Trapped Water and Tiny Holes - Numberphile 

We named the ducks 'Sir Quacksalot' and 'Cleopatra'

One of the better duck documentaries I’ve seen. Now I understand how ducks stay on the water’s surface and don’t sink.

I just loved the random jumpcuts to the ducks

They visit Tom because Navier-Stokes is also their favorite equation as they are very interested in fluid mechanics.

you missed out on mentioning that the condition for the holesize is 1/2 the value you calculated because 1/2 a wavelenght fits in a hole. so the holes have to be smaller than 8.5mm.

Is the inclusion of ducks in a video about tiny holes blocking water intentional, since that partially is how they stay dry as well? Either way, don't care, cute ducks.

Hence the common expression, "like water trapped in a duck's colander."

"So air is obviously less dense than water, but what else is less dense than water?" "A duck."

Dr. Crawford: “and here we have rho-t…” Me, in Homer Simpson voice: “mmm, roti…”

Obviously they're coming to the fluid dynamics guy. In lack of a pond, he's the next best thing.

I think the actual physical boundary of a hole size would be λ/2, because a hole of this size can still fit a half of a wave with length λ so the wave itself can exist. This gives us an upper limit for a hole size equal to 0.85cm

As soon as I see Pi in an equation, I wonder "Can Matt Parker reverse derive it from this?"

One of my favorite Numberphile videos. Also, nice touch hiring the ducks for comedic relief

Of course the ducks showed up to class; fluid dynamics is their favorite subject.

When you read the equation and immediately know who's the professor explaining it. Love it

I love how despite the video being so maths- and physics-related, the ducks have essentially stolen the entire video.

my favorite part about this was the random ducks lol. i really liked how you guys kept bringing them up

That agrees with my intuition and observations made playing as a kid. Consider a narrow bottle: if you upturn it, it does not pour out smoothly but goes _glug_ _glug_ _glug_ in bursts. A little smaller is a straw. A straw is your example reduced to a single hole. A normal straw has no trouble holding water if you cover the end. But look at a super-wide straw, like what you get with pearl milk tea, or a piece of pipe that's larger than a straw. The water holds but is fragile. When disturbed, you see a blob of water going down and the surface going _up_ at the same time, looking like the sine wave. I see now that this leaks only once the amplitude of the wave is enough to allow the concave end to pinch off a bubble.

I'm being honest: I mostly clicked on the video because of the duck.

I think you're missing out of the fact that you are creating a vacuum above the water in the glass which helps the air pressure outside the glass push the colum of water up. It's important that the seal between the edge of the glass remains sealed. If the holes in the sieve have some height, like the larger screen has, then each of those pockets somewhat become their own columns, but if any air gets past the boundary you've eliminated the upward air pressure. An interesting addition to this would be to add a pressure guage to the bottom of the glass to measure the air pressure in the pocket. To further make the experiment consistent, the glass should be completely submerged, and the barrier should be applied when submerged, so that the volume in the glass is uniform for each test. I'm not saying the conclusion is necessarily wrong, just that the experiment doesn't fully isolate what you're testing. You would also need to make sure that there aren't any containing surfactants which would help break down the surface tension.

Typical! Leave a mathematician alone with a physics problem and they will make it much more complicated than it really is. Casting my mind back to my A level physics, We did a whole week talking about the pressure caused by a curved surface with surface tension acting on it. One thing I remember was that it’s inversely proportional to the radius of the spherical surface. The derivation must have been easy, otherwise we wouldn’t have been able to do it! It’s the reason why you need nucleation points to release dissolved gasses, and lots of other fun things. It was a whole chapter in our text books!

I am going to challenge you to repeat the experiment with a hole in the top of the glass to allow air coming in. My prediction is that the system wouldn’t hold water regardless how small the holes are. I believe that in your current experiment, the major force upwards on the fluid is a partial vacuum in the air in the glass. For air to come into the bottom through the sieve, that could collapse this force, surface tension would need to be overcome which then becomes a function of hole size. My intuition is that this is the primary effect at hand.

I'm not sure why in case 2 we have the rhs

What would happen if you repeated all the experiments but with a glass that has a hole in the bottom (maybe pluggable, so you could release the plug only when the glass is upside down or a similarly functioning valve)? Would the air pressure below the water still push the water up more than the pressure of the air inside the glass would? This could be more about the difference in pressure between the air inside the glass and the one outside, making it look like a "vacuum" is preventing the water from falling, maybe.

I wonder if you could "force" the water-air boundary into a standing wave pattern, which has a smaller wavelength than the holes, perhaps using some kind of acoustic wave. Wouldn't you be able to keep the fluid in the glas, even with bigger holes in the bottom?

What's most fascinating to me is that the condition of minimal hole size does not depend on the mass of the fluid above it. That to me is just unintuitive. If I had a ton of water up there surely it would squeeze out of the holes on the bottom.

12:43 is anyone else confused about why the inequality seems to be the wrong way around? because if you solve the one equality for lambda, you get lambda > ..., not less than.

