Re: [Phys-l] question about Bernoulli

[John Denker <jsd@av8n.com>]

On 11/19/2010 11:31 AM, curtis osterhoudt wrote:
> Not sure how applicable this is "at the molecular level", but a heuristic (at 
> least in the limit of incompressible fluids) explanation is nice, and doesn't 
> rely on the mathematics, or at least on the notion of streamlines and the like.
>
>    Consider a pipe of cross-sectional area A1, which connects to a section of 
> smaller cross-sectional area A2. Continuity (which students can understand at a 
> gut-level, at least for incompressibles) says that the speed in the vicinity of 
> A2 must be higher than that near A1. For this to have happened, the linear 
> momentum of a given fluid parcel is higher at A2 than at A1, and so the pressure 
> at A1 has to have been higher than at A2.

That's the right idea.

The only way to make it better would be to remove the restrictions:
 -- That idea works at the heuristic gut level *and* at the precise,
  formal level.
 -- That works for compressible fluids *and* for ... hmmm, wait a
  minute, there are no incompressible fluids.
 -- That works at the continuum-fluid level *and* at the molecular
  level.

Bernoulli's principle is essentially a force-balance proposition,
which (as always) is the same as a momentum-conservation proposition.

There /may/ be other valid ways of thinking about it, but I've
never seen any.  (I've seen plenty of invalid ways.)

The wikipedia article gets several fundamental points wrong.

Practically every aeronautical engineering book talks about
"incompressible fluids" ... which is compleat lunacy.  The
usual /simplified/ Bernoulli equation contains ρ0 which
is constant /by construction/ ... even though the actual
density ρ is not constant.  The simplified Bernoulli equation
is routinely (and correctly) applied to air, which everyone
knows is highly compressible.  And (!) when you look at the
more general expression, the compressibility doesn't even
enter into it!  The only thing that appears is γ, the adiabatic
exponent (aka the ratio of specific heats), which is something
else entirely.  You cannot infer γ from the compressibility or
vice versa.

For more on this, see
  http://www.av8n.com/physics/bernoulli.htm