Re: Bernoulli's relationship

[John Denker <jsd@MONMOUTH.COM>]

At 08:24 PM 11/8/00 -0500, I posted the following flow pattern as a step
toward addressing deeper issues raised by Robert A Cohen's question about
Bernoulli's relationship:

                            /
                            /-----_____
                            /          -----_____
                            /                    -----
        -----_____          /
                  -----_____/    Direction of flow ==>
                            /
                            /
                            /
                            /
        ====================================================
                               Figure 1

This is the simplest flow-pattern with which one can address the question
that was asked.

My assertion that you "will have" this solution was obviously
overbroad.  In a real channel with a real propeller, an infinitude of
more-complex flow patterns is possible.  As I explained at 09:37 PM 11/8/00
-0500, figure 1 is the flow pattern that you get if you idealize the
propeller as a pump.  In particular, imagine implementing the pump as a
bucket-wheel that just raises the water and gently dumps it into the
channel on the downstream side.

If OTOH you imagine a pump plus a _nozzle_, things can get quite
interesting.  For example, at 06:10 AM 11/9/00 -0600, brian whatcott
alluded to:
> possibility of the downstream hydraulic jump

A rough diagram of such a solution is:

                            /
                            /                    _/\_
                            /                __--    ---____
                            /            __--               -----
        -----_____          /        __--
                  -----_____/    __--
                            /__--       Direction of flow ==>
                            /
                            /
                            /
        ===========================================================
                               Figure 2


At 05:34 PM 11/8/00 -0800, Leigh Palmer wrote that figure 1 was "contrary"
to his results and suggested that I must have "forgotten about continuity".

Continuing at 02:52 PM 11/11/00 -0800, Leigh Palmer was even more explicit
and said:
> simply seeing that the answer was unphysical was sufficient.
> I didn't have to check to find the origin of the final error.
...
> I'm very sorry that I haven't the macho pilot's background you
> have, John. I don't think that disqualifies me or my argument. My
> experience with cleaning rain gutters with a running hose and my
> application of physical reasoning are probably more valuable
> qualifications, anyway.

First of all:  I hope phys-l participants do not care about my background
or anybody else's background.  If a ten-year-old with no qualifications
whatsoever makes a physically-correct argument, it is a physically-correct
argument.  Qualifications don't matter.

This makes physics very different from many other aspects of life.  Life is
complicated, and in non-physics fields (e.g. politics or religion) there
are many assertions that cannot be checked in any scientific way, and must
be checked in non-scientific ways, such as checking the credentials of the
person making the assertion.

I hope we can conduct phys-l discussions based on physics, and avoid
religious arguments and ad-hominem arguments.

Secondly, I am not a macho pilot.  A big part of my job as a flight
instructor is to train people _not_ to act macho.  Machismo is incompatible
with safety.
  http://www.m-w.com/cgi-bin/dictionary?va=machismo

Thirdly, experience with solutions unlike figure 1 does not mean that
figure 1 is unphysical.  As I said before, many solutions are possible.

Until somebody makes a _scientific_ argument to the contrary, I will
continue to believe that figure 1 is a perfectly reasonable element of the
solution set.

Leigh went on to write:
> This isn't a pump. There is no outlet pipe here, something that
> is probably necessary to the case you cite here. Surely the outlet
> pressure must be higher if the velocity of flow is lower.

That's almost diametrically wrong.  An outlet pipe, serving as a nozzle,
leads to the solution shown in figure 2 rather than figure 1.  I strongly
suspect that most people use a nozzle for cleaning gutters.

The reason I chose figure 1 as a representative solution was because the
original puzzle specified an open channel. Therefore nozzles, while
certainly possible, would depart from the spirit of the specification.

> I know that my model for the propeller does not admit of this possibility,
> but do you really believe that a propeller in an open channel will
> diminish the velocity of flow in the channel?

Maybe, maybe not, depending on unspecified details of how the "propeller"
is constructed.  In the limit of the idealized bucket-wheel pump, there
absolutely will be lower velocity downstream.

I also think that the original question had to do with the relationship of
height, velocity, and pressure in the channel, and that the details of the
pump are irrelevant.