A 2D Heron's Fountain Behaves Weirdly

(upbeat music) - A while ago I made a video 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 going to make the Heron

fountain

. But before we get to that, I want to say a couple of things about the bell siphon video above. 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 bowl traps.

I was talking about tankers, specifically UK tankers, which happen to be completely different to tankers anywhere else in the world, it seems. But anyway, when people say toilets are like greedy bowl traps, they're actually referring to the toilet bowl, which I didn't realize. And in fact in the UK toilets are not greedy bowl traps either or at least not what I have seen. Although I have seen toilets in other countries where when you flush the water level in the bowl goes up and up and up, which, frankly, is terrifying the first time you see it for someone who has only seen a toilet like the one in the United Kingdom. .

And then anyway, it gets to a certain point and everything slips away. That's a greedy cup siphon. For completeness, here is a collection of the most common toilet mechanisms around the world. This is the one you get in the UK. These two siphons are the types you will find primarily in North America. This one here, I actually saw several of these when I visited The Hague years ago, and you can find them all over Europe, here and there. There's a little platform there and that's what you poop on and then it gets knocked off the shelf.

It's called flushed toilet or (speaking in foreign language), but I call it the shit on the shelf. The main benefit is that it won't splash on you because it doesn't have to fall as far, but the downside is the horrible smell. The other example that people have shared with me is laboratory glassware called a Soxhlet extractor. It's really smart. You have this chamber that is repeatedly filled with solvent and then drained and filled with solvent and drained. Then 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 source. This is the traditional La Garza

fountain

. And when I say traditional, I mean traditional in the sense that this is the one you would do as a school project, not the one Heron himself did in Alexandria in the 1st century. But anyway, the way to start is to simply prepare the first container. The

heron

fountain is really cool because at first glance it looks like it could be a perpetual motion mechanism. Of course, it's not because those things don't exist. But the fact that the water level in the upper basin remains fixed while the fountain is running is disconcerting.

Here it almost seems as if a law of physics is being violated. Balls always roll downhill, they never roll uphill unless pushed. And it seems like something like that is happening here. But we can actually convince ourselves that no laws like that are violated simply by looking at this sped-up version. Look, you will see that all the liquid in the middle container drains away while the bottom container fills with liquid. In other words, the liquid goes from a position of high gravitational potential energy to a position of low gravitational potential energy or, in our analogy, the ball rolls down the hill. but how does it work?

It's actually very hard to tell by looking at this version of the Heron fountain because all the tubes are embedded inside the containers and are quite thin and obscured. It would be better if we could separate the tubes and flatten everything into a two-dimensional version. It turns out that prototyping two-dimensional versions of hydrodynamic mechanisms is quite complicated. One of the difficult parts is finding the right thing to place between the two sheets of glass or acrylic. I found this wiring to be pretty good. This is the type of wiring found in the walls of a house.

It's nice because you can bend it any way you want and it will stay there, and it has this fixed width. (soft music) I definitely had some failures. A lot of my problem had to do with making everything airtight, and in this case, airtight. For the most part I was using the wrong adhesive, but in the end I got it to work. (liquid splash) The 2D version doesn't last long, which makes sense. You can hold a lot more liquid in three dimensions than in two, but it's much easier to see what's going on. Here, side by side, you have the top container, the middle container, and the bottom container.

And you can see that several tubes have been moved to the sides but in any case the mechanisms are equivalent. Then, the liquid from the upper container falls to the lower container through this tube. And so the bottom container fills with liquid and as it does so it pushes air out of the bottom container and into the middle container through this tube. And because the middle container is filling with air, that pushes the liquid out of the middle container up through this tube, and that's how you get the fountain, or in this case, the drip. So, generally speaking, the liquid in the middle container ends up in the container below.

It's just that it goes through the source and that's the explanation. But to me, there is a gap in the explanation because, that explanation would still apply if everything was full of liquid and there was no air and it would be like this: the liquid from the upper container falls into the lower container. under gravity, and that forces the liquid out of the lower container into the middle container, and that forces the liquid out of the middle container into the upper container. But of course, that would never work. Intuitively we know that when the system is full of liquid it will be in equilibrium.

It's like this tube filled with liquid. The water level at each end is fixed, equal and in equilibrium. So the fact that there is air in the mechanism is clearly important. To find out why, let's simplify even further. Let's look at some important distances. Here on the left, you have the distance between the surface of the water in the upper container and the surface of the water in the lower container. You also have the distance between the surface of the water in the top container and the surface of the water in the middle container. And you have the distance between the surface of the water in the middle container and the surface of the water in the bottom container.

And you will 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 have a column of water at the top. It is a tortuous column, no doubt, but from a pressure point of view it is equivalent to a vertical column of water of this height. And similarly, at the bottom you have this winding column. It is equivalent to a column of this height but it is a column of air. So let's simplify that into a U-shaped tube with the same characteristics.

I'm holding both ends of the tube here to keep everything inside from moving. But you'll see that I've recreated what happens inside Heron's fountain. You have a column of liquid on one side and a column of half liquid half air on the other side. Clearly, the liquid-filled column is heavier than the half-liquid, half-air column. Then they both push down, but the heavier one will win. So when I release my fingers, the liquid column on the left will push down until it reaches equilibrium. And there you do it, that's what you see. And really the reason the Heron fountain works is because the gas is less dense than air.

So, looking back at Heron's source, we have a situation of unequal pressure. So the pressure of this column of liquid on the left is greater than the pressure of this half column of liquid and half column of water on the right. So the pressure here is greater than the pressure here. Which means 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 that is as tall as the column of liquid on the left. In other words, you need a source of liquid on the right to achieve equilibrium so that the water pressure on both sides is the same.

We clearly don't have as high a source in this case, which I assume is due to friction. So a

heron

fountain essentially

behaves

like this simple U-shaped tube where there is an imbalance between the mass of liquids on each side. You even get the same font effect here. The difference is that Heron's fountain creates a physical barrier between the liquid and gas parts of the right side. And it also creates a reservoir of liquid and gas to keep the system running for a long time. There are some mistakes when you create 2D versions of things. For example, I originally had several tubes running roughly through the center of each container, but of course, in 2D, a tube in the middle actually splits a container into two separate sections.

So I had to move the tubes to the edges to make it work. I actually made the same mistake in the original greedy cup siphon video, as you can see, look, I have to try to evenly distribute the liquid as I pour it here so that it rises equally on both sides. If you have any other ideas on hydrodynamic mechanisms that could benefit from the 2D treatment, let me know in the comments because I now have the entire production process under control. (liquid gurgling) There are a handful of YouTube channels that I make sure to watch everything they do.

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