How Viscosity Changes Everything

1:14:50they start accounting for viscosity. And so to now kind of quickly recap again, we we are trying to build this understanding of what is the construction of the Navier Stokes equation. We understand DM Bear started with this idea that the divergence equals zero. >> Uh Oiler has come in and used the framework of Newton's second law of motion to try to apply it to fluids. Maybe cheating off of Denar's test >> maybe a little bit >> maybe >> um and added for the force side of the equation uh pressure >> uh >> and just an external force and an external >> gravity or stirring or whatever >> or whatever it might be. And he was

1:15:30like, I did it. Look at me. I'm so brilliant. >> Yeah. >> However, >> when applied in practice to a ball like we just described, it does not actually describe real world systems. And what's interesting is the it's funny because the original prize competition that Oiler was a judge on was about resistance and drag and then he came up with the solution without thinking >> without thinking about resistance and drag. Am I like missing something? >> Exactly. So, so what he ended up missing was another force that he hadn't accounted for. Remember, right now we have two forces. We've got the difference in pressure. Yes. >> And we've got just external forces, whatever they might be, stirring, gravity, whatever. There is a third

1:16:12force on that side of the equation that is missing. And that has to do with viscosity. >> Okay, >> that's the resolution to Dalen Bear's paradox. So, in Oiler's ideal fluid, there is no viscosity. This is called an invisid flow. Um the fluids can slide past one another. These particles can just slide past one another and they can slide along the surface of an object without any friction >> and the flow divides very smoothly in front of the object and then it rejoins at the end of the object. There's no wake, there's no net resistance, there's no draft. Beautiful mathematics. It's different than uh uh diver like it's different than um the problem we were

1:16:53saying where uh water uh like form a incompressibility, >> right? This is different. This is still incompressible, right? Because the flow is coming in and then it's going out. >> It's just that they when they interact with each other, it just rolls off. >> Yeah. They just they just roll off. Okay. Yeah. They're just rolling off one another. It's not especially useful. We haven't taken into account friction and we have to take into account something called viscosity. Viscosity is simply the measure of how thick, sticky, and resistant to flowing a liquid is. >> So, we're going from thick with two C's on the left here to thick with three C's on the right. >> Yes. >> Okay. >> Yeah. Vegetable oil, not that thick,

1:17:33right? You drop something, it immediately goes through. Uh motor oil, also not that thick, but slightly a little bit thicker. Um, honey on the other hand, you drop something and it takes forever >> to get all the way down because there's so much resistance to flow >> in honey. >> The things as they try to move past each other are make it harder. >> Exactly. Yeah. And so for low viscosity like thin liquids like like oils, um, the particles are moving past each other with very little friction. But for high viscosity, there's a lot of resistance to that movement. And there in lies the clue in how we can mathematically describe this because we need we need to have a mathematical language to describe what is viscosity. >> So let's go back to our traffic analogy.

1:18:14>> Okay. >> Okay. Let's consider a multi-laneed highway >> like this. So we've got a shoulder that's 0 miles hour obviously cuz you're on the shoulder you better be stationary. You don't want to be one of those guys that like tries to cut in line and then the the cops catch him. So let's look at let's look at what exchange of traffic from one lane to another will do because in normal traffic obviously cars are going to change lanes. >> So I've colorcoded these cars based on their lane and based on their velocity. >> Sorry. And what I'm realizing earlier you specifically showed us only one lane example. >> Yes. >> Which was fine. >> Yeah. >> At the time.

1:18:54>> At the time. But now we've got multiple lanes and now the flow is going to crisscross >> in between which is how real liquids okay >> how real liquids do it right now what I've done is in in the initial let's say it's a four-lane highway um the the lane that's fastest is going at like 80 mph and the lane that's slowest is going at 20 mph. So it goes 80 then 60 then 40 then 20 and then you've got the shoulder. four lanes and it's in increments of 20 mph and then things start shifting around. Okay, you actually wouldn't notice a change in the flow because let's take for example the lane that's moving at 60 mph. It's

1:19:37going to inherit some faster cars from the 80 mph lane and it's going to inherit some slower cars from the 40 mph lane and the things are going to average out. >> Mhm. >> Right. Um, and so if I were to look at a graph, and that's on the bottom there. If we look at a graph of lane on the x- axis and the speed on the y- axis, it's a line because it goes from 0, 20, 40, 60, 80, and later on it's still going to be a line. >> There's been no change. >> Meaning that the velocity field has not changed, right? It's still big up top, small down below. And the velocity field has not changed from one time to another. Meaning there's no force here. >> Okay? Even though there's some viscosity, there's no force.

