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17: Navier Stoked (Vector Calculus and Navier-Stokes Equations)
964 Plays7 years ago
From our first breath of the day to brushing our teeth to washing our faces to our first sip of coffee, and even in the waters of the rivers we have built cities upon since antiquity, we find ourselves surrounded by fluids. Fluids, in this context, mean anything that can take the shape of its container. Physically, that means anything that has molecules that can move past one another, but mathematics has, as always, a slightly different view. This view is seen by some as more nuanced, others as more statistical, but by all as a challenge. This definition cannot fit into an introduction, and I’ll be picking away at it for the remainder of this episode. So what is a fluid? What can we learn from it? And how could learning from it be worth a million dollars?
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Transcript
Fluids in Daily Life
00:00:00
Speaker
From our first breath of the day, to brushing our teeth, to washing our faces, our first sips of coffee, and even in the waters of the rivers we have built cities upon since antiquity, we find ourselves surrounded by fluids. Fluids, in this context, mean anything that could take the shape of their containers. Physically, that means that anything that has molecules that can move past one another. But mathematics has, as always, a slightly different view.
00:00:27
Speaker
This view is seen by some as more nuanced, others as more statistical, but by all as a challenge. This definition cannot fit into an introduction, and we will be picking away at it for the remainder of this episode.
Introduction to Fluids and Navier-Stokes
00:00:38
Speaker
So what is a fluid? What can we learn from it? And how can learning from it be worth 1 million dollars? All of this and more on this episode of Breaking Math, Episode 17, not VA stoked.
00:00:57
Speaker
And now we'd like to take a quick moment to announce our sponsor, brilliant.org. You may have noticed that we're diving into a lot of calculus in this episode, and while we do, well, we hope we do a good job of explaining it, you may like to dive deeper into those topics. And that's what brilliant.org can help you with. They have courses on everything from calculus to other topics like trigonometry and even general problem solving.
00:01:20
Speaker
I actually used it a lot when I was teaching seventh grade math too, that they have a lot of really fun problems on very specific topics. Like we could go to the standards that we were on and use some random problem from Brilliant. It was awesome. And they help you with the answers. So it's more like tutoring too, which is something that I am very familiar with.
00:01:39
Speaker
Well, the wiki is really cool that you can see other people's answers and analyze whether or not they know what they're talking about, which is a really good learning experience, because sometimes they're just full of crap. But most of the time, most of the time, it offers a really good discussion. Oh, yeah, definitely. It's a wonderful website. So if you'd love to learn, and we know you do because you're listening to us, if you'd like to support the show, please check out brilliant.org slash breaking math. There's nothing to lose by your ignorance. Can we say that? Sure. Good. Now back to the show.
Significance of Navier-Stokes Equations
00:02:07
Speaker
I'm Jonathan Baca.
00:02:09
Speaker
And I'm Amy Lynn, filling in for Gabrielle Hesh, who's out on an important business today. So what are we going to be talking about today? We're going to be talking about the millennium problems, um, on the, on a series of episodes. And today we're going to be talking about the Navier-Stokes equations. Uh, going to warn you that there's going to be a little bit of a scary episode for some of you. We're going to be talking about calculus, vector calculus, and even differential equations. But don't turn off the podcast. We're going to break down each of these concepts so that they're easy to understand. Well, easier.
00:02:38
Speaker
Yeah. So these problems are worth solving just in the world of math, but they're also worth solving in the world of, I guess, money. Yeah. There's a million dollar prize, right? If you solve one of the millennium problems.
00:02:53
Speaker
Yeah, it's the Millennium Prize Problem set by the Clay Mathematics Institute in 2000. It was following a tradition from the century before. Six have yet to be solved.
Applications in Weather and Turbulence
00:03:05
Speaker
As of 2017, only one has been solved. As far as I understand it, I think it's a very complicated proof from what I've heard.
00:03:12
Speaker
Lame. I like, I like the easy proofs. Well, not easy, but elegant. That's what I would say. So, okay. So what, let's talk about the Navier Stokes. I'm sorry. I keep saying it the American way. Is it Navier Stokes?
00:03:24
Speaker
I believe it's Navier-Stokes, but I'm American as well. Okay, so we'll alternate between trying to say it correctly and just going with what comes out. So what are the Navier-Stokes equations? What would a solution to those mean? We do understand and we know how to model things based on the Navier-Stokes equations.
00:03:43
Speaker
When you're modeling fluids for 3D simulation, it's roughly based on the Navier-Stokes equations, sometimes Euler equations, which we'll be looking at later. But a solution would mean simplifying them significantly. It would allow us to predict weather a lot better.
00:04:02
Speaker
mean more efficient hydraulics, new types of machines that we haven't even dreamt of, and even possibly mean a solution to the problem of turbulence, which Feynman called the most important problem in classical physics.
00:04:19
Speaker
I've taken, I think, two classes in all of my math. One was specifically called geophysical fluid dynamics. I don't feel like I learned very much from it, but I do remember being kind of blown away by how difficult it is to mathematically model fluid motion. Like it didn't seem like it would be that complex, but it's unbelievably complex.
00:04:38
Speaker
Oh yeah. And I think maybe a little intuition as to why this so complex is if you've ever been in a pool and
Historical Insights: Archimedes and Al-Jazari
00:04:45
Speaker
if you've ever dropped a pool toy in a pool, like even a ball, it doesn't take a straight path towards the ground. It wiggles around and wobbles. It follows a very complex motion. And that's just a ball. When you're just dealing with, um, with no air, like an ideal physics problems, it's, you could predict where the ball is going to be in a hundred years.
00:05:05
Speaker
So we're going to be talking about these amazing equations, where they came from, where they are, and where they're going.
00:05:12
Speaker
and a little bit about why we care about them. Cities are often built on rivers. Agriculture, beyond the small scale, depends on the mobilization of water, and boats travel very efficiently. For these reasons, and probably others, humans have always studied the mechanics of fluids. It is in this history that we see how the idea that would eventually become the Navier-Stokes equations took form.
00:05:35
Speaker
So I think one of the first fluid dynamics or would it be fluid dynamics or fluid mechanics was invented or discovered by Archimedes of Syracuse. Is that where you submerged the crown to figure out volume? Oh, yeah. And I think we should go back a little bit and discuss how that worked and why that worked just really quickly. Do you want to tell the legend?
