Re: Bernouli formula

[John Mallinckrodt <ajmallinckro@CSUPOMONA.EDU>]

On Fri, 16 Nov 2001, Ludwik Kowalski wrote:

> 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?

Interestingly enough, (but perhaps only to me) the type of work
being talked about here is NONE of the seven types that I am
always going on about because it ignores the work done on the
system by one of the external forces, namely gravity.  Instead, it
treats that work by *pretending* that the potential energy the
system shares with the Earth belongs exclusively to the system.

I'm not saying that the answer is wrong, of course, but here are
two other (I would argue, more proper) ways of deriving it:

1. Use the work-energy relation that says "the work done on the
system by *all* external forces as calculated in an inertial
reference frame is equal to the change in the *total* energy
(internal + bulk kinetic) of the system."  During a brief period
when the ends move through a volume dV, that work is

   dW = (P1-P2)dV -Mg(dH) = (P1-P2)-rho(h2-h1)dV

where P1 and P2 are the pressures at the incoming and outgoing
ends of the system, M is the mass of the system, and dH is the
change in height of the center of mass of the system rho is the
(fixed density) and h1 and h2 are the heights at the incoming and
outgoing ends.  The walls do no work, of course, because they do
not move.  The change in the total energy is

  dE = d[(M/2)Vcm^2] = (rho/2)(v2^2-v1^2)dV

where v1 and v2 are the speeds at the incoming and outgoing ends.
Note that the internal energy of the system does not change as
a result of our assumptions that the fluid is nonviscous and
incompressible.

Set the two quantities equal and obtain the Bernoulli equation.

2. Use the "pseudowork-kinetic energy" relationship.  The result
is the same, of course, and the math for the energy side is
identical, but the work side is more difficult to follow through
because one has to work out the results of dotting the pressure
related forces on the system with the motion of the center of mass
and one also has to show that the walls do no pseudowork on the
system.

I'd stick with number one in this case!

John Mallinckrodt      mailto:ajm@csupomona.edu
Cal Poly Pomona          http://www.csupomona.edu/~ajm