Re: Bernoulli (horizontal)

["John S. Denker" <jsd@MONMOUTH.COM>]

kowalskil wrote:
> We are talking about a Venturi pipe for an ideal liquid,
> not viscous and not compressible.

All liquids (indeed all substances) are compressible.  If
they weren't compressible, there would be no way to balance
the energy equations.

In discussions of Bernoulli's principle, many authors
throw around the word "incompressible" without explaining
what they mean, and I suspect some authors themselves
don't know what they are talking about.

Bernoulli's principle can be expressed by an equation
wherein the compressibility does not appear, but that is
*not* because the compressibility is zero.  The equation
in its simplest form (the form you usually see) is already
correct to first order in dV/V, and it can easily be
generalized to second order, and indeed to all orders.


> 1) Suppose we have an ideal horizontal tube of constant cross
> section. No work is needed to maintain its liquid in motion at
> a constant speed.
>
> 2) The same tube is bent upward near the exit. This time we
> must have a force to push water at constant speed. Suppose
> the volume ejected during a short time interval (at very
> small kinetic energy) is dV and the output is at an elevation
> h, with respect to the axis of the horizontal pipe. The work
> done must be dW=rho*g*dV*h, where rho is the density. It
> is equal to PE gained by the dV parcel of water. It is
> also equal to P*dV, where P is the pushing pressure applied
> by the piston at the pushing end. This is not the same as
> P*dV for a gas whose volume changes.

Actually, it is pretty much the same P*dV concept.  There is
some P and some dV at one end of the tube, and some P and
some dV at the other end.  If you add up all the PdV
contributions, with due regard for signs, you get the amount
of PdV energy transferred to the fluid.

The bookkeeping is a little different if you are measuring
things in the lab frame, as opposed to using a frame comoving
with the parcel of fluid, but the physics is the same.

PdV is not always the whole story.  For a complete description,
you need to include other energy transfers, such as gravitation
acting on a non-horizontal flow, electromagnetism, or whatever.