Trapped Water and Tiny Holes - Numberphile

Tom Crawford shows why water doesn't fall through a sieve with small enough holes.
More links & stuff in full description below ↓↓↓
Tom Crawford's website, with links to his work and other outreach: tomrocksmaths.com
More Tom videos on Numberphile: bit.ly/Crawford_Videos
Tom on the Numberphile Podcast: • The Naked Mathematicia...
Numberphile is supported by the Mathematical Sciences Research Institute (MSRI): bit.ly/MSRINumberphile
We are also supported by Science Sandbox, a Simons Foundation initiative dedicated to engaging everyone with the process of science. www.simonsfoundation.org/outr...
And support from Math For America - www.mathforamerica.org/
NUMBERPHILE
Website: www.numberphile.com/
Numberphile on Facebook: / numberphile
Numberphile tweets: / numberphile
Subscribe: bit.ly/Numberphile_Sub
Videos by Brady Haran
Patreon: / numberphile
Numberphile T-Shirts and Merch: teespring.com/stores/numberphile
Brady's videos subreddit: / bradyharan
Brady's latest videos across all channels: www.bradyharanblog.com/
Sign up for (occasional) emails: eepurl.com/YdjL9

[@ScottFree4all]

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

[@jako7286]

That's easy. It's because they're made of wood! You have to know these things when you're a king you know.

[@ScottFree4all]

I would have liked more explanation about “duck water” and how they just don’t get wet. The same equation probably works for this.

[@RadioactiveLobster]

@@jako7286 Well I didn't vote for you!

[@Priapos93]

Duxass!!!

[@inamdarsaquib9528]

@@jako7286 nah ! How can ducks be made of non living wood? I heard ducks are hollow. That may be true.

[@TomRocksMaths]

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

[@mme725]

Cleoquacktra is my pun contribution. But nice names and nice ducks. Nice math too, gonna check you channel out after this :)

[@noahcrow8982]

What happens when λ=0 in the ω^2 equation? Can it be zero? If ω=0 What does this mean for the fluid; in the sense how does it differ from when ω is in the form of a+bi? Side note: I am a fourth year physics student and am quite interested in fluid mechanics.

[@nagoshi01]

Is this fluid behavior the same as a Faraday cage, in essence?

[@georgio101]

Anatinae and Cleopatra.

[@erdbeerprienz]

I just loved the random jumpcuts to the ducks

[@condor6222]

it's a numberphile video with a pinch of ADD.

[@BizVlogs]

Sorry if this sounds pedantic, but that’s not really a jump cut. Just a “cut” or “cutaway”. A jump cut is when you cut between two different shots of the same thing, giving the impression that the object has instantly “jumped” from one position to another.

[@DavidBeddard]

More pedantry. There was nothing random about the distribution of duck clips in the video. This is a mathematics channel; understanding randomness matters. 🤓👍

[@cavemandanwilder5597]

I enjoyed the jump cuts to the dudes doing math. It was a nice addition to the duck video.

[@jurajvariny6034]

Maybe they wanted the bath. Or they had some insights on Navier-Stokes equation. Or both.

[@mhyzon1]

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

[@S1nwar]

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.

[@rmsgrey]

Which also aligns nicely with the experimental results.

[@GregoMorgan]

@@rmsgrey And that's how the cosmological constant was born

[@S1nwar]

ok to put it less clumsy: half a wave is the smallest standing wave you can have. the endpoints are fixed and the middle part swings. thats also why every single antenna is 1/2 the lenght of the signal its supposed to detect and ultimately ties into why visible light cant resolve atoms for example

[@RobManser77]

Didn’t he cover that in the last couple of minutes?

[@GregoMorgan]

@@S1nwar Not every single antenna no

[@Slothery]

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

[@eplumer]

… but does the duck weigh the same as a witch?

[@jamesepace]

Who are you who are so wise in the ways of science?

[@terencetsang9518]

Very small rocks!

[@user-pw6qe7ur4q]

Nice!

[@LukaszSebastian]

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

[@MadaxeMunkeee]

I think you’re right. The boundaries of the holes define nodes and the length of the fundamentals that fit in those holes is lambda/2. So he was actually pretty much on the money

[@JohnDlugosz]

No, you need a full wave. Half the wave is water falling out, and the other half is a bubble forming to go in.

