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# Blowing square smoke rings

My friend Dianna Cowern — aka PhysicsGirl — is a science communicator who makes wonderful videos demonstrating cool physical phenomena. Sometimes they’re simple but baffling brain teasers, like what happens to the water level if you throw a rock off a boat, or why a mirror reverses left and right but not up and down (hint: It’s because it’s not doing either).

And sometimes they’re on much more complex topics, like vortices in pools (seriously, watch that one). Vortices are spinning fluids, which includes both liquids and gases — anything that can flow — and the physics behind moving fluids, called hydrodynamics, is fiercely complex. It’s one of the single most complex topics in physics (add magnetism to it and watch physicists heads explode).

But what’s weird is that the complexity sometimes cloaks itself in deceptive simplicity. For example, you’ve seen water spinning down a drain a million times, or time-lapse videos of hurricanes from space. These are vortices, and they don’t seem that complicated, right? Yeah, no. They’re ridiculously hard to figure out mathematically.

And that brings me to a wonderful video Dianna just put out about smoke rings. There is a fairly simple device, called a vortex cannon, that can consistently puff out air vortices, and if you add a little smoke you get, well, smoke rings. Their motion is already pretty complicated, but then she (and math communicator Grant Sanderson from 3Blue1Brown) made it even more so. They ask a pretty interesting question:

What happens if you put a square aperture over the cannon? What happens to the smoke rings?

OK, first, think about it for a sec. Then watch the video. The answer is extremely cool:

Did you get it right? I bet you didn’t. I got about halfway to the right answer in my head, but no further. The vertices moving ahead and then behind, causing that weird flapping, really surprised me. When I saw it, though, I almost literally slapped my forehead as I realized what was happening.

But the shape change from rounded square to diamond and back again, over and over? And the rectangle changing shape that way? Yeah: I wouldn’t have guessed that beforehand in a million years.

Dianna’s explanation is great, but I want to add something. Just why do the corners rotate faster than the edges? The answer is the Venturi effect.

This has to do with a principle called the conservation of mass continuity, which is related to conservation of momentum. This principle states that, for constant mass, the rate at which mass enters a system is the same as the rate that leaves. This makes intuitive sense; if you have a pipe with water flowing in at some rate, say 1 kilogram per second, you expect the water to leave the pipe at that same 1kg/sec rate.

Seems simple, right? OK, but what happens if you constrict the pipe? You can’t get the same amount of mass through the smaller part in the same amount of time, so what happens? It turns out that the water speeds up. Remember, we’re talking about the rate of water flow, in mass per second. If the mass drops, the velocity speeds up to compensate so that more mass can go through in the same amount of time. That way, the mass flow rate stays the same.

That’s the Venturi effect in a nutshell. What does this have to do with the vortex cannon?

In a normal cannon the aperture is circular. Friction with the side of the cannon slows the air around the edges that flows over it, but the air in the middle isn’t slowed down. It gets out faster, rubbing against the slower moving air around the edge. That sets up the circular ring rotation Diana talks about in the video. Where every molecule of air along the ring is moving at (more or less) the same speed.

But then they use the square aperture. Remember what Dianna said (around the 4:40 mark) about how two vortices near each other can speed each other up near where they’re closest? That’s critical to understand the weird motion.

Near the corner on one side, air trying to force its way through gets compressed by air coming through from just around the corner. That part of the flow gets compressed, so it speeds up a little. It moves ahead of the ring, but the circulating air is self-sustaining; it wants to stay a vortex. A force similar to tension (which I’m guessing has to do with pressure imbalance in the air in the ring) pulls the advancing corners back, but then the middle parts move ahead of the slowed corners, which in turn are pulled back again. This sets up an oscillation, making the smoke ring flap. I suspect the detailed physics is a good deal more complex than this, but that’s the gist.

For the rectangle, I think a similar thing is happening, with the added twist (almost literally) that the shape itself affects the tension of the ring. It wants to be a circle due to tension (a circle is the lowest energy state of a situation like this, and systems tend to move toward their lowest energy state), so the parts that deviate most from a circle get pulled back. In this case, that would be the two long sides. But when they move back the molecules of air give their momentum to the lower energy ones in the short sides, which then react by moving outward. Again, you get that oscillation, but this time it’s in the plane of the ring as well as the corners moving ahead of the ring structure itself.

I know. This is pretty weird and head-hurting, and to be honest that explanation above is a guess based on what I know of the physics and what I saw in the video.