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[Phys-L] streamline curvature +- Bernoulli



It's often said I have a keen grasp of the obvious.
I take it as a compliment, whether it was intended
that way or not.

In that spirit, let me mention /stream line curvature/.
Whenever you see a curved streamline, you know there
is relatively lower pressure on the inside of the
curve. This should be obvious based on simple notions
of F=ma.

With all due respect to Professor Bernoulli, he's not
the only player in this game. There are lots of cases
where you have your choice of using Bernoulli or using
stream line curvature or both.

There are also lots of cases where Bernoulli doesn't
tell you anything you didn't already know, and stream
line curvature is a big win. Examples include:
-- a bowl of fluid undergoing uniform rotation
(velocity proportional to |r|)
-- a vortex
(velocity proportional to 1/|r|)

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

There's nothing wrong with Bernoulli's equation.
It is, after all, a corollary of the basic laws
of motion. It is often misunderstood and often
incorrectly applied, but you could say that about
almost any equation.

In particular, Bernoulli's equation applies to a
single parcel of fluid as it moves along a stream
line. This does not generally allow you to compare
the parcels on one stream line to the parcels on
another. Any discussion of the lift produced by
a wing requires such comparisons. /Sometimes/ you
can do that by combining Bernoulli with additional
information, but sometimes you can't, especially
when the stream lines close on themselves, as in
the rotating fluids mentioned above.

If there's a lot of turbulence, the whole idea of
stream lines goes out the window and you can't use
either of these methods directly ... although you
can take local averages. The stream line curvature
idea survives averaging better than Bernoulli does.

On 12/21/2014 04:42 PM, Bernard Cleyet wrote:

Is the demo. using a card, pin and a thread spool so common, no-one mentioned it?

The kindergarten version of the arxiv above.

Sure, you can do the demo. The problem is, it's hard
to interpret. Everybody knows there is high pressure
in the blow-hole, so even if there is low pressure
elsewhere, it's hard to predict what's going to happen.

The world already has more than enough experiments
that are easy to do but hard to interpret. I've been
trying to think of a clean, easy-to-interpret version
of this experiment, without much success.

The gap of the levitating disk isn't tightly controlled,
so you don't really know what the velocity is ...
although this is relatively easy to fix.

Then there's the friction, which messes things up.

Then there's the fact that the scheme is not overly
practical. I don't know of very many things that get
levitated this way. I prefer to spend my time on
topics that have real applications.

There's high pressure on part of the disk and low
pressure elsewhere. Qualitatively, the fact that the
low pressure can overcome the high pressure tells you
that there must be a goodly amount of low pressure,
but trying to quantify that is a nasty job. First
of all, it's a calculus problem. Whats worse, it's
a small difference between large numbers.

This stands in contrast to the Pitot-static setup,
which is Pitot /minus/ static. The low pressure
term is subtracted from the high pressure term,
so a lot of nuisance terms drop out. There is
no issue with small differences between large
numbers. It's just a much better geometry.