[@ekster3283]

@@JohnDlugosz that doesn't happen right? Do the experiment, you will not see half a bubble going in.

[@LukaszSebastian]

@@JohnDlugosz you're wrong - as they said in the video, the whole point is that the wave of length 1.7cm can exist - plus water can fall through one hole, and air can rush in through another.

[@enalaxable]

Important omission: the pressure of air inside the glass is less than 1atm. So it would modify gravity force by the difference in pressure. Just to remind that if no holes and the glass has a light but hermetic bottom it will hold the water without any support. So more than λ/2 is possible.modified gravity g-> g-ΔP • Surface_upper /mass_water

[@leppeppel]

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

[@unvergebeneid]

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

[@mokopa]

That was my first thought, too, they come to bask in his fluidic vibes

[@alveolate]

he's always navier-stoked about navier-stokes

[@Triantalex]

false.

[@PeppoMusic]

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.

[@tronalddump2267]

Fry them to make delicious fried duck 🦆🍗😋

[@polus2494]

@@tronalddump2267 I wish I were as cool as you.

[@tronalddump2267]

@@polus2494 thank you

[@obiwanpez]

The ducks have oil on their feathers, which helps to keep the hole sizes small.

[@MrJdsenior]

@@obiwanpez And is not water soluble.

[@samuelpierce9962]

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

[@xXClawtoothXx]

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

[@idjles]

The answer is yes

[@xevira]

He can also Parker Reverse Derive it using very loose definitions of reverse and derive.

[@jonidcrushfire]

@Francesco Rondina To be fair, Matt Parker has an official paper written using his name to specifically mention whether something works or not in maths, lol

[@lars3509]

Parker Reverse Derive, PRD, is now officially a thing? Is a non-Parker Reverse Derive some method that actually calculates Pi?

[@MrJdsenior]

Have you ever NOT seen Pi in a scientific equation? Kidding, but it IS amazing how often it shows up, often where you wouldn't expect it to. Just remember, pi are not square, pi are round. Sorry, couldn't resist with that 'joke' that is probably moldy by now. Should have, didn't. My bad.

[@alexakalennon]

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

[@JohnDlugosz]

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.

[@bcwbcw3741]

Another move is to put a straw through one of the holes in the colander. When this happens, since the surface area of the region in the straw interior doesn't change as the surface moves up the straw you on longer require a full wavelength and the surface can go unstable.

[@crashmancer]

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

[@kruksog]

I always laughed at p_0 said "pea naught" because of how much it sounded like "peanut" when said in the flow of class. Sometimes it's the little things.

[@GrahamFirst]

🤤

[@pvic6959]

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

[@garvett6660]

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

[@rif6876]

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.

[@TomRocksMaths]

[@joshuamiller5599]

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

[@R.B.]

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.

[@ando1249]

I agree, if you opened up a hole in the bottom of the glass the water would flow out. The leaking at the start occurs until the pressure drops enough in the glass. This experiment is more analogous to a bunch of straws with the ends blocked holding fluid until you remove the blockage.

[@janvabek3240]

Yes, what you mention is probably another necessary condition of stabilty. I doubt it can hold with any hole in the glass. It would be nice to mention in the video. The argument of waves makes sense as a different condition preventing instabilities to grow. It would be nice to develop the idea more as other conditions are of comparable or maybe even stronger effect.

[@janvabek3240]

@@ando1249 I think that in the case of straws, you need to consider also the surface straw-liquid. I'm not sure it plays a role in this case.

[@netoboneto]

What holds the water is not superficial tension, its air pressure in equilibrium with the water weight. This experiment can be done with a sheet of paper, blocking the glass opening. The only force holding the paper and the water is air pressure. If they try this experiment with a open tube instead of a glass it wont work.

[@chriss1331]

I tried this experiment using a plastic bottle with the bottom cut off. It held the water up fine with the lid on, but as soon as I opened the lid there was a hiss and the water rushed out. Air pressure definitely seems to be important here.

[@tahashah7658]

Tom Crawford is the reason i watch these fairly complicated topics man...fluid dynamics is trippy

[@TomRocksMaths]

[@RayosMcQueen]

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.

[@wecantry4393]

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

[@pvic6959]

why is no one talking about the duck that came to visit lol

[@TomRocksMaths]

[@nagoshi01]

Basically a Faraday cage but for fluids? This is awesome

[@citrus4419]

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

[@TomRocksMaths]

The edition is very well done. Love the content, as usual.

