Re: Bernouli formula

[Ludwik Kowalski <kowalskiL@MAIL.MONTCLAIR.EDU>]

"John S. Denker" wrote:

> At 02:40 PM 11/15/01 -0800, kowalskil wrote:
> > Bernoulli formula
> >
> > The first impression is that KE decreases at the
> > expense of PE. But this is not so.
>
> Actually it is so.  That's the point.
>   http://www.monmouth.com/~jsd/how/htm/airfoils.html#sec-bernoulli

Thanks for the reference; I looked into it again. But it did not
help me to answer questions I asked in the context of what
I am about to teach next week. I want to answer questions
about an ideal liquid in a tube, for example, when both KE
and PE are increasing. Air around aa airplane is more difficult
to analyze, in my opinion. Here again is the message to
which JohnD was replying:

Bernoulli formula (incompressible and nonviscous fluid) is:

P+0.5*rho*v^2+rho*g*y=const.

Most textbooks derive it in the same way. There is a picture
of a pipe which is narrower on the left and progressively
wider on the right. The right side is at a higher elevation
than the left side. A massless piston on the left is pushed to
the right with F1=P1*A1. The massless piston on the right
is pushed to the left with F2=P2*A2. Presumably both
pistons move horizontally without acceleration.

In other words, the force on the left does the work
W1=P1*vol, while the force on the right does the negative
work W2=-P2*vol. The net work, P1*vol-P2*vol, is then
compared with dKE+dPE. That leads to the above formula.
What kind of work are we talking about? Should it be the
work done on the CM of the small volume ("particle of
water") as it is moving from one end of the pipe to another?

The first impression is that KE decreases at the expense of
PE. But this is not so. Think about a pipe which is wider at
the lower end than at the upper end; in this case both PE and
KE will be increasing. In fact, water gains kinetic energy
even when PE=constant (when the tube becomes narrower
but the average elevation does not change).

The CM work contributes only to changes of KE. Which
work contributes to changes of PE? How to interpret
this process in terms of the First Law? Are all forces
conservative? Where is "the system boundary?"
Ludwik Kowalski