Material derivative (acceleration)

Okay, in this segment, I will talk about the acceleration in a flow. And in order to do that I will demonstrate this will be in the material derivative, okay? So, but first, let's just get the business and talk about the acceleration. And the first thing I'm gonna write will be fairly simple to you because you know this for a long time now, okay? Maybe not this part. I'll talk about it momentarily, okay? But the acceleration is basically the time rate of change of the velocity, right? And again, the velocity is also a function of the displacement vector as well as time. The question is, how do I get this, okay? So, I want to just point out from there. The first thing I need to do is go back one more step. And then, the one more step, what I do is I start by the displacement and I get myself, and if you remember this, the velocity, which is a function of the displacement and the time, will be the time rate of change of the displacement vector, okay? And the displacement vector that I have, so let's start by that. So my goal is to first do this, accomplish V so I can take it over here and plug it over there, okay? So the displacement is, again, you know this, x i plus y j plus z k. Question: what is i, j, k? So i, j, k are, i is the unit vector in the x direction, j is the unit vector in the y direction, k is the unit vector in the z directions, respectively, right? And what's the magnitude of these? Well, I said unit; that means one. The magnitude is one. Are these scalars or vectors? Well, I just said they're vectors, okay? So now you know what they are. This will enable me to express the component of vectors in x, y, and z directions easier for me, okay? Let's get to business and calculate my velocity. So then the velocity will be D t, so it will be a function of the space as well as time. The good thing is, now as this is only a function of time, right, so this will be a plain old derivative, which will be much more easier for me to handle, d dt of the displacement vector, okay? And I have myself over, you can, you. Everybody knows this part. So it's going to be dx dt in the i direction plus dy dt in the j direction plus dz dt in the k directions, okay? By convenience, nothing more, nothing less, I call this u, I call this v, I call this w. So now what happens is, basically let's write over here, these are important things that I really want you to know. u is the velocity vector component in x direction, v is pretty much everything is identical over here in y direction. If I go to w, will be the velocity vector component in the z direction. I will use this a lot, so I would like us to be on the same page. So this is kinda important for me, okay? So now if I just rewrite this, you get yourself velocity vector as a function of the displacement and the time will be u i plus v j plus w k. So I have some accomplishment that I want to share with you. I obtained this. So the second line that I have over here has been accomplished, and the only thing that's left is I take this in and plug it in over here and call it a day and see what happens, okay? Now the acceleration, which is a function of the displacement and the time, will be the rate of change of velocity, which is also a function of the displacement vector and the time. So now I have some issue over here. You may, actually. Let's close this bracket before I forget is. Do you see the issue there? Look at this. Huh, interesting. I also didn't close the parenthesis there, okay. Do you see this is only a function of time, so this was a plain old derivative. So now look at this. This is a function of both the displacement and the time, so now I need to do something called chain rule of differentiation. Do you remember that? I do teach mathematics, so I know this. Here's how it's gonna look. It's gonna look like, the chain rule says this: del V del t plus del V del x times dx dt plus del V vector del y dy dt. Last but not least, this will be del V del z dz dt, okay now? All right now, so I would like you to look at this. I would like you to look at that, and I would like to look at this, and I will go up here. What did I say dx dt is? I said it u. Well, why don't I just plug it in? And I said this for v, and I said that this w, right? So now if I properly rewrite this, what I will get myself is the acceleration is del V del t, so that I was not able to do anything about it, it's going to be u del V del x plus v del V del y plus w del V del z, okay? And this unsteady component you see, you can see why I call it unsteady, right, is a function of time. This is referred as the local acceleration. And the acceleration that I have as a function of the distance or the space is called the convective acceleration, okay? It's just by convenience, this kind of representation is referred as the material derivative. So I will call this capital D V D t, okay? So again, let's discuss that for a minute or two. So basically this D D t is, you can imagine this like I have blah blah over here, that would be del of blah blah del t plus u del blah blah del x, right, let me not make a mistake here, del v, I mean, v times del del y plus, let’s write over here, w times del whatever that is del z. So you can see this is called the material derivative, okay? So now, you know the nomenclature. One thing that I want to highlight is you see, do not confuse this D, which is the material derivative, with the displacement. But the displacement is a vector. You look at the arrow on top of it, okay? So be careful about the differentiation. Some people, some books call this x, but then I am confused with, well, x is direction, right? So there's no easy way out of this. This is the way that I find a little bit more manageable because there's a clear difference between a derivative, material derivative, capital D, versus the displacement, which is this vector itself, okay? You can actually simplify this further if you want to. I personally don't prefer it too much for my undergraduate students because it makes it a little bit harder for them to see. But I do this for my graduate students. So plus the velocity dotted with gradient of this. This is the gradient operator from the calculus, you've seen it. I don't think anybody remembers, but you've seen it. So now, let me look here. How many equations do I see here? You may be asking me, "Excuse me, what are you talking about? I see one equation." Is it? I should put this here. This is the acceleration vector, right? So now maybe I'll give you a benefit of doubt and say that's why you were surprised, okay? So now let me ask this question once again. How many equations do I see over here? Okay, there's no way out now. So instantly, the answer is three. I've seen three equations. Why? It's a vector. Vectors have components. So I can write this: my acceleration the x direction, if I'm using Cartesian, y and z. If I'm using polar coordinates, I'm going to go ahead and write this in r, theta, and z. I will go ahead and write this ax, ay, az because, as you will find out, this will come in real handy in module number 9. We will use this to obtain Euler's equation, and I will mention actually I'll put a link to this video so you will be able to follow what I just said. Let's write the components of these accelerations. Let's start, a x will be del u del t plus u del u del x plus v del u del y plus w del u del z. a y, on the other hand, will be del v del t plus u del v del x plus v del v del y plus w del v del z. Last but not least is the acceleration in the z-direction, that’s gonna be del w del t plus u del w del x plus v del w del y plus w del w del z. Okay, so typically, are these zero? Well, obviously it depends on your case. If this is fluid statics, absolutely; that's why I didn't discuss that in modules 2 and 3. But now I'm going to do dynamic, fluid dynamics, right? So things are moving; there are accelerations, so I need to come in and I need to write these down, okay? Also, there's a different version of this for the cylindrical polar coordinates as well as the spherical coordinates. That's beyond the scope for an undergraduate fluid mechanics class, but you can access this from any book, undergraduate book, they may have it in the appendix of it, or if you have a graduate book, it will definitely be there because we use it, okay?