Gas Dynamics Equations

variational formulation
1

Hello,
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.

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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: http://membership.agu.org/files/2014/01/da-Silva-and-Pasko-2013-Dynamics-of-streamer-to-leader-transition-at-reduced-air-densities-and-its-implications-for-propagation-of-lightning-leaders-and-gigantic-jets.pdf

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?

2

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

3

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!