00:06:06
Speaker
Well, I only have a cursory knowledge of it that he was able to determine the volume amount of gold in a crown by submerging it in water and measuring the displacement.
00:06:16
Speaker
Yeah, what he did was he put the scale itself underwater and he weighed an equal amount of gold above water where air is very light, so he can't tell the difference. And when he submerged the two, the gold and the crown had different buoyancies. So that's how he saw that the crown was not made out of pure gold and the man was executed.
00:06:40
Speaker
Huh. See, I hadn't heard that part. I guess we just got the boiled down version that didn't involve murder in elementary school. Just the part about you can put things in water and figure out how much volume they have. It's not ancient Greek history without a little bit of murder. That's true. That is a fair point.
00:06:55
Speaker
And what's really important, I think, about this and tell me if you agree is that for the first time you have, well, you have a little bit of calculus on the amount of volume displaced by the water, volume itself being like a principle closely related to calculus. And you have a physical and you have physics related to that, the amount of force that is caused by the buoyancy.
00:07:22
Speaker
Well, I think it's kind of a stretch to say that it's related to calculus, because I feel like, at least in my mind, calculus is the math of movement. And I don't know that there's a direct analogy, but I suppose there's rate of change of displacement. But it's definitely one of the first words we're trying to scientifically measure something or do it in an entirely different construct than had been previously done before. And I think that creativity is certainly the foundation of a lot of mathematics.
00:07:51
Speaker
And one interesting thing that he actually did with his principle of buoyancy is a sort of proto calculus where he would divide shapes into a bunch of different slices that had levers submerged in different amounts of water. You can actually read it online, the method of mechanical means. We don't have time to go into that. See, that sounds a lot more like calculus. Yeah. He would prove everything with calculus and then reprove it with traditional Greek geometry.
00:08:18
Speaker
Traditional Greek geometry was a lot of, uh, a lot of work. It was cool, but it was a lot of work. Oh yeah. It's definitely different than the sort of math that we have since I guess, what, the 1800s. Certainly the geometry we have since then. We don't do it though. I feel like the Greek geometry was much more philosophical and logic based. And now it's a lot more about like, what's the circumference?
00:08:41
Speaker
And maybe that's why Hero of Alexandria invented the first steam engine and never did anything with it. He called it the Aliopile. It was a steam that made this ball rotate very, very quickly. It's not like he didn't know that he could do anything with it. He invented a windwheel, but he did nothing with Aliopile. We could have had steam engines for 2000 years now.
00:09:06
Speaker
I wonder if it was like his Van der Graaff generator that it was just a cool thing he had in his office in the background to make him feel like he was a cool scientist. Oh, yeah. I mean, I do know that hero of Alexandria, one of his things was like he had a coin operated holy fountain water thing. It was a lot of novelty things. All right. So here's here was a jokester. Guess he couldn't sell that one.
00:09:27
Speaker
So basically, between then and about a thousand years later, there's not much fluid dynamics or fluid mechanics that was really invented. But I think, and I just discovered this person while researching for this episode, and I think that I'm going to try to pronounce his name. His name, I think, is Badi Azzaman Abu Isma'il Ibn Ar-Razaz Al-Jazari.
00:09:57
Speaker
Your internet was not blocking out. That's just how many syllables there are. And I have probably butchered that. I know we have some listeners in the Middle East. If you could write in and tell us how well I did, that would make me feel very good.
00:10:15
Speaker
I think from now on, we'll just call him Al Jazeera. All right. Yeah. We'll go with the, that might be too white or too American, but we're going to go with the very last name because we can pronounce it more than anything. So I'm looking at the list of things you have printed here. He's done about a dozen amazing things. It's eight, but it's close to a dozen.
00:10:34
Speaker
Oh yeah, I mean, he invented, or at least not necessarily invented, but refined the camshaft and the crankshaft. He invented a chain pump, which is an improvement over the traditional Archimedes screw that everybody learns about in elementary school, and that was to provide irrigation.
00:10:57
Speaker
And this was actually a chain pump driven by water itself. So Zoe basically diverted a river uphill, which is a really cool feat in my opinion. Yeah, super helpful. So where was he? He was, I believe, Baghdad. Okay. If I'm wrong about that, then I will feel very bad.
00:11:16
Speaker
And timeframe, do we know about the golden age? So I think about the eighth century, ninth century, somewhere around there. Nice. I have no idea what a camshaft is. Is that embarrassing? I don't know what that is.
00:11:30
Speaker
Camshaft I believe that's It's it moves it translates rotational motion into sinusoidal linear motion like a pin or a rod up and down with Yeah with the two it's basically an off-center wheel that moves a a pole and it's useful for all sorts of things and
00:11:55
Speaker
Is that similar to, uh, what's the, the one that's in the water, the water wheel? Maybe. Is that what it's called? Is that all it is? I think it's good. It's called water wheel. I mean, that's usually the grain and refineries back in the, you know, what 18th century, 17th century.
00:12:10
Speaker
Oh yeah. A camshaft is more like, um, let's say you needed to have like a pounding motion for some reason or a, so instead of being a direct, like a very steady motion, it's a quick up and down motion that's caused by rotational motion.
00:12:27
Speaker
Okay. So this is the kind of cool stuff that Al-Jazari was bringing to the table. He also had a, one of the cool things is he built an automatic waitress who had come out through a set of doors, pour tea from a selection of teas and liquors and go back. So, I mean, he probably had a lot of money at his disposal, but he did some cool things with it.
00:12:53
Speaker
That reminds me of Doc Brown and Back to the Future 3 with this giant machine that built ice. I'm not familiar. You haven't seen Back to the Future 3 where he goes back to the West? I thought there's only one Back to the Future movie. Oh my God, there's three. What have you been doing with your life, Jonathan? I've been writing this outline. Fair enough. You need to, I think for the next episode, you need to go watch all of the Back to the Future movies. You're missing out. We'll do. All right. So we, uh, we're in the what, eighth century is that what you said?