1:20:18>> Right. Okay. Yes. >> Right. >> Yes. >> There's still like lane changing happening, but there's no force. There's no change in the flow of traffic. >> Mhm. >> Now, let's consider a different type of profile. A type of profile where the shoulder is again zero and I've got the two slowest lanes both at 20 mph. And then I've got a lane at 30 mph, but the faster lane is still going at 80. Imagine that. like the the carpool lane is still going at 80 and then the rest is like still very very slow and the carpool lane has like a barrier so there's nothing happen and then all of a sudden the barrier gets lifted. What's going to happen? Well, the carpool lane is going to slow down a lot but also the other lanes are going

1:21:00to speed up because stuff from the carpool lane is going to come in and you know sort of push that lane forward. This is a bumper car scenario. Okay, not real but but you get what I'm saying? Yeah, >> like like before we had this curvy profile which we see on the lower left where um again we we don't have a line because it's slow slow slow then really fast. >> Yes. >> And with subsequent time it's going to level out. >> It's going to level out >> because of the diffusion of cars. >> Here there is a force right because I've changed the velocity profile. If I change the velocity profile that's an acceleration which means there's a force

1:21:40lurking in here. Mhm. >> Right. There's a viscous force that is lurking in here. >> And this is what's it's what's interesting is um just going back to our previous example uh because basically the system was evenly distributed in the the initial state >> based on how the variables would end up mixing. >> You get you had no net effect, no force. >> No force. >> In our second example here, yeah, >> uh it was sort of uh a not evenly distributed system. Yeah. And so when you look over time and then there's the mixing. >> Yeah, there's a mixing. There's going to be a change. >> There's some diverg not diver. Let me not use the word divergence. There's some change. >> There's a change over time. >> Over time, >> which means that there is a force. And

1:22:21notice in the first example, there was a straight line >> which means I had a positive derivative, right? Like the slope is positive, but the second derivative is zero. There's no curvature. >> In the second version, there's a curvature, right? Which is why when I inherit the fast stuff and I inherit the slow stuff, the fast stuff is way faster than the slow stuff, which is why I'm speeding up. >> The curvature is the origin of the force. >> That's the key. >> Yes. >> And so that is the form that the viscosity is going to take mathematically. And if we go in the next slide, we'll actually see what the viscosity term looks like. >> It looks like uh dell squar. del square

1:23:02means a second derivative in some sense with respect to space and there's your viscosity the mu the the Greek letter there that's telling you how viscous it is like honey would have a very high value water would have a very low value and that's the force the forces again has to do with the velocity field itself >> this is always key we want to describe everything except for maybe the external force as a property of the velocity field itself >> of whatever the system as I use in a colloquial sense as a whole >> um is a material idea here which is why we have all these are all derivatives because we're looking at the system level >> yes exactly and like it kind of reminds

1:23:43me of like when we were talking about the springs right the force on the spring had to do with the position in this case the force on the spring has to do with the velocity field itself >> right >> yes >> and on the right hand side just for those who are engineers and physicists I want to show you a a similarity between the viscos velocity term and the heat equation where if you have the heat equation and you if you have a temperature profile as time moves forward everything gets smoothed out because of thermal diffusion. The same thing is happening with viscosity. Viscosity is smoothing out the velocity differences right and that's what we saw was happening with the traffic analogy. There was a lot of high speed here and low speed here and viscosity was

1:24:25smoothing it out and making the flow a bit smoother. >> Right? I'm starting to see the connection between the words incompressible and smoothness that I keep seeing as it relates to the whole Navier Stokes piece. >> Yeah, we're we're we're starting to see that. So, the viscous term becomes a smoothing operator in some sense, right? Um and now which feels very natural as a property of trying to measure a liquid because in this context it's the it's it's sort of this contiguous thingy when you look at it from the vector field perspective. >> Yeah. Exactly. And and we don't want that contiguous thingy to have like little pockets of really high fluid or

1:25:05sorry really high velocity because if there's a little pocket of really high velocity well that velocity is going to spread out. >> Right. And that's what viscosity does. >> Right. Right? That's why in honey, it's really hard to have a small pocket of really high-speed honey. On the other hand, for water, it's it's maybe a little bit easier to have a pocket of high-speed water, >> but then it just depends on what time time. Yeah. Anyway. >> Yeah. Yeah. >> And exactly. And so now we can solve Dal Bear's paradox because now we have um viscosity in mind. So we can apply this thing called the no slip condition. This is something that we've um looked at with hypersonics and we've looked at in the FIFA episode where if you've got a

1:25:46object that is moving through a fluid um very close to the object, the the fluid is not going to be moving at all. It's going to be stuck to the object and there's going to be a boundary layer where it's going to transition between being stuck to the object and moving with the rest of the fluid. And that boundary layer causes the drag >> because the object is dragging the fluid with it. >> Right? And this is how we solve Dal Bear's paradox because now if we look at for example a golf ball >> that is moving in air the fluid is sticking to the golf ball and so it's creating a wake behind it because the

1:26:26golf ball is moving through the fluid and as the fluid sticks to it the the the part that's right above the golf ball is going to be moving with that sticky layer but it's also going to be moving kind of with that. And so you get all of these dynamics happening because of viscosity. >> There's this transitionary space between the surface of the object that is moving through the fluid >> and then the rest of space >> and in proximity to the surface of the object there is a transition where the fluid around the object is going to move on some gradient from sticking to the object to being a part of the larger system. >> Exactly. Yeah. Exactly. And so that's how we get drag. And so this is the resolution and this is the contribution

