Re: pressure-energy density

["Eugene P. Mosca" <mosca@nadn.navy.mil>]

In a recent message I wrote:

> The Bernouli equation is that
>
>               P + (1/2)mv^2 + mgh
>
> where m is the mass per unit volume, is constant along a streamline of a
> viscosity-less incompressible fluid undergoing laminar flow.  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.

To which Leigh Palmer replied:

********************

Thinking of the pressure as a potential energy density is perfectly
reasonable here. It should not be reified, however. Thinking of
energy as "stuff" leads to some cognitive errors as we have seen in
the energy flow discussion. Always keep that caveat in mind.

I notice that you did not ask if one could think of mgh (I'd prefer
rho*g*h) as a potential energy density. Surely that is no different
from pressure. Why are you more comfortable with one than with the
other? Both are formal expressions that represent non-localized
energy densities. Only the kinetic term can lay any claim to being
local.

*******************

Leigh,  

Your question is a good one.  It gave me considerable pause.

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.  

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.