00:13:22
Speaker
Yeah, but we're going to fast forward all the way to the 17th. Does that mean my favorite? Is that Newton and Leibniz? That is, and specifically Newton this time. Yeah, the physicist. And one thing that he studied, I mean, he studied everything, but including alchemy, but was the friction caused by water. So this is an essential component
00:13:50
Speaker
of how fluids interact. I think we should talk a little bit about the properties of fluids before we go on. We should probably also mention friction. I think everyone's familiar with friction in the sense of when you get a rug burn, that's from friction, right? That you scrape your skin over the rug and there's heat that comes from the friction of the movement on the floor. I didn't realize, I mean, I guess I did, technically I did, that there's friction in fluids.
00:14:19
Speaker
Oh yeah, I suppose anything that translates, anything that can slow down creates heat through friction, I guess. And I guess fluid is no exception, even though, and I'm not sure if it was Newton who did this experiment, but someone showed that stirring water made it warmer. So not sure if that was before or after Newton, but
00:14:44
Speaker
That's one way of showing that friction is, even though it's much less part of, I mean, we use lubricants all the time for things like doors, but there's still friction involved. You can't get rid of it completely. Well, that's why all the physics classes when you're in high school say, presume a frictionless surface because it's just too complicated to include friction in our calculations.
00:15:06
Speaker
Oh yeah, with non-linearity, if you listen to the episode Reaking Chaos, you get a chaotic system with certain degrees of non-linearity, which are very easy to come across when you add things like friction and viscosity. It's the resistance to shear stress.
00:15:24
Speaker
Sheer stress being when you rub your hands together, you have a shear stress between the two or when you twist a pencil apart. There's a little bit of shear stress there. So you'll notice if you have a very viscous fluid, soap. When you get soap, when you push it, a lot of it moves.
00:15:44
Speaker
with it, and only liquid hydrogen, and I think some bizarre physical things are actually inviscid fluids. Well, liquid nitrogen, how could you even tell it evaporates? Oh, no, liquid helium. Liquid helium, okay. Yeah, so it's... Well, I feel like that'd be a similar issue. Because it would evaporate before you could even, like, could you get something cold enough to be able to determine its viscosity?
00:16:11
Speaker
Oh, well, the reason why you could tell it's, um, inviscid is because the, uh, the vortices that it forms don't stop. So you can stir it and it'll keep going forever. That's awesome. Yeah. So if you ever need a cup of coffee, make it with liquid helium. You'll sound funny too, when you drink it.
00:16:32
Speaker
Yeah, and that's the only bad thing that'll happen. I feel like you'd have an eternally flushing toilet with liquid helium. You'd never have to flush it and just keep on going.
Newton and Calculus in Fluid Dynamics
00:16:41
Speaker
And that's actually a very odd way of bringing up one important thing about fluid dynamics is that they do not violate Newton's laws or any laws of conservation, even though they may seem to. And the reason why is you can think about it
00:16:58
Speaker
this way. If you have a train that's going around on a circular track, if you're on the train already, it takes very little energy to get to somewhere else, but to get off the train is what takes energy. So it's only change in rotational energy or rotational movement that takes energy, at least in certain contexts. So staying on the merry-go-round isn't hard, it's getting off of the merry-go-round that's hard.
00:17:22
Speaker
Oh, yeah. And starting the merry-go-round. There's definitely something about the difference between rotational and linear motion, but I think that's a philosophical thing for another episode, definitely. Well, and it's all about perspective, too, right? Yeah, physics is all about what you make it. Although my students try to use arguments like that all the time, and you still have to do the homework I assigned.
00:17:45
Speaker
Now, one cool thing about Newton's study with fluids is because he was one of the inventors of the calculus. He studied like if you do the experiment at home, get a bottle of water and you poke holes near the top, near the middle, near the bottom.
00:18:06
Speaker
and the water will flow much more quickly out of the bottom than the top. So this is a way of studying a fluid dynamics thing with a new type of math that had just been invented in 1666 or 1674 if you're talking about Leibniz.
00:18:22
Speaker
I mean, I guess that makes sense that there's more pressure at the bottom. And so there would be more, more liquid or being pushed out with more pressure. Oh yeah. Yeah. It's all about the, the weight. Um, yeah, definitely. And, um, but we didn't really develop the math to describe that situation until Newton.
00:18:40
Speaker
Yeah, and using just this and also the study of how water flows through, if water's flowing from one container into another container through a small hole, the water will contract after it's left the hole into the new thing. So basically, the water almost seems to be attracted
00:19:08
Speaker
to itself and this is a very unintuitive, I think. So let's imagine you're in a giant swimming pool and there's no swimming pool next door and they're connected together by a wall. If somebody cuts a hole in the swimming pool and you go through, you're gonna go through and then feel compression after you go on the other side for a moment.
00:19:32
Speaker
And this is the beginning of the formalization of fluid dynamics. Well, isn't that one of the properties of water that makes it so that we can even exist on Earth? That it's polar, right? That there's a negative end and a positive end of the molecule itself. And that's why surface tension is the thing.
00:19:53
Speaker
Oh yeah, but I think that just has to do with the meniscus. This has to do with the vena contracta, I think it's what it's called. And it happens with any fluid that's going from one container to another. It causes like a compression after it's left the first container, seemingly paradoxically.
00:20:15
Speaker
That reminds me of like heart motion, like fluid being pumped through your heart. That sounds familiar that. Oh yeah. The heart is the heart. I think is one of those examples where evolution was, is a lot smarter than we are right now. I mean, you're only three, three cells away from a blood, uh, from a capillary at any, uh, any point in your body. See, we don't need to, we don't need to frighten our audience, Jonathan. Well, too late.
00:20:42
Speaker
We are talking about Newton and Bernoulli now. All the famous mathematicians could possibly be more terrifying. The Bernoulli who studied a lot of fluid dynamics is Daniel Bernoulli, one of the many Swiss members of the Bernoulli family that somehow we're all brilliant mathematicians. Don't know how that happens. But Euler, I don't know, do you want to talk a little bit about just kind of the type of person that Euler was before we go into his work on fluid dynamics?
