- The liquid at the bottom here is clearly more dense
than the liquid on top. We know that because it
has sunk to the bottom. But surely if it is more dense then when we swirl the container that more dense liquid at the bottom should be flung out to the sides. But instead it seems to
bunch up in the middle like an inverted Whirlpool. To me that's really counterintuitive. But if at this point you are thinking actually, no that's what I
would've expected to happen, that is not counterintuitive. That's probably because you
have had life experiences similar to this, specifically if you've made
tea using loose leaf tea, or if you've added sugar to a drink. When you stir tea leaves in a mug, they tend to gather in
the middle of the mug. That's why this is often
called the Tea leaf paradox. I'm calling it the
inverted Whirlpool paradox because I think it sounds cooler. You may also have experienced this effect when adding sugar to a drink. But in any case, maybe life
experience is telling you that the heavier thing
moves towards the middle when you stir it. But if you think about it, surely the heavier thing
should move out to the sides. Like that's how a centrifuge works. You spin the liquid and the heavier stuff moves out to the sides. If you're still not convinced
that this behavior is weird consider what would happen if I replaced the less dense top fluid with
an even less dense fluid, specifically air. In this case, look, the more dense fluid does
indeed fling out to the sides. I've actually told you what
these fluids are, have I? The fluid on the bottom is water. I've dyed it brown
fittingly with tea leaves, and the top fluid is white spirits, you might call it mineral spirits. White spirits are made of
short chain hydrocarbons. So a lot like oil it's hydrophobic. It doesn't mix with the
water at the bottom there, and like oil it's less dense
than water so it floats on top. (indistinct) might actually be
an oil technically speaking. Now we're not used to oil
being as runny as this. But anyway, I tried the same
experiment with mineral oil and it just doesn't work as well. The water and the oil
mix into this emulsion where you've got droplets of one liquid suspended in the other. So the demonstration doesn't work as well, probably because mineral oil and water are closer in density than
white spirits and water so they take longer to separate. In any case, if you
wanna try this at home, I stick with white spirits
instead of other oils because otherwise you might not get a nice boundary between the two. Thanks by the way to Sora Bradgeput for the idea for this video. So why does the more dense
liquid bulge up in the middle? Well, one clue is to test what happens when you spin the whole
thing on a turntable. Interestingly, the bulge doesn't appear. It's not that easy to
see with the water dyed and the white spirits clear. So in this version,
I've switched it around. The white spirits are
dyed with acrylic paint and the water is clear. You can see how the white spirit
bulges down into the water. In other words, the water
is thrown out to the edges as we would expect. When I stop the turntable, the liquid continues to spin for a while. And at that point, the bulge appears. So it seems as though the tea leaf paradox or the inverted Whirlpool
paradox, as I'm calling it, arises when the liquid inside
a container is spinning but the container itself is not. Interestingly, it was Einstein that described the solution first and his reasoning goes like this. First of all consider
the bulk of the liquid, the white spirits in this case. This liquid is essentially in orbit, I'm being a bit loose with language there. Orbit has a strict technical
definition, I'm sure, but go with me. In simple terms, when
the liquid is stirred it is flung out to the
sides of the container. That's why the level of the
liquid rises up the walls. And that makes intuitive sense, but let's formalize that a little bit by considering a packet of
fluid inside the container. This packet of fluid has a
velocity in this direction. And if it wasn't for the fact that it had an external force acting on it, it would carry on in a
straight line, but it doesn't. And that's because there is
a force acting perpendicular to the direction of motion, a force pointing towards the center. So instead of traveling in a straight line it travels in a curved line forming an orbit around the
center of the container. And it's actually really easy to see where that inward force comes from. As you already know the
pressure that you feel when submerged in a liquid goes up, the deeper you go into the liquid. Like if you dive into the
ocean, the deeper you go, the more pressure you feel
from the water around you. And if you look at this
container from the side you can see that an object placed here will feel pressure due to this depth. And an object placed here
will feel less pressure due to this smaller depth. So you have high pressure
here and low pressure here. And as you know, fluids flow
from areas of high pressure to areas of low pressure. And I'll state this now because
it becomes important later, that force on this packet of fluid is derived from the difference in pressure between these two points, not the absolute pressure
at any particular depth. For the packet of fluid
to move in a circle, the inward acceleration and
the velocity of the packet need to be balanced. The details aren't important, but for those of you who are interested, it's just the equations of orbital motion. If the acceleration towards the center is equal to the square of the
velocity divided by the radius then you get circular motion. If the velocity were too high
it would go like this instead. And if the velocity were too small, it would go like this instead, it would collapse in towards the center. And we know that for the bulk of the fluid this condition is met, the velocity and the acceleration towards the center are balanced. We know that because liquid is not escaping from the container, so in bulk, the acceleration
