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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.

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  • $\begingroup$ 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? $\endgroup$
    – Solar Mike
    Oct 31, 2021 at 13:01

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No, the logic flows the other way around. Bernoulli's equation is derived from conservation of momentum plus some special conditions, not from conservation of energy plus some special conditions.

As you've noticed, however, Bernoulli's equation contains several of the same terms as the steady flow energy equation. This means that, in the conditions where Bernoulli's equation applies (which include incompressibility), conservation of momentum ensures that a bunch of the terms in the steady flow energy equation cancel, leading to a radically simplified energy equation in which the only form of energy storage one needs to consider is internal energy, and the only form of energy transfer one needs to consider is heat (i.e. not work).

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  • $\begingroup$ "Taking his discoveries further, Daniel Bernoulli now returned to his earlier work on Conservation of Energy. It was known that a moving body exchanges its kinetic energy for potential energy when it gains height. Daniel realised that in a similar way, a moving fluid exchanges its specific kinetic energy for pressure, the former being the kinetic energy per unit volume." en.wikipedia.org/wiki/Daniel_Bernoulli. It can be gotten at myriad ways once you know what you are after. $\endgroup$
    – Phil Sweet
    Oct 31, 2021 at 16:24
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    $\begingroup$ The way the logic flows today depends on what you were taught and in what order you were taught it. Very little of that follows from how things were originally puzzled out. $\endgroup$
    – Phil Sweet
    Oct 31, 2021 at 16:38
  • $\begingroup$ "ABSTRACT In this work we thoroughly explore the meanings of dissipation (sometimes referred to as viscous dissipation) and stress power. To do this we utilize the Cauchy momentum equations and the first and second laws of thermodynamics. First, the generalized engineering Bernoulli equation (GEBE) is derived from the Cauchy momentum equations and it is clearly shown to have nothing to do with a balance of energy." sor.scitation.org/doi/abs/10.1122/1.550746?journalCode=jor So choose your engineering religion carefully and respect the choices of others. $\endgroup$
    – Phil Sweet
    Oct 31, 2021 at 16:41
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    $\begingroup$ @PhilSweet My Latin isn't great, but about three years ago I spent quite a while staring at Bernoulli's Hydrodynamica, and I convinced myself at that time that Bernoulli had derived his equation starting from conservation of momentum, although he certainly does also mention conservation of energy (or at least of vis viva).... $\endgroup$
    – Daniel Hatton
    Oct 31, 2021 at 20:24
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    $\begingroup$ ... You're right that Bernoulli's equation can be derived both ways, but I don't think I'm being entirely subjective in saying that the auxiliary assumptions needed to derive it from conservation of momentum are much more natural than the auxiliary assumptions needed to derive it from conservation of energy. $\endgroup$
    – Daniel Hatton
    Oct 31, 2021 at 20:30

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