00:21:09
Speaker
Well, I mostly know Euler from his, is it, it's Euler's constant, right? I know E that everybody's familiar with pi as an irrational number, the 3.14, on and on, that Euler's number is E, which kind of shows up naturally. It's actually the natural base. It comes up in mathematics, whereas pi kind of comes up naturally in circles and in geometry.
00:21:33
Speaker
Euler's constant E comes up in calculus and logarithms and it just kind of... Understring growth. Yeah. Yeah. And it comes up that there are certain patterns in nature that settle to E and things like that. So that's one of my favorites. That's probably my favorite number actually is E.
00:21:52
Speaker
Yeah, and it's weird the natural logarithm was actually invented before E was discovered, which I think is always weird. I think it was Napier who discovered it or someone like that. So, logarithm before exponential?
00:22:07
Speaker
Yeah. That's crazy. Yeah. I mean, I think it was some, it was less than a decade between the first and between the discovery of the natural logarithm and when they're using it to do a slide rules. Um, people were pretty desperate for fast multiplication. That's awesome.
00:22:27
Speaker
But yeah, Euler, one of my favorites, just to show you how much thought he put into things, when he was going blind, he would have his nephews, who were probably very annoyed by him, do the writing part of his work for him, and he would fall asleep by calculating cortex.
00:22:46
Speaker
So really interesting kind of guy. And he generalized the theory of fluids using partial differential equations. And we've talked a little bit about differential equations on other episodes, but I think it's good to do a little refresher since we will be talking about them for the remainder of this episode. Well, before we get into differential equations, though, maybe we should talk about calculus in general.
00:23:06
Speaker
Absolutely. Calculus was a branch of mathematics, glimpsed by Archimedes, formalized by Leibniz and Newton, and is used to this day in the study of almost all modern mathematics. Although avoided by many students due to the introduction of scary-looking symbols, calculus is a very simple field of mathematics with rates of change. A rate of change is anything that goes from one point to another. This can be over space, such as with hills,
00:23:34
Speaker
with time, such as with velocity. One common rate of change that we see every day is represented by this pedometer. When we're going at a constant speed, the needle does not move.
00:23:47
Speaker
So calculus literally means, I was just looking this up, it literally translates to small pebble used for counting and calculations like an abacus. Uh, so it's, it's counting with very small things. And what's kind of interesting is that the concept of calculus went from infinitely many small pieces into something that's continuous, which I think is really beautiful. That was one of the first times I ever felt like I understood in any way that a human being can understand it, but the concept of infinite.
00:24:15
Speaker
Oh yeah, and that was what attracted me to calculus before I learned it. I was kid obsessed with the symbol infinity and seeing limit x approaches infinity, which is part of calculus. That was fun. You could tell what kind of people we are because we think that kind of things are fun.
00:24:38
Speaker
We do. They are. You don't know. But we're going to tell you. So there's two major branches of calculus. We have differential calculus and integral calculus. So we're going to talk about differential calculus deals with differentiation, otherwise known as derivatives. And then integral calculus deals with integrals, which that one's a little more self-explanatory. But let's talk about what a derivative is.
00:24:59
Speaker
All right, so yeah, we definitely already mentioned that it's a rate of change with respect to another variable, but you brought up during the break a good example of a ladder.
00:25:12
Speaker
So that's one of the more common, most hated problems in a beginning calculus course. If you think about a ladder leaning against a wall and it starts to slip along the ground so that it's falling, like dragging along the wall as it's slipping on the ground away from the wall, the ladder's shape doesn't change, but the velocity with which it's falling or the rate of change with which it's falling along the wall is a different rate than the rate that it's going away from the wall.
00:25:41
Speaker
So the change at the top of the ladder is actually a different rate than the change at the bottom of the ladder. And those are related by the shape of the triangle that it forms with the wall in the ground.
00:25:50
Speaker
Yeah, so it's relating different rates of change. Another example I think that sort of shows a little bit of this kind of thing is when they built the pyramids, they got stone out of the quarries at a constant rate of
Role of Calculus in Understanding Change
00:26:07
Speaker
change. So they didn't speed up, they didn't slow down. But because of the shape of a pyramid,
00:26:12
Speaker
It would have taken about seven-eighths of the time to build the first half vertically of the pyramid, which must have been a nightmare for the people who are managing stuff for the overly demanding Pharaoh. Because first of all, you'd have to explain to him what seven-eighths, you know, of the timeline would be. Nobody likes fractions.
00:26:32
Speaker
Although it must've been very nice to see the pyramid go up very, very quickly during the last three years or so. True, true. But yeah, so those are derivatives is literally just a rate of change over any measure. And the one that we're probably most familiar with is something like speed or velocity where you have a change in distance over time that, you know, if I walk a mile over the course of half an hour, I'm going two miles per hour. So the rate of change is two miles per hour.
00:27:00
Speaker
Now, one thing you said is over half an hour. Would you like to talk at all about what an instantaneous rate of change is? Were you divided by an infinitesimal amount?
00:27:11
Speaker
So that's essentially what calculus is, right? So my average rate of change, I walked an hour in, wait, I walked a mile in half an hour. So my average rate of change is two miles for every hour. But occasionally I'm going to have to cross the street, which means that I'm going to be jogging to avoid whatever, you know, driver is not paying attention to me. And there's going to be times where I lollygag because hey, look, there's a butterfly.
00:27:32
Speaker
So the average rate of change is just over the course of the entire trip. Instantaneous rate of change is how fast I'm going at any given moment. And that's what a derivative can calculate, is that instantaneous rate of change.
00:27:45
Speaker
And if you'd like to know a little bit more about this, I suggest looking up the secant rule. It's basically saying you get a stopwatch, you measure somebody's path, you click it faster and faster until you're clicking it infinitely fast, and then you get their instantaneous rate of change. But it's a little bit more formal than that, but we don't have time to talk about the secant rule.
00:28:11
Speaker
Well, what's kind of interesting is a secant line is where you take any given function. So if you can just imagine some random curvy line, if you take two random points on that line and connect a line through them, that's called the secant line. And it's an estimate of the slope or the line that that function forms. And if you get those points closer and closer and closer and closer together, eventually you'll get to what's known as the tangent line, which I think more of us are probably familiar with from geometry.
