Lagrangian and Eulerian Descriptions

Okay, so far what we've done: in module one, we introduced some important definitions and concepts that I need for module two and module three. And in module two and three, we focus on the fluid statics case, okay? Now, for mechanical engineers specifically, we are more interested in things that move, okay? So I'm mainly interested in how the river flows, different boilers, different cycles, thermodynamic cycles—there's always a flow, right? If I'm looking at the solar energy, there's a heat transfer fluid, so it's transferring energy from the panels all to the storage, right? So I need to analyze these things. In order to do that, I may need to look at the fluid kinematics. What kinematics means is basically deals with the acceleration and velocities that I will obtain, okay? It will not be focusing on how this flow is generated—that will be the subject of module five, module six, module seven, and so on and so forth, okay? In this one, I want to set the tone for the rest of the modules. This module is brief, okay? So there's not a lot going on; it's more like to ensure everybody is on the same page. And let's write over here—in statics, this is what we did—we sum up the forces, if you remember, and I call this ma, right? Everybody knows this part. I said this a is equal to zero, so this whole thing, whole thing, the summation turned out to be zero. Let's analyze your house, okay? So I have my house over here. Okay, so this is a pretty bad drawing over now. Maybe if I put over here a doorknob, it would be much better. Okay, this is perfect now. Anyway, so what's going on in here is—in your house—so think about what's going on. So this is not a static case. Like right now, in this room that I'm recording this, I have the AC on, okay? There's some cool air coming into the room that I'm in, and also there is an inlet for the air conditioning system. So it pulls up some air into the system, it cools it, then blows it back to me, right? Similar to the refrigeration cycle. But here's the thing: the air particles that travel within my house, it will bring its mass, momentum, and energy into the system, and that's how it gets cooled, right? So I said these three things—those three things are important: mass, momentum, and energy. So I need to analyze these things, okay, for my house as a some system or control volume. We'll talk about these definitions, okay? I don't want to get ahead of myself too much. So the first thing I want to talk about actually is the reference frame. Many of the books, including the textbook that you're assigned to work for my particular class, is focusing this a little bit differently than me, but that's okay. It's just a different take than what I'm gonna do, okay? Even the names are the same, but let's look at the Lagrangian reference frame, and this two is the Eulerian reference frame. So actually, if you look at the official definition of it—not from the fluid mechanics but from physics—here's what it says: fixed on a particle. Again, now I'm going to focus onto the fluid side, and this particle will be a fluid particle in my flow, okay? So that's the definition of this reference frame. The Eulerian reference frame definition is fixed at a certain origin outside the flow. Actually, you know, when you read these or I'm going through this with you, you may realize that—boy, what is going on over here? It looks complicated. You know this; you've known this for a long time now, okay? I'm just revisiting these things. So basically, let’s say I have something like this, say bounded flow, so like a bend. In module eleven, we'll talk about and analyze these things, okay, for viscous flow specifically. Anyways, so there's an inlet over here, there's an exit over there. Right now, there's a flow, so you can see it's flowing over here. So I have a particle over here—this particle is flowing, it's just like this, and leaving my system. Again, I don't want to be giving too much definitions because I didn't cover them yet. So here's what the Lagrangian says: Lagrangian says that, you know what, I'm gonna put my reference frame with the flow. So let's say that I'm using x y as an example, obviously. So this will be stuck to a particular point, and what I'm gonna do is—there's another point over here, right? All the fluids is made of, you know, an infinite number of particles. So it's like looking at this relatively—my origin is moving as well. And if you track this over time, you will see that this is not gonna be a function of distance, but it's only gonna be a function of time, okay? So this part is important. So it's only a function of time because I look at my origin—I'm moving with the flow; I'm not jumping from here to here it my origin, okay? The second definition; convenient and much more what you're used to, okay? So what I'm gonna do is—I'm gonna call this x, I'm gonna call this y. So you note something—my origin is outside of my flow. So now I'm not moving along with my flow. And what is the difference over here is that now if I pick a point, I'm gonna represent this over time. What will happen is this is now a function of time as well as position. Do you see? Because origin is fixed. I'm not tagging, that's the definition that we use, I’m not tagging onto the particle, okay? So that's something that I would like you to know, and I'll give you an example in here. It'll be a pretty brute example, but it'll do it. So let me draw this, and I'll ask you—the hardest part about this is actually drawing it, okay? What am I drawing over here? I hope you see where I'm going with this. Okay, oh yeah—let's put the doorknob. Okay, let's put the wheels. What is this? This is one nice bus, okay? Let's put some windows, right? So what's going on over here is—let's say that I have somebody sitting over there, okay? And I'm gonna call this person B. Let's put a head to it as well. So this person is gonna be B, okay? And what is happening is, there's a person walking, although they are not supposed to. It's moving this way. I'm gonna call this point C, or rather the person C. It's moving, let's say that speed that I have over here is one mile per hour. So, this person is moving this way, one mile per hour, and the bus is going with, let's say, 50 mile per hour, as an example, okay? So the bus is going 50 mile per hour, and this person C is moving 1 mile per hour to the left, and the person B is sitting down. And I have a person outside over here which is waiting in a bus stop, okay? Obviously, if the bus is going this fast, it's gonna miss it, but that's beyond the scope here. So, I have a person over here. So my question to you is this: what's the velocity of this guy? You can say this way: you know what, I am this person; I'm sitting right over in the bus, and it seems to me that this guy's coming to me at minus 1 mile per hour, right? So if I write V C B, based on the B point, it's gonna be 1 minus 1 mile per hour, right? As simple as that. So the bus speed doesn't really matter, okay? So now, in the second definition, let me look at this by V C A. Again, my person that are over here, although he or she shouldn’t, going this way with one mile per hour, fairly slow, and this person is sitting over there. So according to this person, what is the speed of this guy? And that would be 49 mile per hour because to this guy, to this guy, the bus is going 50, and this guy's going that way, so it's gonna be 49 mile per hour. And it's coming towards this person, right? You see it's completely different from that, right? I'm talking about the same velocity of the same, and this is what's happening over here. This is how I typically teach and relate things; it's a bit different than the book, but that's quite alright. And this V C A is this, okay, and the V C A is basically the Eulerian representation. And sometimes we call this, let's call this Eulerian in here. And the other one is minus one, right? So in here, that's the material or Lagrangian, the Lagrangian description, okay? So this is the Eulerian, this is the Lagrangian. And you can see, obviously, this is like the V C B type of, okay, you're looking relative to each other. One thing important here is that the mathematical expressions that we are aware of, such as the derivatives, are all also defined in the Eulerian reference frame, okay? So when I have a Lagrangian reference frame, I will get myself something called material derivatives, okay? And this definition will come in and fairly soon, okay? This is the material derivative versus this is the regular derivative that we are all used to, lowercase d, right? That's the what I'm talking about. And I do ask this question when I teach to the students face to face, "which one do you think is easier?" Except for a person or two, everybody kind of says, "this is the one that I'm more comfortable with. I mean, I wanna put my axis over here and see what happens," okay? And what we're gonna do in this particular module, part of this module, not the segment, but this module, is we are gonna actually introduce a theorem where transports to properties from one domain to another, so it would be fairly handy. But at this point, what I want you to know is, if I want to express the properties of things with this frame, it is actually shorter lengthwise, okay? And if I'm talking about the Eulerian reference frame, things are going to be simpler, but the mathematics will be longer, so it has advantages and disadvantages.