MHD Spinning Rod

Valtteri

00:52

Hello?

KDN

So if we were to worry just about charged particles spinning along with the rod, we would expect the positively charged ions to move upward

excuse me, downward

Valtteri

Really?

KDN

while the negatively charged ions should move upward

Valtteri

Yes

KDN

I assume this is what you are after... you are assuming that the magnetic field is parallel to the earth's surface, yes?

Valtteri

00:54

yes, essentially

no

KDN

and that the axis of rotation through the center of the length of the rod is perpendicular to the earth's surface?

Valtteri

Perpendicular

like

both the rotation is parallel

-both

KDN

so that means that if one end of the rod started out pointing East, and it were rotation counter-clockwise, that it would then point north, then west, then south, and back to east, right?

Valtteri

yes

KDN

Then the v cross B term of the lorentz force goes to zero when the rod is oriented east west (where the velocity of the ends of the rod is north south) because v is parallel to b at these points

Valtteri

00:56

the movement is perpendicular to the magnetic field

KDN

Do I have the correct picture?

Valtteri

hmmmm, I don't think so

let me put in more words

the rod is places parallel to the ground, assumed flat

KDN

as it might lie on a table?

Valtteri

the magnetic field comes out of the surface, perpendicular to it

yes

KDN

So we are at the magnetic north pole?

Valtteri

00:58

well, essentially

KDN

(Or south)

Ah... okay, so not near the equator, where the magnetic field is approximately parallel to the "tabletop"

Valtteri

the direction of B would dictate the direction of the particles, south meaning opposite to north

yes

and the rod would spin on the "table"

so that one end would turn from east -> north ...

KDN

Ok. If the magnetic field is coming out of the table top, so that the magnetic field is parallel to the axis of rotation of the rod... the v cross B direction is toward the end of the rod for positively charged particles, and toward the center of the rod for negative particles, right?

Valtteri

no

if it spins on the center

hmmm

other particles would move towards other end

and opposite end for opposite charge

in the absolute center

all particles would stay completely still

thus no movement

KDN

Well, no Lorentz force anyway

Valtteri

01:03

true, for v would be 0

KDN

The fluid dynamics of a spinning body are incredibly complex

Actually, v isn't even necessarily 0, but it isn't non zero due to the rotation of the rod, per se

Valtteri

well

the speed of the particles is greatest at the ends

KDN

Yes

Valtteri

and lowers towards center

and the center doesn't move

KDN

To be more precise, you actually only know the speed of the rigid rod walls, not the fluid inside

Valtteri

01:05

well, to make a fine point, thats true

KDN

So you know that on average the fluid is moving fastest at the ends, but you don't know that the fluid flow at any particular point is fastest at the ends

Valtteri

wait

I am talking about the movement towards the direction of the rotation

not towards the end points

ie. flow

KDN

I'm afraid I don't quite understand.

Valtteri

well

I mean

the rod is spinning at a constant velocity counterclockwise

thus

the velocity of a particle inside the rod would be same as the rod itself at the point it is

KDN

That is where I disagree

If you have a fluid inside, the fluid is free to flow within the rod.

Valtteri

01:09

true

but take a bowl of water

KDN

Thus, although you know what the average velocity of the fluid throughout the rod must be, you don't know that at a particular point in the rod exactly what the fluid is doing.

ok.

Valtteri

with a goldfish

and hold the bowl

and stretch your arms out, holding it

and start turning

when you turn at a constant speed

the water inside moves with the bowl?

KDN

Ye

s

Mostly

Valtteri

how does it move?

or rather

does the goldfish feel any force?

KDN

Well, because of centrifugal effects, the water further away from you rises slightly higher than the water level near you, right?

Valtteri

01:11

true

KDN

The water must flow for that condition to occur.

Valtteri

but I assume that the rod is completely filled and water is non compressible...

but

the water would clearly "want" to move to the ends

it cannot all be there

KDN

Well, then at the very least, the direction the fish perceives as "down" changes, as does the effective water pressure, right?

Valtteri

so there has to be currents inside...

KDN

Well, there don't have to be currents just because the water wants to be there

Valtteri

01:15

don't there?

