Let's start with euler equations: https://en.wikipedia.org/wiki/Euler_equations_(fluid_dynamics)#Euler_equations
They do describe the system that I want, but they produce wave-like processes that have finite propagation speed and hence severely limit time stepping during numerical simulation. However, in my case I don't really need the wave part, I simulate relatively slow flows and I'd like to assume that sound speed in my flow is infinite. This gets into the area of Navier-Stokes equation, but I have not found a formulation of compressible navier-stokes with temperature terms.
I'm pretty sure there has to be a standard name for a system of equation that governs a process like this. Can anybody please suggest what to look for?