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https://en.wikipedia.org/wiki/Bernoulli's_principle#Derivations_of_the_Bernoulli_equation

In the first derivation, using Newton's second law, it is claimed that the effective force on a parcel of the fluid is ##-Adp##, but in my opinion, it should be ##-Adp-pdA##. I see no the reason why the cross-sectional area should vary less than the pressure.

In the second derivation, using conservation of energy, which seems to be more common in texts, it is assumed that the only forces acting on a volume of the fluid between two cross sections doing net work are the forces caused by the pressures at the cross sections at the ends, and gravity.

But how do we know that these are the

*only*forces acting on the volume doing net work? Why can't there be inner forces in the volume doing net work?

Actually, it seems to me that my two objections are related. Clearly, if the cross-sectional area is constant, then the inner forces pressing on parcels of the fluid cancel each other out, by Newton's third law, together with the fact that all parcels have the same velocity, so these forces do no net work.

But if the cross-sectional area is constant, then so is the pressure, making Bernoulli's equation meaningless in this case.