatives by product rule and total derivative for compressible Navier-Stokes equation
For the one-component compressible Navier-Stokes equation in connection with the ideal gas law we might do the following changes:
Momentum balance:
\frac{\partial \left(\varrho v\right)}{\partial t} \approx \frac{\partial \left(\varrho^{n_I-1} v\right)}{\partial t}
Mass balance:
\frac{\partial \varrho }{\partial t} =
\frac{1}{RT}\frac{\partial p}{\partial t} + \frac{-p}{RT^2}\frac{\partial T}{\partial t}
Energy balance:
\frac{\partial \left(\varrho u^e\right) }{\partial t} \approx \frac{\partial \left(\varrho^{n_I-1} u^e\right) }{\partial t}
with n_I-1
: last iteration step