Temperature of something is a measure of the average kinetic energy

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russ_watters
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I've written and deleted and rewritten several long responses here and now I'm getting frustrated - maybe I'll come back and finish going over your questions later, but for right now...

I think the issue may be one of dealing with differing principles/versions of Bernoulli's equation. There are more advanced versions of Bernoulli's equation/principle (not developed by Bernoulli) that deal in compressible flow and that may be what you are describing. However, there are a couple of problems here:

1. A waterfall (or white water rapids) has no continuity - no streamlines. It can't, under any circumstances, be considered an application of Bernoulli's principle. Friction, viscocity, and potential energy dominate, and these are not adequatly covered by Bernoulli's principle in any of its forms. In particular, since it is open to the air and static pressure never changes, there can be no compressibility effects (even assuming water was more than a little compressible!).
2. In a venturi tube, air does experience compressibility effects that become relevant above about 220 mph. These compressibility effects result in ideal gas law implications for the flow: change in temperature with pressure/density changes. But these effects are explicitly excluded in the standard form of Bernoulli's equation: the flow is assumed to be incompressible. So when you say:
Please consider it to be an idealised Bernoulli case...
To me, "an idealized Bernoulli case" is the one Bernoulli derived and it explicitly discards the effects you are describing. Some of what you are describing is valid for more advanced versions, though. Read the wiki on Bernoulli's principle, paying specific attention to the descriptions of the two forms they describe: the compressible and incompressible flow forms.
http://en.wikipedia.org/wiki/Bernoulli's_principle
 
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There is an analogy to be drawn between the situation described and the simple free electron theory of electrical conduction.When a current flows there is a low drift velocity which is analogous to the bulk velocity of the falling water and this drift velocity is superimposed on the high thermal velocity of the charge carriers this being analogous to the high thermal velocity of the water molecules.With the waterfall there is a conversion of bulk kinetic energy to thermal energy.Because of the high specific heat capacity of water the temperature rise is not very high and this was first shown experimentally by James Joule.He carried out numerous experiments including measuring the temperature at the top and bottom of a waterfall as mentioned by mgb phys.James took a thermometer on his honeymoon but don't we all?
 
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I think the 5 lines I listed above is a mechanism for cooling in an idealised Bernoulli restriction, true for ANY fluid, compressible or incompressible, which gives and receives heat energy by changes in translational motion.

Thereafter, additional complications arise from compressibility which Russ says produces cooling, and for real gases the Joule-Thomson effect which is cooling for most gases, plus turbulence and friction etc etc

Count Iblis, you say temperature goes down, I suppose you mean the temperature mesured in the frame of the motion, ie the man astride the molecule is holding a thermometer.
But what would be the temperature on the walls of the tube? Can we start by assuming billiard ball energy echanges between wall molecules and fluid molecules.
 

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