[@youdonknowme2471]

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

[@AdamBomb5794]

Study at Oxford then :P

[@TomRocksMaths]

[@Pianoblook]

​@@TomRocksMaths woah, didn't know you had your own channel! Just subscribed :D I love your arm tattoos btw - I'd be very curious to learn the significance of the bands, numbers, etc

[@glennnicholls8510]

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.

[@scott_the_engineer]

I love the videos with Tom because he always blows my mind with crazy uses of the Navier Stokes equation

[@TomRocksMaths]

[@bunnyben5607]

This was a mindblowing episode

[@willyanteixeira]

fantastic another approach of treating particles as waves

[@mussalo]

Tom is handling the formulas in such a way I was never taught in school and it's so amazing! What I mean is that many times he doesn't straight away jump into the equation, moving the variables and plugging numbers but he understands the relations in it, recognizes the parts that matter and deals with them only. I love how he takes out the parts that he wants to manipulate and understands what parameters those parts need to fulfill, before even pluggin in the numbers!

[@theloganator13]

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. :)

[@MatthewAHaas]

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

[@TheRogueRockhound]

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

[@bullcompost]

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.

[@jeffkaylin892]

Yes, the partial vacuum is probably the biggest factor.

[@Mr.Beauregarde]

I have never felt so compelled to try a thing immediately. I didn't even make it to the math before I made it to the sink. Thank you for this

[@TomRocksMaths]

how did it go??!

[@Mr.Beauregarde]

@@TomRocksMaths the water stayed in the cup!

[@relt_]

animals in numberphile videos are a surprisng, yet welcome addition

[@JesusP7]

As usual, complex subjects are so nicely explained that makes me want to dive deeper into the reasons! I had no idea of this "magic trick"! Great video

[@TomRocksMaths]

[@alexwolffe7805]

A mathematician doing an actual experiment? THAT is incredible.

[@TomRocksMaths]

[@kieronparr3403]

More a demonstration than an experiment

[@HaloProGam3er]

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!"

[@mudkip_btw]

I love the clever conclusions fluid physicists can draw from simple models, thinking outside the box

[@mudkip_btw]

And ducks

[@JamminJosh7]

Loved the format of this video (deriving formulae from everyday life), would be great to see more like this!

[@oggiisme2458]

He’s back again with more fluid mechanics!!

[@TomRocksMaths]

you know it

[@CSGraves]

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!

[@Priapos93]

Waiting for the extras! extraas! and loving Tom Crawford!

[@TomRocksMaths]

[@yasurikressh8325]

I love that i can just pause the video and get lots of book recommendations to try out

[@chrisweddle2577]

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

[@TomRocksMaths]

[@darikdatta]

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

[@koenth2359]

It bugged me that he did not even try to explain the physics of it all. I have the idea that the pressure difference is actually almost perfectly balancing the weight of the water and that the surface tension is just preventing air bubbles to go in. Must be, because the height of the water column isn't even part of the equation

[@Henrix1998]

@@koenth2359 it's possible it leaked so little water that there wasn't a significant drop in pressure

[@koenth2359]

@Henrix98 Indeed, only a very small amount is needed to accomplish that. (For a 10 cm water column only about 1% of the air volume)

[@Forka137]

​@@koenth2359 I've done some experiments like the ones in the video and i reached the same conclusion as you. Also, @Henrix98 there is no drop in pressure since whenever it leaks water it's because its volume was replaced by air. In order to decrease the pressure inside the glass you would need a 10 meter column of water.

[@knightofsaintjohn9345]

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?

[@zenithparsec]

It seems to make it work you need to constrain the nodes. I think you'd just be doing the equivalent of blowing on it, possibly quite musically.

[@Musicdude14z]

I was wondering if you just mounted the glass in some kind of arm that could vibrate at a specific frequency if you could get the water to stay without any covering on the bottom of the glass! Seems like if you could get the fright frequency based on the size and shape of the glass's opening you should be able to form standing waves of any desired length, right?

[@MushookieMan]

It's been done. Shaking a fluid can make it sit stably over a less dense fluid. It can even make air bubbles sink. Check out the nature article Floating under a levitating Fluid. I've also seen videos of this effect, but I don't know where.