00:28:37
Speaker
Oh yeah. So the, if you put a book on a basketball, the, um, the, the lines that go on the surface of the book that touched the surface of the basketball are tangent to that basketball. I think that's actually the technical definition of a tangent liner. That was the original definition. Is it a line that intersects a circle exactly once? Oh yeah. And, um, and I think they appropriated it for more complex curves in the what 1700s and stuff.
00:29:07
Speaker
That's, yeah, that sounds about right. I mean, it would have been Newton, right? Oh yeah, Newton and et al.
00:29:15
Speaker
So okay, so that's a derivative rate of change over time that one I think we can we can wrap our heads around because most of us have seen Slope in middle school in high school. We know rate of change, you know distance over time the opposite of that both in understanding it and actually Calculating it is much more complex It's a fairly simple concept that an integral is essentially the reverse operation of a derivative and
00:29:39
Speaker
Instead of, you know, we're given velocity and then we've, you know, velocity over time would be acceleration. It's your change in velocity over time would be acceleration. If we go the other way, velocity, the reverse of that velocity over time or velocity is distance over time. So the integral of velocity would be the distance. Yeah. So your odometer is the integral of your speedometer. Um, or at least it measures the integral over time. Dude, it never even occurred to me that my car could do calculus.
00:30:08
Speaker
Another example of the integral is if you take the surface of the sea, subtract the sea floor from it, so you have this basically the height at every point, and you take that integral with respect to the map, then you get the volume of the sea. So the integral of the surface of the ocean? Minus the sea floor. That makes sense.
00:30:33
Speaker
So anywhere you have an area, anywhere you have two things multiplying by each other. So if you have miles per hour times hours, you get miles. You get miles per second per second times seconds. You get miles per second. Don't know why I chose those units. It's very quick.
00:30:52
Speaker
Yeah, so the, I mean, definitely touched upon it is that the integral is, yeah, like the opposite of the derivative. And if you take the integral of the derivative, you get the original thing back.
00:31:04
Speaker
Well, in the most basic and you actually inspired this analogy. So I think we just about everybody knows velocity equals distance over time, right? So that in my mind is a derivative. So distance divided by time. If we multiply by time, then distance is velocity times time. That's essentially integration. That's the reverse operation, but it's still the same formula that we're talking about. Just rewritten.
00:31:28
Speaker
Yeah, so if you if you understand that division is the opposite of multiplication, then you will understand the fundamental theorem of calculus. There's a little bit more involved for formally, for example, derivatives are one of those things that have.
00:31:45
Speaker
closed form solutions basically all the time. Integrals can be very difficult or even impossible to solve using existing functions, which is very interesting. We don't have time to go into that, but it's a fascinating topic. And oftentimes frustrating. All the time frustrating. Besides pulling the ladder out from your friend and watching them fall with some applications of calculus.
00:32:12
Speaker
Well, let's see, if we're talking mathematical applications or maybe physics applications, I mean, calculus is kind of the foundation of science. I mean, it's really, and all branches of mathematics use it. It's almost like the higher level algebra, that algebra is a foundational mathematics for every other thing you're ever going to do. Just basic algebra is included in that, which I'm sure all of you out there are shuddering to hear that.
00:32:39
Speaker
But calculus is kind of the, it's like the algebra of higher level math. It's the foundational math that you need to know to be able to apply and analyze other branches.
Practical Calculus Applications
00:32:48
Speaker
In fact in elementary school you probably learned a little calculus without knowing it. Newton's first, second, and third law, well first and second especially, an object in motion will stay in motion, or an object at rest will stay at rest, is saying that the derivative is zero. That's all that's saying. Originally he stated it using calculus, not English.
00:33:08
Speaker
That's interesting that no one ever told me that in English. I know the laws in, or no one, no one told me the laws in math. You would have thought that me a mathematician, they would have told me, but that's the first time I've heard that. That's interesting. And, um, using, uh, what's called, um, variational calculus, which we definitely don't have time to get into, you could solve problems like what's called the brokistochrone problem. Basically it's saying, imagine you have a flexible piece of a cardboard and you're rolling a ball down it.
00:33:38
Speaker
What shape does it have to be so you can put it anywhere on the cardboard and it'll still take the same amount of time to reach the bottom? You could solve that using calculus. So you could do magic with calculus, basically. Yeah. So remind me, what are the three laws? Because I've remembered, I think I remember them all.
00:33:54
Speaker
An object in motion will stay in motion. An object at rest will stay at rest. So those two are the derivative closed system is zero. And the third law is about interacting bodies. Any action has an equal and opposite reaction, which is saying that the sum of all actions is zero.
00:34:15
Speaker
So I definitely remember dealing with vectors when dealing with that law in physics in high school. So the third law. Okay, so let's talk about what a vector is. Before we transition into vector calculus, what is a vector?
00:34:26
Speaker
One thing that you could definitely do is understand a vector. Just by the nature of being a human or a dog, dogs are one of the only animals that understand symbolic vectors, pointing. Well, pointing is part of the vector. You have a direction, but you also need magnitude. So let's say there's a point in space and it's windy at that point.
00:34:54
Speaker
the direction that the wind is going and the speed that the wind is going represents a vector. That's my example. That reminds me there's a character in the kids movie Despicable Me whose name is Vector and he's very excited because he I don't remember if he has it or he talks about having direction and magnitude and he's very excited about it.
00:35:15
Speaker
That's a hidden joke, apparently. I love despicable me. Um, so a vector. So let's see an example of a vector. Like I said, most of my examples are in pure mathematics, which aren't super helpful. I know for sure that vectors are used a lot in air traffic, uh, that the, the travel of planes and their, their trajectory is, is mapped by vectors.
00:35:39
Speaker
if you throw a ball, the motion of that ball at any point in space is represented by a vector. So, I mean, if you have rate of change and that's happening in free space, that's a vector. So when we talked about derivatives, we talked about speedometer, but a speedometer is very, it's very flat. It just goes, it just either 60 or 30 or 45 or whatever. With a vector, you have
00:36:06
Speaker
unless it's a one-dimensional vector, which it can be, but you have multiple degrees of freedom. So if you're in a plane, if you're taking off, you're going vertically, very slowly, but forward very quickly. So that's an example of a vector. And to add vectors together, you just use the Pythagorean theorem. They're actually quite simple.
