A 2D Heron's Fountain Behaves Weirdly

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This guy just gained a new subscriber. Nice post OP super cool

👍︎︎ 1 👤︎︎ u/Dloms45 📅︎︎ Dec 22 2020 🗫︎ replies
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(upbeat music) - I made a video a while back about the greedy cup siphon or bell siphon in which I made a 2D version to help with the explanation. And it occurred to me that other hydrodynamic mechanisms would be easier to explain with a 2D version. So in this video, we're gonna do Heron's fountain. But before we get to that, I wanna say a couple of things about the previous video on the bell siphon. Well, two more examples, basically, of bell siphons in the real world. In the original video I said that toilets weren't really a good example of greedy cups siphons. I was talking about the cistern, specifically cisterns in the UK, which it turns out are completely different to the cisterns in every other part of the world, it seems. But anyway, when people say that toilets are like greedy cup siphons, they're actually talking about the bowl of the toilet which I didn't realize. And actually in the UK toilet bowls aren't greedy cup siphons either or at least not that I've seen. Though I have seen toilets in other countries where you flush the toilet and the level of water in the bowl rises and rises and rises, which, frankly, is terrifying the first time you see it for someone who's only ever seen a UK-type bowl. And then it gets to a certain point anyway and it all drains out. That is a greedy cup siphon. For completeness, here's a collection of the most common toilet bowl mechanisms around the world. This is the one you get in the UK. These two siphon ones are the types that you'll find mostly in North America. This one here, I actually saw several of these when I visited the Hague years ago, and you get them all around in Europe here and there. There's a little platform there and that's the thing that you poo on and then it gets flushed off the shelf. It's called a washed-out toilet or a (speaking in foreign language) but I call it the turd on the shelf. The main benefit is that you don't get splash back 'cause it hasn't got as far to fall but the downside is just the horrendous smell. The other example that people have shared with me is laboratory glassware called a Soxhlet extractor. It's really clever. You have this chamber that repeatedly fills with solvent and then drains away and fills with solvent and drains away. So you can put a solid in there that you're trying to extract something from and it will be repeatedly soaked with solvent through the greedy cup siphon mechanism. But anyway, this video is about Heron's fountain. This is the traditional Heron's fountain. And when I say traditional, I mean traditional in the sense that this is the one you would make as a school project, not the one that Heron himself made in Alexandria in the first century. But anyway, the way to get it started is simply to prime that first container. A Heron's fountain is really cool because at first glance it looks like it could be a perpetual-motion mechanism. Of course, it isn't because those things don't exist. But the fact that the level of water in the top container remains fixed while the fountain is running is perplexing. It almost feels like a law of physics is being violated here. Like, balls always roll downhill, they never roll uphill unless they're pushed. And it feels like something like that is going on here. But actually we can convince ourselves that no laws like that are being violated by simply looking at this sped up version. Look, you'll see that all the liquid in the center container drains while the bottom container fills up with liquid. In other words, the liquid is going from a position of high gravitational potential energy to low gravitational potential energy, or, in our analogy, the ball is rolling down the hill. But how does it work? It's actually really hard to tell by looking at this version of Heron's fountain because all the tubes are embedded within the containers and they're quite thin and they're obscured. It would be better if we could separate the tubes out and flatten the whole thing into a two-dimensional version. It turns out that prototyping two-dimensional versions of hydrodynamic mechanisms is actually quite tricky. One of the hard parts is finding the right thing to put between your two sheets of glass or acrylic. I found that this wiring is pretty good. This is the sort of wiring that you find in the walls of a house. It's good because you can bend it into the shape that you want and it will stay there, and it has this fixed width to it. (mellow music) I definitely had some failures. A lot of my issue was to do with making the whole thing watertight and, in this case, air tight. For the most part I was using the wrong adhesive but I did get it working in the end. (liquid splashing) The 2D version doesn't last very long, which makes sense. You can hold a lot more liquid in three dimensions than you can in two, but it's much easier to see what's going on. Side by side, here, you've got the top container, the middle container and the bottom container. And you can see various tubes have been moved to the sides but in any case, the mechanisms are equivalent. So the liquid in the top container is falling into the bottom container through this tube. And so the bottom container is filling up with liquid, and as it does, it's pushing air out of the bottom container into the middle container through this tube. And because the middle container is filling up with air, that's pushing liquid out of the middle container up through this tube, and that's how you get the fountain or in this case, the dribble. So overall, the liquid in the middle container ends up in the bottom container. It's just, it goes via the fountain and that's the explanation. But for me, there's a gap in the explanation because, like, that explanation would you still apply if the entire thing was filled with liquid and there was no air and it would go like this: liquid in the top