Gas Dynamics Equations

I am trying to implement gas dynamics equations to describe the air heating and expansion due to the passage of the electrical current through the stem’s cross section.


The equation (1) accounts for mass transport.
The equation (2) describes the transport of momentum.
Equations (3) and (4) describe the energy transport.

Here’s the link for a full description:

I couldn’t find similar topics and I’m struggling to find weak forms for these equations and to implement the model.
May anyone help me with this?

Compressible flows are challenging. Consider looking into discretisations of the compressible Euler and compressible Navier-Stokes systems as a basis for your desired formulation. These are likely more extensively covered in the literature.

E.g: Solving Compressible Flow Navier Stokes

Thank you for your response.
I already made researches on Navier-Stokes systems and I also implemented Gallimberti simplified model but it didn’t help much.
I will consider reading more about these topics anyway, thanks!