00:36:33
Speaker
Although sometimes that math can get really ugly really fast. It's interesting now that I'm thinking about it back to when I took vector analysis in college and any classes pertaining to linear algebra that I always kind of thought of vectors as being fixed and then when I took vector analysis and that was no longer the case when we were dealing with things like vector fields where there was a rate of change of not only direction but magnitude of acceleration basically or velocity. The wind was changing speed.
00:37:02
Speaker
That that blew my mind that that was really hard for the math for that was really hard you look at any I mean the fluids which we're talking about today I mean the motion of people in Grand Central Station the Antivore antivore disease when they accidentally find their own pheromone trails all this stuff can be modeled by vector fields That sounds like a lot of work
00:37:27
Speaker
Oh, absolutely. And that's another good point about vectors, is that they're a building block. They're not a BL and end-all. And so you could do things with vectors and calculus. Do you want to maybe get into gradients? Well, basically, what a gradient is, is it points towards the steepest ascent or descent, depending on if you take the negative or the positive of the gradient.
00:37:54
Speaker
So if you're a mountain climber, you're interested in the gradient of the mountain because that's the way to get to the top of the quickest and I guess the most perilously in some cases. Yeah, and actually you could see a gradient if you look at those altitude maps that have all the... You've probably seen them. They have a bunch of squiggly lines that represent different altitudes.
00:38:18
Speaker
If you connect the lines that are closest to each other, those are the vectors that point in the direction of the gradient. So what is, is the gradient kind of a, that's like the path of least resistance kind of thing? I think that's the path of least resistance when it's applied to a system of actions. But it's definitely the path of steepest ascent and descent.
00:38:46
Speaker
So most efficient then? Oh yeah, definitely in some contexts. And the reason why it's important is because, like we touched upon, derivatives are used for optimization. So when a derivative is zero, let's say you're on a roller coaster and you're measuring your vertical speed.
00:39:07
Speaker
When your vertical speed is zero, you're either at the top of a hill or bottom of a hill. And you can extend that to any number of things. So you can always optimize with derivatives. Not always easily, but you can always do it.
00:39:19
Speaker
Well, one of the best analogies I have for the derivative being zero is when, so if you think about throwing a ball in the air, it's kind of a parabola shape. It's got a curved shape that there is a moment in space where the ball stops and then starts to fall again. So it's traveling up and then for a moment, instantaneous moment, it stops and then begins traveling down. In that moment, the derivative of its motion is zero. It is no longer
00:39:43
Speaker
moving in that exact moment. That's where it goes from going up to going down. It changes direction, and in order to do that, to go from a positive direction to a negative direction, it has to go through zero.
00:39:55
Speaker
And the one dimensional gradient is just called the derivative. Gradient is just extending it to multiple dimensions. So if you, if instead of throwing a ball, like in the previous example, you're making your bed and you throw the bed sheet on the point where the sheet is not moving is going to be, if you take the gradient of the motion with respect to, with a height with respect to the motion, it's going to be zero.
00:40:25
Speaker
Oh. See, that makes a lot more sense than what they tried to teach me in vector analysis. Where were you 10 years ago, Jonathan? You were probably still in middle school. I actually did not go to middle school. Fun fact. How did you manage that? Oh, I read Goosebumps. It made me scared of middle school, so I just skipped right to high school. I don't know that that's much better. But moving on to the curl.
00:40:53
Speaker
Move on to curl. What is curl, Jonathan? How swirly things are.
00:40:58
Speaker
So you stir a cup of coffee, the curl is the vector that points towards the swirls. So it'll be a vector that's coming out or into your cup of coffee. And that's, so yeah, an eddy is an example of a curl or a vector or where a curl vector would show up. It's just a way of measuring swirliness, which has obvious applications in the mathematics of fluid dynamics.
00:41:28
Speaker
Well, so what are the, I mean, I guess what are those applications or why do we need to know the measure of that vector? It has to do with, because we're going to talk a little bit about Eulerian and Lagrangian fluids in just a little bit. But basically, you have to deal with these parcels of water that move around. And if it's going towards the wall, for example, it's going to flatten out.
00:41:55
Speaker
And just like in the latter example, you have non-uniform motion. And anytime you have non-uniform motion, you have curl.
Vector Calculus: Curl in Fluid Dynamics
00:42:03
Speaker
It's also the same principle why electricity traveling through a wire deflects a compass because you can measure the wire in this case would be the curl of the magnetic field, I think, but that's a little bit of a contrived example.
00:42:24
Speaker
The example that's coming into my head of why this is important, like why we care about this is something along the lines of dealing with getting fluid from one place to another. So something like an oil pipeline or something like even just plumbing in a residential area, getting water to a residential street, that that requires a fair amount of mathematics to be able to do that in a way that's safe.
00:42:45
Speaker
Yeah, the Romans were an example of, they thought that doubling the size of a pipe, for example, of an aqueduct would double the amount of water that went through, and somehow they managed to build an empire. But if they knew the curl, they could have done things a lot more efficiently with their... And anytime you want to change direction, you're going to have to deal with the curl.
00:43:05
Speaker
And when you're dealing with turbulence, which is what happens when you, I mean, you've seen, you see turbulence all the time. You look at clouds. Those are an example of turbulence. You have to deal with the curl. Well, and I know for sure that, or I know that once you get to a certain point in engineering that, or when you're going fast enough that you no longer treat air as like the way that you, you calculate how you move an air becomes a fluid calculation.
00:43:32
Speaker
Essentially, you're moving fast enough that air becomes similar in the way that it interacts with your object as it does in fluid. All of these examples have to do with just the study of torque, which at this point in time, when the curl and things were being developed, it replaced a lot of simple machines, basically.
00:43:59
Speaker
So these are reasons to care about the... These are the reasons why we need to know about the curl. If you're measuring something as swirly as liquid or gas, you're going to need to know how to measure that swirliness. That sounds funny. I hope that that's something that is said in graduate level mathematics courses. Let us measure the swirliness. Swirliness 235.
