Spinning Liquids (physics)

[A] A quick way to derive it is to regard the surface as being made up
of point particles that may slide about on the particles that lie just
below them. In steady motion, a droplet moves neither up nor down the
surface, so the component of its weight projected onto a plane tangent
to the surface has to provide the corresponding component of the
centripetal force.

Some notation: we'll suppose that the droplet has mass m and is at
distance r from the axis of rotation; the surface of the liquid is at
height y(r); the angular frequency of rotation is W; the acceleration
due to gravity is g.

If the surface (at r) is inclined at angle A to the horizontal, the
condition for equilibrium will be

[tangential component of weight] = [tangential component of cent. force]

=> mg sin(A) = m W^2 r cos(A)

=> tan(A) = W^2 r / g.

But tan(A) = dy/dr, so we can integrate with respect to r to get

y(r) = W^2 r^2 / (2 g) + constant,

which is the equation of a paraboloid.

Although that's a typical textbook argument, it might be slightly
puzzling to some. E.g., the surface droplets are really in contact
with neighbouring ones in the surface: why didn't we have to think
about the forces between them? [Glib answer: it's a fluid, so the
droplets will "squish" upwards if those forces aren't exactly zero.
Less glib answer: you would need a pressure gradient for there to be a
net force from the surrounding drops; but all drops in the surface are
at the same pressure (atmospheric), so there is NO pressure gradient
within the surface.]

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A more sophisticated alternative is to use Bernoulli's theorem,
for the special case of an incompressible, potential flow:

(v^2)/2 - gy + [pressure]/[density] = constant.

At the surface, the pressure is constant (atmospheric pressure), and
in a rotating body the velocity at distance r from the axis is v = Wr.
Hence Bernoulli's theorem immediately gives

y = W^2 r^2 / (2 g) + constant.

A likely objection to argument is that Bernoulli's theorem is
normally applied only to different points on the SAME streamline, but
we're applying it to points at different distances from the axis, and
these lie on DIFFERENT streamlines. But, as it happens, for the
special case of a potential flow you CAN compare different
streamlines. It gets technical, though -- at about the level of an
undergraduate fluid mechanics course.

And there are other ways, too. They'd use more mathematical
formalism (calculus of variations), but less prior knowledge.

Personally, I'd stick with the elementary argument [A] and retire to a safe distance... good luck.

Thinking about that again, version is wrong. I gave the gy term
the wrong sign, and rotational motion can't be described by a scalar
potential anyway -- so the argument's doubly screwed.

x

When I find a liquid spinning I just wait for it to cool down. Then I can drink it.

Did you know that when a male person passes water there is an arrangement inside the organ by which the process is facillitated that spins the column of liquid so that it is possible to sign one's name with it in a snowdrift.And add the full stop.

As ladies are unable to match this incredible feat it has been suggested that it is the definitive explanation of male superiority and the reason for art being the preserve of men.

See what I get when I post something serious!




x

You were just bragging smorgsie. Showing off in your special field in order to leave us all gasping in admiration which, I must admit, worked on me.

But there's more to life than that. Serious is always a bit straight-faced.
If you do too much serious the corners of your mouth turn downwards and you are stuck with the chaps who are left after the ladies whose mouth corners turn upwards have had first pick.

My serious posts are few and far between, spendius...

And here's me thinking that this would be tucked away in some corner a A2K and no-one would ever read it (except quincy), and besides, there are very few 'blonde' jokes on the Science and Mathematics Forum.

Who says blondes can't do fisiks?

...and my mouth is always turned up

...but I have a penchant for guys with turny down mouths

Serious can be VERY sexy...

x

It must be the challenge to cheer them up I suppose.

You strike me as a lady who thrives on challenge.

What were "fisiks". I've forgotten.

Things like maffs


x

Only in this uni-sex world can fisiks become so watered down as to resemble watered down maffs and co-mingle in the greyness. From a linguistic point of view. All that sharp jagged sound in fisiks and all the soft treacle pudding langour in maffs.

I've been reading Anthony Burgess so I'm a bit yinyanging and I'm also off to the pub.

Mine's a Bloody Mary - cheers!

x

Didn't she come to a sticky end?

exactly...

smorgie-

What's that grey blob that keeps pulsing as the heart beats down upon it in your avvie supposed to mean?

Do you really see heartless men under your driving in such an undignified role.

Is it a "rolling pin" joke?

Not at all, spendius!

It's merely a bouncing heart and shadow, 'cos I'm so full of love.

I don't think men are heartless, I like them just as much as another gender!

Sometimes, I even let them sleep in my bed!

Smorgs, wave off this piffle. Glad to see another facet from y'self.
Not that the first facet wasn't fine...

smorgie wrote-



Looking at the new avvie I recommend taking it easy. It can be overdone selfishly.Nice as it is.



From what I have seen in a long career in publand there are few prizes handed out for such achievements on account of their ubiquitiety, if that's a word. Pity us poor men!

Don't quite know how to take that spendius?

But I have removed the avatar - I was just practicing...

smorgie-

One can give children so much love that they find it difficult to do without it when they grow up and prove impossible for someone else to deal with and especially someone who has been through the same process.

Practice more. I can hardly wait.