Isn't the Bernoulli equation just a special case of the Steady-flow equation?

This question is about the following two energy balance equations (assuming no energy input/output):

• Bernoulli: $$\frac{p}{\rho} +\frac {v^2}{2} + zg = const.$$
• Steady-flow: $$u +\frac {p}{\rho} +\frac{v^2}{2}+ zg = const.$$

So as everyone knows the Bernoulli equation only applies to incompressible fluids in steady state. When comparing the Bernoulli equation with the Steady-flow equation, the former seems to be a special case of the latter when the internal energy is kept constant. But the steady-flow equation applies to compressible gases as well, so I conclude that all we need to do to apply Bernoulli for compressible flows is to consider the internal energy term. But searching for Bernoulli for compressible flows online seems to result in much more complicated equations, so I am confused and unsure if I am missing something.

• Why not think of the history? Bernoulli was describing in 1738 a way to look at the behaviour of a fluid and the form we now use came later from further work by Euler. And why don't you just derive it from Newton's second law? Oct 31 '21 at 13:01