[@marchaustein1429]

@saint john completly off topic is this profil picture the knight from aoe3 1st campaign?

[@knightofsaintjohn9345]

@@MushookieMan damn... you mean to say I am *not* the first person to think of this? And here I was thinking the youtube comment section was the source of all novel scientific research. :,D

[@mikebel74]

Epic. Well explained and quite entertaining.

[@TomRocksMaths]

[@CobetcknnKolowski]

Great maths video! Love the duck intermissions!

[@abhijeetsarker5285]

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

[@TomRocksMaths]

[@nowonmetube]

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

[@chalichaligha3234]

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?

[@Rattiar]

I really appreciate Tom explaining the "trivial" case of water beneath air. Just like he said, it was a great little sanity check to make sure I was following along. And I was! But it was great to confirm. Excellent teaching moment, there.

[@TomRocksMaths]

[@serhancinar5218]

Fantastic video

[@Henrix1998]

Everyone's favourite guy is back!

[@TomRocksMaths]

[@samuelvilz]

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

[@TomRocksMaths]

Amazing! send pics?

[@Plut0-YT]

The duck updates cracked me up ^^

[@ultraL2]

same lol

[@MrMisaToman]

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.

[@londonalicante]

More water pressure does make a big difference. That's why he uses an upside down glass, which means the water pressure at the interface is zero (and the air inside the glass is at a slight vacuum). Make a hole in that glass, air will get in and the water WILL fall through.* The point is, in the absence of a pressure difference, if you constrain the surface to small holes, you limit surface waves and the water can't spontaneously invert with water moving below the air. * If the holes are REALLY small, the water won't fall through even under pressure and there's a different equation (young laplace equation) relating pressure and radius of curvature of the water surface. Water will actually rise up a small capillary, and the smaller the radius the higher it will rise. Capillary radius and radius of curvature are not necessarily the same, they are related by the wetting angle.

[@jsincoherency]

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.

[@fireballninja01]

if so, it might have something to do with density. The larger the volume of water above, the denser the water at the interface, thus resulting in stronger surface tension

[@gizmogremlin1872]

I was kinda thinking the trapped air inside the cup is also perhaps having some influence. So if instead of a glass, would just a through cylinder hold the water as well as the cup is my question.

[@henryD9363]

@@fireballninja01 surface tension of a liquid is independent of density I think. Although mercury has a very high surface tension. So does molten solder. ????

[@charlesdick1133]

It's probably be added to g in the equation given its acting in exactly the same direction. If the force was applied at an angle then just trig. But again, it'd be with g

[@kristentocherspoon6034]

I would agree with this, I don't think N-S is relevant here.

[@Jordan-zk2wd]

An extremely simple non-math B-plot is a great way to create engagement tbh. I dunno what you wanna do with that info, but it was nice to learn more about the ducks and it helped me enjoy the maths even more : )

[@adzaaahhh]

There's no ducking the question here... only understandable, well-articulated nswers. Great bit of applied maths in action, well done sirs!

[@TomRocksMaths]

[@rickyheath7607]

I was just wondering about this!

[@Shiasu-sama]

Some really cool stuff under the stairs there :D.

[@TehPwnerer]

This stuff should be taught in pre-school that would have been amazing to see as a child!

[@aryst0krat]

Neat how this turned to a bunch of math! It starts out feeling like a Steve Mould video.

[@matteogauthier7750]

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

[@grahambishop6807]

I was thinking the same, but when he showed the image with the whole glass, he said that the hole size must allow an entire wave to form with the given wavelength, so the whole wave forms in the hole, not just the half that is below y=0, like I initially thought. Not sure I understand correctly, but that was my thought process

[@grahambishop6807]

Have to admit that I am finding myself agreeing with you though

[@cjk32cam]

The whole glass is a special case. You’ve only got one hole, so conservation of volume prevents it containing just a half wave.

[@DrewNorthup]

Ahh, yeah. If you've ever built antennas from scratch before you know that the critical factor here is the size of 1/2 wave.

[@DrewNorthup]

@@cjk32cam Conservation of what? Wave resonances don't give a damn about anything like that.

[@paulstewart6293]

Thanks !

[@oyibechibundu628]

Tom is my favourite teacher in the whole of numberphile

[@TomRocksMaths]

[@TomLeg]

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.