Are we saying that the pressure of the water above the interface doesn't matter? Intuitively, I'd expect that a sieve can't hold up a mile-high column of water because the pressure would be enough to push through the surface tension. Is my intuition wrong? I'd love to see a demo either way.

The numberphile thumbnails and titles are getting wild these days. Btw, Tom Rocks Rocks.

What about the pressure differential between the trapped air and the atmosphere?

Also it’s worth pointing out that the hole size will be half the wavelength, I think?

Does the trick work if the glass is open on both sides? I don’t know how you’d do that, but my first thought when I saw it was the vacuum created by the empty space in the glass like when you hold your thumb over a straw

What about water pressure? The upward pressure that surface tension can provide is constant, but by increasing the water column the downward pressure can increase arbitrarily, forcing water through the holes regardless. Surely this should factor into the equations?

For those wanting to read more about this, it's called a Rayleigh-Taylor instability. This combined with the Kelvin-Helmholtz instability are why nuclear explosions create mushroom clouds. :)

13:15 - missed opportunity to use g=π² approximation…

This guy should be featured in more videos! He's one of my favorites now. And he applies math and physics to real world, not just recreational math.

Ehhhh... not so much, guys. First off, the reason the 1cm holes didn't work isn't because the 1.7 cm is an upper bound (the correct solution _is_ an upper bound, but it isn't 1.7 cm). I haven't done the maths myself, but as a physicist, I _strongly_ suspect that the hole size constrains a _half-wavelength_ not a whole one. Each _node_ must be at a solid boundary (at the edge of a hole), especially for the ones with just a thin wire separating the holes. With the bigger strainer with the distant holes, they might have to be full wavelengths per hole, but that's a significantly more complicated question. But my biggest issue isn't with the analysis... it's with the interpretation. The water's surface tension is most definitely _not_ what is holding the water in. That's actually completely backwards. The surface tension _and the water's weight_ are instead holding the atmosphere _out._ What supports the weight? The atmosphere below. Why? Because the 'air' in the top of the glass is under vacuum. So ultimately the water is held up by the glass's rigidity providing a pressure vessel to create a vacuum that holds up the water. All the surface tension has to do is be strong enough to prevent the air at each hole from bursting through, and _that_ is why waves are so important at that boundary layer. The surface tension, I'm pretty sure, is actually providing _stability_ more than force, which I believe is why you have to be so careful in how you lift the experimental setup off of the plate. If you weren't so careful, or if you shook it around after, you'd destabilize the wave structures and the air would find paths to form whole, separated bubbles, and it would all spill out. If the glass didn't have anything to do with it, and it was really just surface tension holding the water up, then you'd be able to hold a sieve full of water without the glass. Or with an open tube. Or a thin plastic cup. But none of these would work; the sieve by itself would just pour through. The tube would as well. And the thin plastic cup would be crushed by the atmosphere thanks to the vacuum inside.

Tom is literally my role model, wish I could meet him

I think Tom messed up the "

Basically a Faraday cage but for fluids? This is awesome

The duck were sent by the duck secret agency to observe human advancement of mathematic of fluid. They have to be on point ;P

Tom Rocks is Disney princess confirmed. Also, ducks are cute and pretty funny. Had them as pets for a bit.

The ducks are perfect for my ADD. thank you. actually helps me better pay attention to the maths.

Brilliant video I loved it. I think the wavelength is 2 holes. So you will be fine as long as your holes less than 8.5mm which seemed to match up with your plastic tray. I love the way that pi creeps into just about everything. Thanks Tom for a great video.

who else did not only not succeed but also did smash some glassware whilst trying to recreate the experiments themselves?

Just a small thing to point out. The first inequality on top being

So enjoyable to watch! Can you help me understand this? So if the RHS is less than 0 our system leaks. And we found that RHS

Interesting that it depends on the density of the water and not the weight. Does that mean if I had a giant mass of water above I could still hold it easily? Does it depend on the shape of the container (cup)? Pardon my ignorance I'm not familiar with fluid dynamics

I am not shocked that Tom drinks Huel.

If the glass wasn't closed on the bottom it wouldn't work.

Why didn't the height of the water column factor in? I would think that would determine the pressure difference across the interface. I'd love to see the derivation of that equation

Numbers....Equations...Greek Letters....Blah Blah Blah! DID YOU HAVE DUCKLINGS THIS SPRING AND JUST HOW CUTE WERE THEY?

My favourite Numberphile video of all time! Can we have more duck videos please?

I didn't see any NS equations? This seemed just like simplified Fourier series, and some mechanics physics and physical assumptions (which don't all seem legit, like the boundary having no mass when the boundary must consist of at least 1 H2O molecule thick).

I was hoping eggs would be mentioned. Those are covered in tiny holes too (to allow air to come in), and even though there are holes, the contents of the egg does not leak out.

I find it difficult to understand why the height of the water in the glass does not affect whether or not it passed through. Surely the water pressure has an effect?

