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Re: Bernoulli's relationship



At 06:26 PM 11/8/00 -0500, Robert A Cohen wrote:
Suppose you have an infinitely-long W-shaped canal:

Each trough contains the same amount of water.

The same infinite amount of water? Comparing infinities is a tricky
business. See below for a counterproposal.

In one trough, there is a propeller pushing the water down the trough.
...
1. With the propeller on, what does the surface of the water look like in
that trough?

You will have a ramp, plus a step at the location of the propeller.


/
/-----_____
/ -----_____
/ -----
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-----_____/ Direction of flow ==>
/
/
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/
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The slope of the ramp depends on viscosity.
The height of the step depends on the vigor of the propeller.

In other words, where along the trough is the surface of the
water raised above or below the surface of water in the stationary trough?

That depends on hitherto-unspecified details of the boundary
conditions. There is no particularly obvious way to enforce the boundary
conditions at infinity for the infinitely-long troughs.

Therefore I suggest rephrasing the question in terms of a
figure-eight-shaped apparatus consisting of two loops. Each loop has a
straight section that adjoins the straight section of the other loop. Let
the propeller sit in the middle of one of the straight sections. I believe
this captures the essential physics in question.

Then, to answer the question: everywhere upstream of the propeller, the
height will be relatively low. Everywhere downstream the height will be
relatively high. The height will cross its average value somewhere on the
far side of the loop, roughly halfway around from the propeller.

2. How does the water pressure (at some level) inside the moving trough
vary with distance along the trough (and how does it compare to the
pressure at the same height in the stationary trough)?

To be specific, let's pin down the notion of "at some level" and confine
attention to the floor of the trough.

Also I will assume that "pressure" means "static pressure" not "dynamic
pressure" or "stagnation pressure" or ... whatever.

I assume this is related to the height of the water along the trough.

Yes, it is related. The name of this relationship is called "equality" (in
the appropriate units) or "proportionality" (in some other units).

The height of the surface above the bottom tells you the static pressure at
the bottom.

3. If at some point, there is a hole in the boundary between the two
troughs, into which trough would the water flow and why?

It will flow from the high-static-pressure side into the
low-static-pressure side, Other Things Being Equal. You can of course mess
this up by using a funny scoop-like hole, if the hole is unbiased then
static pressure tells the story.

==============================

Remarks: The foregoing answer is the obvious answer, and the only answer
consistent with relativity and symmetry and other basic principles.

The foregoing simple answer does not mention the word "Bernoulli". I am
quite aware that there are more-complicated ways of analyzing the situation
that shed an interesting light on the meaning of Bernoulli's principle, and
on the rules governing correct application the principle. If somebody
wants to ask that question, we can discuss it.