KDN

Water wants to be at the bottom of the ocean rather than the top, but that alone is not enough reason for there to be currents

Valtteri

the force isn't great enough to win over friction

KDN

So that's where the answer starts to become "it depends"

Valtteri

yeah :D

definitely

hmmmm

KDN

In plasma physics, where you talk about magnetohydrodynamics (a two-fluid approximation of plasmas), a special type of instability (that will cause flow) arises called the magnetorotational instability

Magnetorotational instability

The magnetorotational instability or MRI is a fluid instability that arises when the angular velocity of a magnetized fluid decreases as the distance from the rotation center increases. It can also be known as the Velikhov-Chandrasekhar instability or Balbus-Hawley instability in the literature; not to be confused with the Velikhov instability which is the electrothermal instability. The MRI is of particular relevance in astrophysics where it is an important part of the dynamics in accretion discs. Balbus and Hawley were the first to realize the astrophysical importance of this instability...

Valtteri

01:17

Do you agree with the principle that the particles should move to opposite directions?

KDN

I agree that the oppositely charged particles will experience opposite Lorentz forces if they are moving at the same velocity

Valtteri

indeed

KDN

However, since the ions you are talking about are sitting in water, they are colliding with that water and so their motion is heavily influenced by the behavior of the water (and to a much weaker extent, the motion of the water is influenced by the ions)

Valtteri

and I ask if that would cause the particles inside a rod filled with water to collect to the opposite ends as the rod is rotated about its center

KDN

The answer is likely not

Valtteri

01:19

what if you crank up the magnetic field?

KDN

It would provide a motive force that would lead to instability, however, a complex circulatory flow is likely to arise

Valtteri

I see

of course

KDN

There might be a small gradient, with more positive charge at one end than the other

This depends on a lot of things, however

Valtteri

I am a mathematician, so I am likely not the most informed person to discuss with...

KDN

For example, it depends on the Reynolds number, which depends on the flow speed, the rod diameter, the fluid viscosity...

It may also depend on the Prandtl number and the fluid temperature...

It gets wicked complicated

Valtteri

01:22

what about in ideal situation?

KDN

On the other hand, if your tube is very skinny and your magnet is strong, then yes, you will see an accumulation of charges

It will be opposed, of course, by the building ion gradient, which will tend to want to keep the ions well mixed

Valtteri

if there is only lorentz force and gravity and centrifugal force?

KDN

Well, the Lorentz force includes an electric field term.

As the ions do start to separate to the magnetic term, the electric field will increase, and the system will balance

Valtteri

yeah, the full equation

KDN

turning up the magnetic field (or the rotational velocity) will push this further, but you will never push all the charges all the way to the ends of the rods

Valtteri

01:26

I was more concerned with the fact that at the start half of the ions are on the other side and half on the other

and as the rotation start

there are opposing forces

lorentz and centrifugal

KDN

The magnetic field will tend to drive them apart, that's true. Centrifugal force won't have anything to do with the ion drift, however, if your container is really full

Valtteri

I see...

KDN

The increased "effective gravity" will be opposed by a pressure gradient that leads to force balance

If the container isn't full, on the other hand, then there will be bulk fluid flow (ions and water) towards the ends.

Valtteri

it does get complicated

KDN

Otherwise, you will expect to find more chloride ions at the center, and more sodium ions at the ends

(for a counterclockwise rotation)

yup

real fast

Valtteri

01:30

hmmm

what if they were just charged steel balls inside insulating coating and we rotated a tube filled with only these in a strong magnetic field?

KDN

If you want to get really complicated, water is diamagnetic...en.wikipedia.org/wiki/Diamagnetism#Curving_water_surfaces

That is also very complicated

Valtteri

Yeah

its half past 3 AM here

KDN

The steel balls are much more dense that the water, so they will tend toward the end of the tubes due to centrifugal force

ah...

Valtteri

I think I go sleep

KDN

Suffice to say, you would need a really huge charge on those balls for the magnetic field to make a difference

good luck with your problem

Valtteri

01:33

I doubt it will keep me awake

KDN

No? :)

Valtteri

night and thanks for the chat

KDN

sure

Valtteri

how do I leave?

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