This is a cute analysis, yet the water isn't staying in the colander without the glass. The low pressure air in the glass is doing a lot of work here, canceling out most of the weight. Now, same thing generally happens in normal case, water at the bottom air on top, with pressure of the fluid column providing resistance to gravity, so nothing new there, but because air cavity here is shared between all the cells, at a minimum, you have linked oscillators. So I don't know how reliable this analysis is.

The explanation at the start of the video bothers me too much to not comment on it. The water is held in the glass by atmospheric pressure, not by surface tension. Surface tension just prevents formation of instabilities (bubbles) on the interface. Obviously, if it was held by surface tension, then the water would not flow through the sieve after you lifted the glass.

Why should we assume that "m" (in F=ma) is negligible to conclude that F=0? The rigorous argument would be a=0 then F=0 at the boundary.

This is a neat trick and great way to explain surface tension, well done sir, well done.

i love the ducks being curious about the rubber ducky. I'm just imaging them being like "Hey guys, come check out this big yellow duck that never moves!"

Commenting before seeing the whole episod, but isn’t the biggest difference between the two larger sieves that one is made of metal (hydrophilic) and the other made of plastics (hydrophobic)?

First off the ducks are coming there because they get fed! Next I performed this colander/strainer experiment with my children after seeing this video. thanks for making me a magician!

The ducks be like: "Hubert! Hubert! Those humans are in our weekend cottage again!"

The ducks are to Tom as Audrey is to James Grime

13:50 Wouldn't the hole size need to be smaller than HALF the wavelength? The borders of the hole are fixed in place, so they are nodes, but between two consecutive nodes there is half a wave.

Would this work with a pipe? Is the vacuum at the top of the glass taken into account?

So I'm a math dummy who subscribed to this channel for some reason. I found myself having to try and use music/sound analogies to try and understand some of the concepts at work here. The mention of the sine wave as a basic building block to other waveforms is basically the same as partials used in additive synthesis being combined to build a more harmonically complex timbre. And then the bit about the long wavelengths not being able to travel through certain smaller holes... reminded me of reading how if you were to play the lowest notes on a marimba eroica in a small enough room, the wavelength of the sound would exceed the dimensions of the room and be inaudible. Also, ducks!

Everyone's favourite guy is back!

The condition formula for Lambda seems wrong to me: how can it be independant of the level of the liquid? Tge higher the level the higher the presure, so the wave length must correspond to that

A mathematician doing an actual experiment? THAT is incredible.

I actually tried this experiment like 20 times and it doesn't frickin work! Idk what am i even doing wrong lol

Practical Physical demonstrations! None of that, give me brown paper, animations and theory.

I see many comments about ducks. But none mentions the huge plastic duck right in the background...

Can you stop the water pouring through with a duck? And if you can, does that mean that Tom Crawford is a witch?

Ok now you've introduced the ducks they have to be recurring characters in future videos or I will be very sad

Came for the math, stayed for the ducks

He’s back again with more fluid mechanics!!

I think the white plastic thing failed because he slid it to the side of the plate and it tilted at the uplifted edge of the plate. If he had a handle on the white plastic mesh it might have worked.

Featuring: mystery ducks

They're coming for the bread, but they're STAYING for the math!

So beautiful derivation 😀......Very satisfying.

animals in numberphile videos are a surprisng, yet welcome addition

The ducks were patiently waiting their turn for professor's office hours.

Another factor is that when you move the glass ( and you can't hold the glass perfectly still) the water gains momentum, straining against the interface, increasing the likelihood it will break through. It would be interesting to take the colander and give it a little jiggle and see how much movement is required to destroy the interface. (later) Interesting to see that some people are arguing that some motion might improve the retention, while I'm arguing the opposite .. for different energies and frequencies, of course.

Ducks aren't great at english, they must have heard there was some pondering going on

I think you accidentally included some math videos in your Duck Vlog

Just use a gorilla. The gorilla will push the duck through those tiny holes with his great strength.

Those ducks are trained spies sent by the undergrads to get a hold of all those prelims on the shelf!

Constants and variables in equations are just like Pokémon, you gotta catch'm all

"above the lighter fluid", so youre saying your house is full of propane I see

rho rho rho your tea gently through the sieve until the vacuum increases above it and surface tension wins.

kinda sussy how it cuts off everytime before the water falls, ngl

i reckon the ducks heard you say ρₜ again and again thought you had (indian) bread

They should have also tried to test water off a duck's back while they were at it.

2:05 sounds so underwhelmed. I like to think i know a bit about science but i still was legit amazed. 😀

I love this guy. Seeing how well blackwork tattoos suit him made me get my first and only tattoo 💚

This is so cool! I’m so glad that you had Tom come out! I recently found his channel on youtube :)

...love the video-bomb by the duck at the very beginning!!

Mathematicians need to stop trying to do physics. "F=ma, but the boundary is infinitely thin, so m=0, therefor F=0" Um no... the mass of the water is considerable, but since it isn't moving, a=0.