[@caitlinbeck7281]

I watched this video with my cat. He appreciated the ducks and would like to see more of them in future videos. Thanks!

[@9merk_]

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

[@X_Baron]

Yeah, I feel like there's something missing from the explanation. A critical piece has been cut out or something. It suddenly goes from "right hand side must be positive" to the opposite and no reason is given!

[@SirIsaacTheRed]

Really liked this! Then again, I am partial to fluid mechanics, especially when demonstrations and your presentation style are involved Tom 🙂

[@TomRocksMaths]

[@leviathan7477]

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

[@ragnkja]

No, because then you have another force pushing the water down: atmospheric pressure.

[@VladimirNicolici]

@@ragnkja Kind of, but not entirely true. First of all, the same atmospheric pressure also pushes from below, so the two forces cancel out. But when one side of the glass is closed, a vacuum suction force starts to form as the water is "trying" to descend due to gravity, and this suction helps counteract the gravity that is pulling down on the liquid. At some point you get an equilibrium between the two forces, as the more the liquid is trying to descend, the more the suction is pulling it up. So, with the glass closed as one end, all you need is for the interface to be stable enough to prevent the air from going up through the holes to replace the water. The mass (and weight) of the water above the interface is not important. You could have a 10 cm tall glass, or a thin 10 meters tall glass full of water, it wouldn't make any difference. However, if the glass is open at both ends, the mass of the water becomes important. If it weighs too much, it will push down hard enough that it will break the surface tension, and there will be no suction force to counteract that. Anyway, if the sieve holes are small enough and the layer of water above them is not very thick, it will actually work even with a glass open at both ends. After all, the holes in a dense sieve remain filled with water after you get it wet, even if you don't use any glass. But of course, after you add enough water above, the holes will start to leak, because at some point the weight of the water becomes too much for the surface tension to counteract. And fluids can do a lot of other weird and wonderful stuff. A liquid can even climb by itself against gravity. If you take a paper towel square and dip a corner of that square in water, the water will start climbing through the paper towel. That's the capillary action.

[@VladimirNicolici]

One more thing to add, that 10 meter limit on the height of the water column I used in my example is not arbitrary, if you go higher than that it actually starts to matter, even with the glass closed at one end. This is the maximum height of a column of water that can be supported by the air pressure pushing from below, when the air pressure is one atmosphere. So you can't use suction to raise a column of water more than 10 meters under normal atmospheric conditions.

[@NateCrownwell]

Really awesome :)

[@jonaszurba4906]

I loved the duck intermissions

[@DominikJaniec]

intriguing idea!

[@kruksog]

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

[@TomRocksMaths]

can confirm this to be true. both the princess part and the ducks.

[@ujjwal3247]

Thats Great!

[@colinstu]

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.

[@killerbee.13]

Eggs have a membrane though, which doesn't allow for the fluid to pass through. It is technically possible to remove the shell from an egg entirely and it still won't leak out (though to do this you pretty much have to dissolve the shell, as anything more violent will damage the membrane)

[@SteveGuidi]

There is some really neat physics and math at play here! We can use the given equation to figure out the smallest hole in a (stationary) submarine that won't cause any problems, or similarly a spacecraft -- accounting for pressure in both cases of course!

[@marklonergan3898]

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

[@chaoticlife5569]

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

[@basnettd98]

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

[@shruggzdastr8-facedclown]

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

[@cmilla111]

The ducks are a perfect addition. Just standing there, slowly realizing this all applies to them.

[@expomath9348]

Toujours un contenu sympa à voir ! Merci Numberphile !

[@leonardromano1491]

This surface tension is similar to what keeps smooth particle hydrodynamics (SPH) codes at low resolution from exhibiting fluid instabilities. While the SPH surface tension is purely numerical, it is fun to see physical surface tension prevent the instability.

[@the_sci3ntist]

Thanks

[@adershvarshnei5198]

I really like Tom's handwriting.

[@TomRocksMaths]

[@MonsieurNab]

ok this is my next party trick for xmas with the family !