00:44:28
Speaker
Is there a scale of swirliness? Is it from 1 to 10? Well, I think that if the radius approaches 0, it gets infinitely swirly. That's a technical term. Infinitely swirly.
00:44:44
Speaker
We only have a couple more things about vector calculus and bear with us and we're gonna just go through them very quickly. We have divergence and that's important because if you've ever opened a door and then the door slams closed you notice that there's a difference in pressure outside of the room and inside of the room.
00:45:09
Speaker
So you can model that as what's called a source or a sink, which it was literally named after the sink that you wash your hands in by Maxwell in his treatise. So a sink is something that sucks out fluid and a source is something that inputs fluid. And you measured how much that is inputting or outputting using what's called the divergence.
00:45:38
Speaker
I do remember learning the divergence theorem. I don't remember what it is, but I do remember learning it in both vector analysis and the physics part two.
00:45:49
Speaker
Here's a way of explaining the divergence theorem. Let's say you're in a closed room with a bomb. The bomb is going to be a source of air. It's going to explode and it's going to add a lot of air to the room, so it's going to be a source. If you measure the pressure in the entire room, it's going to all increase because of the bomb exploding.
00:46:13
Speaker
So the sum of all sources and sinks in a region, like a volume, gives the net flux out of a region. So if you open a window, air is going to go out that window when you explode a bomb, and that's the divergence theorem. I mean, I don't think I'm going to be real worried about the divergence theorem if I'm in a room with an exploding bomb, but you know, for later, when my ears stop ringing.
00:46:37
Speaker
And finally, we got the Laplacian, which is kind of like the, it's kind of the opposite of the gradient is actually the divergence of the gradient. Don't think about that too hard. If you take the Laplacian of a hilly area, it's going to be positive on the top of the hills and negative on the valleys. And that's what that is. And the reason why we're illustrating these concepts so quickly is just to show you that vector calculus
00:47:03
Speaker
is accessible. It's not magic. It's not something that's unintuitive. It's something that you have the hardware to understand already. And all you need to know is the silly little symbols that go along with it. And I do really wish that we had had this conversation 10 years ago when I took vector analysis, that I feel like I would have understood it a lot better had I been able to talk about it in these words first.
00:47:28
Speaker
YouTube has been a godsend for just my understanding of especially the Laplacian. I don't feel like there was a lot of Laplace on YouTube when I was back in my day, unfortunately for me. After a few centuries of mathematical inquiry and development of calculus and in particular vector calculus, theories of fluids abounded.
Origins of Navier-Stokes Equations
00:47:49
Speaker
Whether it was Euler's description of fluids as a field of vectors, or Lagrange's description of fluids as a set of deformable packets, significant advances in fluid mechanics were made in the 17th and 18th centuries. With the solid foundations of the principles of friction, viscosity,
00:48:06
Speaker
and other properties of fluids, the Navier-Stokes equations, which are used every day for things such as studying ocean currents, weather modeling, and hydrofoil and aerofoil design, began to take shape, and it all started with one remarkable man, Jean-Claude Navier.
00:48:23
Speaker
So it strikes me as interesting. We were researching Navier for this episode and I didn't really know anything about him. And we went to look him up and of course I went to Wikipedia because Wikipedia is the best. And there were, there were what, two paragraphs about him? Something like that. It was, it was basically a stub. It was like, Hey, this guy was born and then he died and he worked with Stokes. And his name's on the Eiffel Tower along with 71 others. That's pretty much. And he was born in Dijon. Yes. Like the mustard.
00:48:53
Speaker
But basically nothing else. And so we looked up, we found just the trick of Jonathan's that I didn't actually know. That if you look up an article in the native language, so Navier is obviously French. So we looked up the French article and then it translated it for us. There was way more to go off of there. Quick, good study tip. If you're studying anything from any other country, especially studying German philosophy, check out the German page, etc.
00:49:19
Speaker
But we are studying the French page and he didn't just study fluid mechanics. He studied what he likes elasticity. He was famous for building things, right?
00:49:31
Speaker
Yeah, he was, well, that's why his name is inscribed on the Eiffel Tower. He's supposed to be one of the kind of the founding, I guess, developers of bridges. Basically, he made bridge building in a very big sense possible. So he understood that there is elasticity and deformation, what's called soft body dynamics versus hard body dynamics existed.
00:49:55
Speaker
So this guy was actually incredibly prolific in his life's work. So on the French page, he is referred to as an engineer, a mathematician, and an economist. So he worked with some really just well-known mathematicians. It's impressive to me that Navier is not one of the mathematicians that is as well known. He worked with, who was his advisor? Fourier, who you'll appreciate if you're listening to this at all, because MP3 compression relies on Fourier's theories.
00:50:25
Speaker
He worked with Cauchy. He worked with, he corrected something of Galileo's. He worked with, or he did, he worked on the work of Laplace. Like he was prolific. He worked on everything and made meaningful contributions to them. Yeah. Amy described them as a 18th century Erdos. Who we talked about on one of our previous episodes.
00:50:46
Speaker
Yeah, so basically what he did was he took oilers and Lagrange's fluids, which only really work in imaginary land, or when you're talking about liquid helium, these mysterious fluids that just flow forever, that have bizarre properties that have no meniscus, meniscus being the product of surface tension. When you fill up a glass of water, it curves up at the edge, that's the meniscus.
00:51:13
Speaker
None of that was modeled before Jean-Claude Navier, at least not very rigorously. And not in a way that was in any way helpful or meaningful. And yeah, Navier's equations are, they were, they're called the Navier-Stokes equations because of Sir George Gabriel Stokes, who we'll talk about in just a second. But Navier was the idea man behind that, although Stokes in his own right was a brilliant human being.
00:51:41
Speaker
Agreed. So Stokes improved on Navier's analysis by inserting a different explanation for the internal friction of fluids, making the derivation of the equation more credible. Yeah, he took what was basically intuition and found the core of it. I think it's a lot like Galileo with he observed that the period of things of the same length
00:52:07
Speaker
is equal, so if you swing one of those incense burners back and forth, it's going to have the same period. He didn't do that with calculus. Newton and Leibniz and et al did that with calculus. So that's what Stokes basically was. And Stokes did a lot of stuff with optics too. He was a polymath of the highest degree too.
