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Re: pressure-energy density



Gene Mosca wrote:

... . 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.

and
....
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.
....

Leigh and I discussed this very issue this July on the physhare list. In
that discussion Leigh took the position that the pressure term in the
Bernoulli equation was a potential energy density, whereas I took the
position that that term represented the work done (per unit volume) on/by
a given fluid parcel as it is pushed around by the surrounding fluid
parcels and as it pushes other neighboring fluid parcels out of the way to
make room for it. Even though the resulting discussion lost most of the
participants on that list, at least I came to understand Leigh's viewpoint.
Under the restrictive (but usual for the Bernoulli equation) conditions of
the fluid being in both an explicitly time independent steady state and it
being incompressible then the pressure term may indeed be thought of as a
kind of potential energy (density) field for the fluid which accounts in a
conservative way for the work being done by/on each parcel by its
neighboring parcels as it moves to and fro along its streamline. If the
above two restrictions are relaxed then this pressure term cannot represent
such a potential energy density as in this more general case the work
done on/by a given parcel due to contact forces with its neighboring
parcels cannot be accounted for as a conservative potential field.

In the more general case the corresponding equation (although not usually
called the Bernoulli equation) is still a spatial first-integral of Euler's
equation and still may explicitly represent energy conservation/accounting
in the fluid. This generalized Bernoulli-like equation holds for the fluid
as long as the fluid is both irrotational and inviscid. The irrotational
condition is needed for the flow to be streamline-type flow, and the
inviscidity condition is needed for the flow to be dissipationless
(isentropic) so all the macroscopic energy can remain accounted for at
the macroscopic level (without having to deal with heat effects). Besides
having a pressure term, a kinetic energy term, and a gravitational
potential energy term this generalized equation includes two other terms as
well. One of these terms contains the explicit time-partial deriviative of
the velocity potential, and the other is the local internal energy density
for the fluid observed in a comoving frame. This internal energy density
term accounts for local concentrations bulk elastic energy stored in the
fluid by local adiabatic density changes, and the time-derivative of the
velocity potential term represents the *explicit* time dependence of the
fluid parcels' inertia as they accelerate (the kinetic energy term accounts
for the inertia effects for implicit time dependence contained in the
comoving flow). When the fluid is in a steady-state then the time-partial
derivative of the velocity potential vanishs and no longer contributes.
When the system is incompressible then the local comoving internal energy
density is a constant term which can be dropped from the equation since it
has the same value throughout the fluid for all time. In this restricted
steady-state flow case then the pressure field is a spatially dependent,
but time independent, conservative field which may then represent a
potential energy density. In the more general case this same pressure term
cannot be a potential energy density.

I don't know overall whether or not its a good or a bad idea to treat (for
pedogogical purposes) the pressure term in Bernoulli's equation as a
potential energy density. It probably is a good idea to at least interpret
it as representing the contribution to the energy conservation for the
fluid coming from the work done on the fluid parcels due to the contact
forces exerted on the parcels from their neighboring parcels.

David Bowman
dbowman@gtc.georgetown.ky.us