1:27:07that Navier and Stokes were trying to capture. >> Um Claude Louise Navier in 1822 he presented an equation on viscous fluid motion where he added that viscous term but he was an engineer as much as a mathematician and um he tried to explain the internal friction by imagining molecular interactions inside a fluid. This was 1822 so everyone thought he was crazy because there was no such thing as atoms back then. I mean there was this Greek concept of an atom by democrat or something but you know no one took that seriously and it's one of these rare um occasions where a good outcome came from a derivation that's based on a questionable premise um now we know it's

1:27:47not questionable obviously and over the following decades we had Koshi Pson and others developed the continuum theory of stress and how stress affects bodies >> and in 1845 Stokes George Gabriel Stokes derived the viscous equations from a completely different direction from this idea of stress and how you can deform objects. Um, turns out it's the same form as Naviier and so now that's why we're called the Navier Stokes equations. They never collaborated by the way. Stokes was a teenager when Navier died. Um but their names are joined because you know different physical arguments but they arrived at these two equations. And now these are the two equations that have

1:28:28been such a headache for mathematicians and physicists all these years. >> And so as I look at now these two equations the one at the bottom which was where we started >> is this idea that uh the divergence always has to be zero for an incompressible fluid. >> Yeah. You can't make fluid and you can't destroy fluid. You can't have a fluid have emanate from a point in all directions and you can't have a fluid all converge on a point from all directions. >> That's what the bottom one is effectively saying. >> Yep. >> And then now the top one which was the more complex one is we're trying to replicate >> um uh force on the right equals match

1:29:09mass times acceleration on the left. in in this construction which was Newton's second law of motion. We're trying to apply it to incompressible fluids. >> Yes. >> Um and in doing so, we constructed the left side of the equation first where we have our mass represented by our volume. >> Yeah, that's the density. The dens the density, excuse me. Uh and then we then multiply that by everything in the larger brackets which is our acceleration which we have the first term being the uh time derivative of the velocity field. >> Yeah. >> Um >> that's like rain coming in and changing traffic patterns >> patterns. Um but we also have to add

1:29:50into that now the convective term which is how an individual car within that vector field is uh uh changing its velocity over with distance. >> Yes. >> Uh over time. >> Yeah. >> And so that's the left side. >> Yeah. >> And that that gives us mass times acceler acceleration. >> Yes. >> Loosely speaking. Maybe there's a few details not quite right. and then on the our right. So there's there's sort of three terms that have complexity on our mass times acceleration on the left. >> It's really two terms I would say. Okay. So so I I so fair within acceleration it's has two component parts. >> Well no well I'm I'm nitpicking here but

1:30:31like it's really two terms because it's density multiplied by dudt. >> Okay. >> And then it's density multiplied by that second term. You know what I mean? Like density is not its own thing. That's a very No, that that's an important distinction and that that's helpful. >> And then on our right side >> Mhm. >> where we're trying to define force, >> we started with the obvious thing which was pressure. >> Yeah. >> And it's negative cuz high pressure always moves to low pressure. >> Then we moved to uh we had we we added our F which is some general some general force. >> Yeah. Stirring, gravity, whatever you wanted to call it. >> And that's where Oiler >> ended up. >> Yeah. Yeah. and he stopped and he stopped

1:31:11>> and we were missing this concept of drag or resistance >> and ultimately both Navier and Stokes from different directions ended up on being able to define viscosity as the missing piece which is again talking about there's a boundary layer between the object in the system and then the rest of the medium external environment and there's some gradient delta there of change and so we need to account for the viscosity because that applies a force as we look at all of this is that now like an accurate description of what we're looking at >> that is the Navier Stokes equation and we have just built it up from first principles you know >> as as someone who did not do any type of

1:31:52mathematics it's a testament to your way of taking this story from start to finish >> yeah it kind of makes sense though right >> it makes total sense like when you understand the underlying systems I think the hardest part for I think the hardest part really is is getting familiar with the idea of a velocity field >> you know Once you have that and you've internalized, oh, it's like a traffic pattern, then all of the other stuff kind of falls together because you can like kind of, you know, figure it out. >> That was the Oilerian perspective was the velocity field is the way we described it earlier. >> Very nice. >> Dambert, dude. >> A little dam. >> Yeah. And so now we can start asking at this point, we've got the Navier Stokes equations. We know what every term

1:32:32means. >> Yes. So now we can start tackling how to solve them, >> right? And because and part of what we also define through all of this is this is an extremely complex problem. >> Yes. >> For all of the nuances of each of the underlying terms which also have their own sub >> uh issues, sub subcategory issues. >> Exactly. And so we've got an equation of motion. Now can we solve it? like can we can we find a general like I give you a velocity field now you tell me what the velocity field is later right what is the flow going to look like later >> and to just come back to the beginning of the episode the reason we want to do this is because with Newton's uh

1:33:13equations that we were talking about part of it's like then you can use it to make predictions >> exactly yeah >> and that's how we do the space stuff and blah blah blah so what we're trying to do is take this equation of motion >> so that we can like what is the who cares and the answer is so we can make >> so we can make predictions We can give you one input, the velocity field, and then you can give me the output consistently that's true every time. >> Yeah, that'd be dope. >> That'd be great.