container falls into the bottom container under gravity, and that forces liquid out of the bottom container into the middle container, and that forces liquid out of the middle container into the top container. But of course, that would never work. Intuitively, we know that when the system is full of liquid it'll be in equilibrium. It's just like this tube filled with liquid. The level of water at each end is fixed and equal and in equilibrium. So the fact that there's air in the mechanism is clearly important. To figure out why, let's simplify even further. Let's look at some important distances. You've got on the left here, the distance between the surface of the water in the top container and the surface of the water in the bottom container. You've also got the distance between the surface of the water in the top container and the surface of the water in the middle container. And you've got the distance between the surface of the water in the middle container and the surface of the water in the bottom container. And you'll see that two of those distances on the right add up to the distance on the left. And you'll also notice that that distance on the left is a column of water, but on the right you've got a column of water at the top. It's a circuitous column to be sure, but from a pressure point of view it's equivalent to a vertical column of water this tall. And similarly at the bottom you have this circuitous column. It's equivalent to a column of this height but it's a column of air. So let's really simplify that into just a U-shaped tube with the same characteristics. I'm holding the two ends of the tube here to keep everything inside from moving. But you'll see I've recreated what's going on inside the Heron's fountain. You've got a column of liquid on one side and you've got a column of half liquid half air on the other side. Clearly the full column of liquid is heavier than the half-liquid half-air column. So they're both pushing downwards but the heavier one is going to win. So when I release my fingers, the column of liquid on the left will push down until equilibrium is reached. And there you do, that's what you see. And so really the reason the Heron fountain works is because gas is less dense than air. So looking back at Heron's fountain, we have a situation of unequal pressure. So the pressure from this column of liquid on the left is greater than the pressure of this half column of liquid half column of water on the right. So the pressure here is greater than the pressure here. Meaning the system is unstable. It will evolve until it reaches a stable state. What does that look like? Well, it looks like a column of liquid on the right which is as tall as the column of liquid on the left. In other words, you need a fountain of liquid on the right to reach equilibrium so that the pressure from the water on both sides is the same. Clearly we don't have a fountain that tall in this case which I assume is because of friction. So a Heron's fountain behaves essentially like this simple U-shaped tube where there's an imbalance between the mass of liquids in each side. You even get the same fountain effect here. The difference is that the Heron's fountain creates a physical barrier between the liquid and gas parts on the right-hand side. And it also creates a reservoir of liquid and gas to keep the system going for a long time. There are a few gotchas when you're creating 2D versions of things. For example, I originally had the various tubes going roughly through the center of each container, but of course, in 2D, a tube in the middle actually partitions a container into two separate sections. So I had to shift the tubes around to the edges to make it work. I actually made the same mistake in the original greedy cup siphon video which is why you can see, look, I'm having to try and evenly distribute the liquid as I'm pouring it in here so it rises equally on both sides. If you have any other ideas of hydrodynamic mechanisms that could benefit from the 2D treatment, let me know in the comments because I've got the whole production process down pat now. (liquid gurgling) There are a handful of YouTube channels that I make sure I watch everything that they make. Now, I don't get much time to watch YouTube these days but when I go to YouTube those are the channels that I look for. And if I find that they haven't put up anything new since I was last there, then I ended up wandering aimlessly around YouTube looking for the next hit. (laughs) You know, the next, like, curiosity hit that we're all searching for. And, you know, maybe you're like me and you have that experience sometimes. And you just wish that someone would say to you, "Look, here's a good video. I promise don't, worry about the thumbnail or the title, I promise it's a great video, you're gonna love it." And the sponsor of this video, CuriosityStream, can actually help with that because it's a curated collection of nonfiction videos, programs, and documentaries, thousands of them, whatever category you wanna go into, pick a video, it will be interesting. I like to look through the collections. Look, staff picks, David Attenborough, award-winning, there's loads of stuff in here. You can do shorts, you know, if you just wanna watch something quick. Oh, look, it's Derek. CuriosityStream is available on your computer, your phone and your tablet, of course, but it's also now available on your smart TV. The promo on this one is actually really good. If you go to my URL, curiositystream.com/stevemould, use promo code Steve Mould at checkout, you get 40% off annual membership. That's just $12 for a whole year. It's for limited time only so check out CuriosityStream today. I hope you enjoyed this video. If you did, don't forget to hit subscribe and I'll see you next time. (upbeat music)
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Channel: Steve Mould
Views: 2,633,342
Rating: 4.9395413 out of 5
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Length: 12min 21sec (741 seconds)
Published: Fri Dec 18 2020
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