[@Quake1dan]

Love your videos! Will we get an extra video showing *how* the forces balance out? I can't wrap my head around it and I studied fluid dynamics at uni :O i'm drawing the vectors but i just don't get it :D

[@millylitre]

I guess the phrase "balance out" was whizzed through in the video. Slowed down a bit, this is my best guess...... "Balance" A (the big one): Atmospheric pressure at mouth of glass, and reduced air pressure above water inside glass, balance the total weight of water in the glass. Zero nett force. The whole body of water does not fall down as a single mass. The vector diagram for Balance A is simply three parallel vertical vectors. "Balance" B (the little one): But perhaps the water could trickle out at one half of a hole, with air leaking in at the other half of the same hole? That leakage can be prevented if restoring forces acting at the interface are strong enough to pull the interface surface towards a central mean position ("the system does not collapse"). "Restoring forces are strong enough", is equivalent to "Restoring forces cause wave motion to occur". So the main subject of the video is the conditions for wave motion to exist. This is where reference to "balance out" gets rather sketchy. In wave motion there ARE accelerations. Forces are not "balanced" .... unless we consider inertial forces as "balancing" real forces, as in the errm... "balanced" equation F=ma. (Maths-wise I can see that. Physics-wise the idea is uncomfortable.) Fundamental analysis of wave propagation uses inertia and restoring forces, and may include concepts of "balance" as noted above. That analysis was not part of the video. Instead one result of such an analysis was presented as a given... a simplified form of Navier-Stokes equation. And that equation does contain inertial terms as densities, and restoring force terms as surface tension and gravity. (So no wave vector diagrams for us here.) I think the reason for mentioning "balance" was to say only that the Navier-Stokes interface-wave-equation has already been derived from first principles, such as F=ma etc. Then the wave equation was used in the video to explore what wavelength of waves could exist, preventing air-in-water-out leakage at each small hole. Balance C... There is yet another balance, occurring at each hole, not mentioned in the video. Atmospheric pressure outside, hydrostatic pressure inside water at each hole, and Laplace pressure due to interface curvature and surface tension. I am still trying to figure out if Balance C is somehow the same thing as Balance B. Fascinating question :)

[@Quake1dan]

@@millylitre thanks for the deep dive. I sill find it hard to grasp even the balance A... On an intuitive level the weight seems to be greater than the difference in pressure (infact the same glass flipped over without a colander pours). This would mean that what holds the system together is an additional force (reasonably the surface tension) that nets the whole sum to zero. But if the surface tension in tangent to the boundary, on a wave shape, it should once again net zero against itself on the vertical axis (unless it is just half a wavelength). But this is not what they are showing here... I'm at a loss here :D Probably I am just struggling to keep up with your explanation as I am not an english mothertongue. I would love to see a vector graph with all the components on the z axis competing against eachother

[@chrisdooph5092]

I think Tom messed up the "

[@physicsjeff]

Indeed. He starts off by calculating the scenario when the wave fails, forgets to invert the inequality sign during manipulation, but then talks about the result as the criterion for success. Might just be due to editing.

[@typha]

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.

[@Henrix1998]

He flipped pb and pt

[@typha]

@@Henrix1998 No, not that bit, later. (check the time stamp)

[@LimitPotential]

The initial inequality was for the unstable solution (arg of sqrt < 0), so he flips it once more for the condition on stable waves.

[@Brmlyklr]

For some reason at 11:49 he started by assuming the square bracket portion is negative. But we just determined that a negative right hand side leads to a complex wave frequency... Then at 12:45 he skipped some rearranging steps and one of those steps involved taking the inverse, but he didn't flip the inequality. So he made two mistakes that cancelled out to get the correct answer. 👍

[@typha]

@@LimitPotential Ooooh, yeah, that's it. Thanks.

[@Adrian-me4qz]

Those ducks were very cute!

[@agmessier]

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

[@RobManser77]

I’m a newbie to this area, but I wondered - is this why mesh/gauze attenuates airflow way more than one would expect? For example, airlines can’t protect aircraft engines from distastrous birdstrikes with a mesh, because it has such a drastic effect on the air going into the engine and hence the performance of the engine. All I know is that pressure builds up outside the mesh, so I guess that recreates these circumstances because the density each side of the mesh/gauze would be different, but what I’m not sure of is where the waves fit in to this situation. You get the same effect with material, so if it’s windy and you’re inside a tent, it’s not windy, despite the holes in the material being millions of times bigger than air molecules such as N2 or O2. ps - brilliantly explained! You make it all seem so simple. 😊

[@Doktor_Vem]

Love the addition of a bit of duck documentation in this video d: