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Re: [Phys-L] Bernoulli's equation ... or not



On 12/15/2014 06:23 PM, Anthony Lapinski wrote:
I was trying to find some practical/relevant problems involving Bern eqn,

7) Suppose you have a bowl of water. It doesn't even have to
be a round bowl; a squarish cake-pan will do just as well.

Put it on a turntable and rotate it at a steady speed. After
a while the water will undergo uniform rotation. The speed
of the water is everywhere proportional to |r| in cylindrical
coordinates i.e. the two-dimensional radius i.e. the distance
from the center when position is projected onto the plane of
rotation.

It has been known since Newton (and possibly longer than that,
I don't know) that the surface of the water is a paraboloid
under these conditions. You could have figured that out on
your own, using basic physics concepts and a little algebra,
but since I am such a nice guy I'll just give you the result:
The height of the surface is proportional to r^2.

Now pick any horizontal plane within the fluid. At any
point you can figure out the pressure, just by considering
the height of water above you. Obviously this will be a
constant plus something proportional to r^2, on any given
horizontal plane.

Since the velocity is proportional to |r|, we see that
there is a contribution to the pressure that scales like
velocity squared.

Explain why this OBVIOUSLY cannot possibly have anything
to do with Bernoulli's principle.

Then explain why Bernoulli's principle is in fact valid,
even though mindless plug-and-chug gives the wrong answer.