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... . Does
thinking of the pressure P as some kind of pressure-energy density to be
encouraged, or does is lead to later difficulties? Since the fluid is
incompressible, it is clear that P does not represent work per unit volume
done in compressing the fluid.
....
I am thinking of a volume element of the fluid moving along a streamline
in a region occupied by the Earth's gravitational field, which is
conservative. Conservative fields have potential energy functions so it
seems OK to me to think of mgh as the potential energy per unit volume
associated with the volume element of fluid. Also, if I think of the
Earth-volume-element system the gravitational potential energy of the
system is clearly a function of the system's configuration, a requirement
for any potential energy.
I am less comfortable with thinking of P as pressure-energy density
because I am unable to construct a picture analogous to the one I just
described for the gravitational-energy density.
....