towards the center is equal to the square of the velocity divided by the radius. So what's different about the liquid at the bottom of the container? Well, these liquids are sticky. Water is sticky. It sticks to things like glass, like the glass at the
bottom of the container. So it will experience a drag force. So now think about a packet of fluid near the bottom of the container. That packet experiences a force pushing it towards the center. Because as we talked about before, fluid is pushed from
areas of high pressure to areas of low pressure. And the difference in pressure down here is the same as the difference
in pressure up here because the difference in pressure is just due to the difference in depth. Here I've shown the
difference in depth in red. And look, the difference
in depth is independent of how deep the packet of liquid actually is. That means that the inward force felt by a packet of liquid on
the bottom of the container is the same as the inward force felt by a packet of liquid
anywhere else in the container. That means the acceleration
of that packet of liquid towards the center will
be the same as well. But here's the crucial difference. The velocities are not
the same because of drag. If you remember in the
main body of the liquid, the velocity of a packet of fluid and its acceleration towards
the center are balanced. So they form a circular orbit. But near the bottom of the glass, the velocity is lower because of drag. So the acceleration and the
velocity aren't balanced. And when the velocity is too
small to maintain an orbit that packet of fluid will migrate towards the center of the container. In other words, what you end
up with is a flow of liquid towards the center at the
bottom of the container. This has been characterized
as secondary flow. You've got the primary flow which is the vortex motion
of the bulk of the fluid. And you've got this secondary
flow at the boundary layer of liquid moving towards the center. And it's this inward flow of fluid that causes the more dense
liquid to bunch up in the middle, or indeed the tea leaves. This explanation also tells us
why the effect is so extreme when you first start
stirring the container but only when the stirring
implement is partially submerged. That's because the stirring action very quickly generates
the difference in pressure but because it's being
stirred from halfway down, it takes time for that vortex motion to propagate to the
bottom of the container. In other words, the liquid at the bottom is moving even more slowly
when you first start stirring. One additional experiment that I did that I thought was really interesting was to use a magnetic stirrer
to move the liquid around. A magnetic stirrer has a
hidden magnet under the base. And when you put another
magnet inside your container and then spin the magnet
hidden under the base, your magnet inside the
container will spin as well. And that will stir the contents. Honestly, I wasn't sure
which way this would go because on the one hand, the liquid inside is spinning
and the container isn't. So we should expect that secondary flow. But on the other hand, it's
being spun from the middle, so which way will it go? Will the heavier liquid
be flung out to the sides? Will it bunch up in the
middle? Let's find out. So it seems like the fact that the liquid is being spun from the center is enough to overcome the
inverted Whirlpool paradox in this scenario. Isn't that funnel shape beautiful though? It's much deeper than
the funnel shape you get when you just spin the thing
or stir the thing by hand. As you might expect, the
inverted Whirlpool does appear when the stirrer is turned off, but the liquid is still spinning. Guess what arrived today
while you were at school? - [Child] What? - KiwiCo. (child screaming) You know, it's rare to find
something that my kids love that I also think is good for them. Like they love chocolate ice cream but they can't have it for breakfast. Okay. You can't have it for breakfast. Whereas KiwiCo, I've been
getting KiwiCo crates since before they were
sponsoring my videos. So I feel like I can
say with some confidence that they've had a really
positive impact on my kids. KiwiCo is a subscription service where a stem project comes
in the post once a month. Everything you need for the project is right there in the box. There's no running to the
shops or anything like that. And there are nine
different subscription lines for every possible age group. It's been amazing to see my kids grow into little makers over the years. Like these days, they fight
over toilet roll chews when they become available because they they've both
got an idea of something they wanna make with it. They're always telling me about
things they want to invent or like asking me how things work. It's amazing how often
those things come up in KiwiCo crates. For example, my son was
asking me how cogs work, I was telling about gears and
cogs and things like that. And the crate they got this
month had gears and cogs in it so I could show them and
they could build something using that thing that we talked about. But beyond all that, it's just a really fun
thing to do with your kids and like super wholesome and educational, all that good stuff. The promotion on this one
is really good, by the way. If you go to kiwico.com/stevemould50, you'll get 50% off your
first month of any crate. The link is also in the description. So check out KiwiCo today. I hope you enjoyed this video. If you did, don't forget to hit subscribe. And if you were to ask the algorithm, like which one of Steve's videos would you most like to watch next? Well, that would be this one. (rhythmic music)
oooh physics!
Tldw: boundary layers
Toward the beginning, he says that he thinks the emulsion forms instead of the inverted whirlpool with the mineral oil because the oil has a similar density to water. However, I believe this occurs because the viscosity of the mineral oil is much higher. Can anyone confirm/correct?
my followup question is: what happens if in a cylinder, you have a greater volume (or even just greater mass) of the heavier fluid than the lighter? does the effect remain the same, or does it turn into a "whirlpool upon a whirlpool"?