00:52:29
Speaker
So, one thing that you might wonder is, these equations have, like, he, um, Navier did a presentation on them in 1822, uh, 14 years before his death. Why has it taken centuries to understand and not even completely understand these equations?
00:52:48
Speaker
Well, as I recall, they're pretty awkward. There's a lot of variables in them. There's a lot of coefficients in them. There's just a lot going on in the equations. It's like a grocery list. You know all the individual things that go into it. You have to buy two pounds of flour. You have to buy pasta. You have all these things. But just giving somebody a grocery list does not tell you, for example, the type of diet that they're on. Or it doesn't tell you that they dislike
00:53:16
Speaker
that they dislike chocolate more than caramel, but chocolate was on sale. There's a lot of nuance that comes from these equations because they are, as we touched on earlier, nonlinear. And why is nonlinearity difficult to solve?
00:53:32
Speaker
Well, I mean, we're talking about so many different, I mean, it's almost chaos, right? If we're talking about fluid dynamics, that there it's, it's borderline impossible to model it that specifically that exactly, I think is probably the right word. Nonlinear definitely makes it much more complex.
00:53:48
Speaker
Oh yeah, if you look at turbulence, which as we said happens when you have clouds or when you're swimming in water and you kind of feel your, if you switch your hand through water real quick and you feel your hands vibrate, that's turbulence. And turbulence has been shown to be extremely dependent on initial conditions and what's called Reynolds numbers.
00:54:09
Speaker
Reynolds' numbers describe how fluids act in certain situations. For example, bordices created by wind passing a pole. So you have, I mean clouds are part of the fluid dynamics equations. Pouring a glass of water and seeing it stay in the cup is part of the fluid dynamics equations. All of this is part of the fluid dynamics equations.
00:54:31
Speaker
So you said earlier that Feynman said that turbulence is the most important unsolved problem of classical physics. Why is it so important, do you think? I think it's so important because, I mean, you run into turbulence practically when you're designing, like we said, aerofoils, things like that. It's unpredictable. And any unpredictable system that shows up as frequently as turbulence does, I mean, as my day job, I am
00:55:00
Speaker
I developed algorithms that rely on simplified models of turbulence to get solar panels to stay on roofs. And if we knew how to do that more efficiently, we could do that cheaper. I mean, that's just an economic reason, but there's a philosophical reason too, I guess, which to me is more interesting.
00:55:20
Speaker
the more the millennial or millennium, sorry, the millennium problem approach of, you know, solving something that's difficult for the sake of it for the pursuit of mathematics.
00:55:31
Speaker
And that's one thing that makes it especially difficult to solve is that it's not very well defined. The whole problem boils down to figure out what the Navier-Stokes equations do. Simplify them so that computers or whoever can understand them better. It's very ill-defined and that's one of the major challenges.
00:55:55
Speaker
So one thing I think we should talk about is not just what the solution to the Navier-Stokes equations do for us, but what it might look like.
Solving Navier-Stokes: Challenges and Opportunities
00:56:09
Speaker
Because obviously we already have the equations, so what would a solution look like? I don't even know if we could find a solution unless we can define the equation better.
00:56:20
Speaker
Yeah, I think that it would have less variables because, I mean, the Navier-Stokes equations, you have viscosity. I mean, if you've ever played with cornstarch in water, that's a highly non-Newtonian fluid. We didn't even talk about Newtonian fluids. But you press down on cornstarch in water and it turns solid. You stop pressing and it turns into a liquid. You can model that.
00:56:43
Speaker
using the Navier-Stokes equations. So they're extremely complicated and it's not even clear that a solution exists that's more simple. So it's like a lot of the light millennium problems like the P equals NP problem.
00:57:00
Speaker
whether or not computers can solve difficult problems more quickly than they currently do. All these problems are research problems that would take any PhD, possibly life, I mean they have taken PhDs lifetimes to solve.
00:57:18
Speaker
To not solve or not solve, right? Well, I mean, maybe the next step in the Novia Stokes is to kind of define parameters, not necessarily to even clearly define it in the most general sense, but to be able to define it better given certain conditions or certain situations.
00:57:35
Speaker
It would also, if we had, and the heart of any theory is predictive ability really. The theory, for example, the conservation of energy means that you can see how fast a ball will be, how fast a ball needs to roll down a hill.
00:57:52
Speaker
to go around a loop-de-loop using some very simple things, which would, it's a predictive thing. And as we've seen recently with the devastation caused by the hurricanes in Texas, Puerto Rico, and other places, if we have a better way of predicting when these things might happen, we could literally save property and life.
00:58:20
Speaker
I always find it interesting when we can connect things back to what we've previously spoken about, and weather research is something that comes up. I mean, the discussion is always chaos theory, that our entire world is a chaotic system. And it's interesting, or I find it exciting, that maybe the Navier-Stokes equations can be a bridge across that chaos, as it were.
00:58:44
Speaker
Fluid mechanics is an interesting field that can claim discoveries being made every year, but a set of 300-year-old equations still continues to baffle the world's greatest mathematicians. But every year the world is getting more educated and less poor. Sooner or later some mathematician will be struck with inspiration such that he or she will have solved the Navier-Stokes equations satisfactorily.
00:59:07
Speaker
and we'll be stuck with a new super hard problem. Or at least one that isn't currently defined as one of the six that will get you a million dollars each. I'm Jonathan Baca. And I'm Amy Lynn.
00:59:19
Speaker
And this has been Breaking Math. We'd like to give a quick thanks to brilliant.org for sponsoring us. We genuinely love their content. All of us here are users of the sites. Anything you'd like to say about brilliant.org? It's just a super fun app and website. Check it out.
00:59:42
Speaker
And if you'd like to directly support the show, you can do so by going to patreon.com slash Breaking Math Podcast. Just the price of a cup of coffee, which you stir using the Navier-Stokes equations, will get you advanced access to episodes when we have them.
00:59:59
Speaker
And if everybody donated just a dollar, it would be enough to keep the show not only going indefinitely, but it would become a primary job for us. And that would be amazing. So thank you for listening. And until next time, row, row, row your boat gently down the stream, merrily, merrily, merrily. Navi Stokes aren't what they seem.