# Scale invariance and state equation for fluid dynamics

I am trying to understand the example provided in this section of the Wikipedia article on scale invariance. In particular where it says

In order to deduce the scale invariance of these equations we specify an equation of state, relating the fluid pressure to the fluid density

The state equation used is $$P = c_s ^2 \rho$$

Q: Why such specification of the state equation is necessary and why that particular choice ensures the scale invariance?

Thanks

• To choose an equation of state has nothing to do with scale invariance, you simply always need it to close your system of equations. Otherwise P remains an unknown, without its own governing equation (if you don't use an energy equation). – AtmosphericPrisonEscape Apr 4 at 2:33
• @AtmosphericPrisionEscape , does it mean that any equation added to solve the system is enough? I mean any $p(\rho)$ or $p(v)$ or $v(\rho)$ etc (supposing that those functions have physical meaning) – user1420303 Apr 4 at 3:04
• Any $P(\rho,v)$ is needed, otherwise not every quantity on the right-hand sides of mass and momentum conservation equations are known. – AtmosphericPrisonEscape Apr 4 at 11:34