Re: pressure-energy density

[Leigh Palmer <palmer@sfu.ca>]

> 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.

Yes, and just *where* is that potential energy located? That is the
crucial point here. The quantity we call "gravitational potential
energy" is really an expression which does the accounting for only
a part of the energy of the system which includes the Earth. In the
exact expression

               GmM
        U = - -----
                r

it is easily seen that both masses enter symmetrically.

Where is the 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.

In that case I hope I've satisfactorily obliterated your previous
picture. The tow terms have common validity.

> The lack of localness of the gravitational-potential-energy density can be
> dealt with, to a degree, by considering the potential field associated
> with the gravitational field of the Earth.

The pressure energy term is dealt with in just that manner by
considering the pressure field in the fluid.

Energy density is merely a way of talking about the mathematical
behaviour. It is a good one, but as always, one should keep the real
physics in the back of one's mind all the time.

Leigh