Discussion of the Reynold’s Transport Theorem

Alright, in the previous segment we had derived the Reynold’s Transport Theorem. It was a fairly long segment, so I decided to separate it into two and talk about discussing what I'm talking about here in this equation in this particular segment. First thing I want to highlight is, when you look at the other books, you may see this is written like this: DB/Dt. So far, so the same. Del Del t triple integral control volume b rho d volume, control surface b rho and this part may be different; W dot and dA. Okay, I'll talk about each of these terms momentarily. But first, let's look at the differences between these two. So you see this is a velocity. This is a W, and I'm gonna highlight that this W is equal to velocity for stationary control volumes. Okay, so majority of the control volumes that we deal with are gonna be stationary. So those are going to be the same thing. But what this W is, it is the flow velocity. So it's right at this in here. It's a velocity measured with respect to the control surface. So what I mean by with respect to the control surface is maybe it isn't moving. I gave the example of a bus. I mean, a public bus is traveling 50 miles per hour. My control volume is the air inside of the bus. Or well then, I'm moving. Okay, so I have to look at the relative movement of the air within the bus. So I'm gonna exclude that 50 miles per hour in that particular example that I gave. Okay. But unless I specifically inform you, I'm gonna use this purple. Let's call this purple-colored equation. So D Dt. So as I mentioned in the previous segment as well, this is the Lagrangian. This, the right-hand side, is the Eulerian. Okay. And you can see this capital D is a derivative sign in the material reference frame or Lagrangian reference frame. And it's called the material derivative. Okay. It is measured with respect to the moving coordinate axis with the flow. Okay. And B, let's write it over here. B will be the mass for module 5. It will be momentum for module 6, and it will be energy for module 7. Okay. And it is an extensive property. The lowercase b that I have on the right-hand side will be capital B divided by mass. And then what will happen is this is called the corresponding intensive property. Okay. So there's an extensive property and an intensive property. So those are kind of what we discussed. This capital B, this lowercase b, and lowercase b. Okay. This is del del T. Okay. Let me give an example over here. So it's looking at how does this triple integral over over time. And in the previous, two segments ago, I talked something called steady. Things for not a function of time. So then what will happen in that particular case is we can think. This is looking at how this changes over time. The answer is it doesn't. As it doesn't, so this term can be vanished. Rho, rho density. We know it. Rho, density. We know it. dVolume. So this dVolume, if I'm using Cartesian, it's gonna be dx dy dz. So it's a triple integral. That is why I have myself control volume and it is a triple integral. Not one, not two. That's why that is three. If I'm using polar coordinates, it's gonna be dr dtheta and dz. If that's how you abbreviate your coordinate axes. Some people call it x as well. Let's assess where we are at. The left-hand side is clear. The right-hand side, the first term is clear, although I didn’t say it's easy. Then the last term is where it gets a little bit more tricky. Okay. b, we know that. rho V dot n dA. So I'm gonna talk about V dot n right after this, but let me, you know. So that for now, this parentheses is a black box for now. Okay. Let's look at the dA. It really depends how you wanna do it, but it will be like dxdy, it can be dxdz, depends on how you determine your coordinate axes, or it can be dr dtheta. But it's always d two dx dy. Because that, I also have double integral. And this is the control surface. And the definition of the control surface is the boundaries of the control volume. And the surface is, if you think about a two-dimensional look at it, two integrals. Volume is three-dimensional, right? When you calculate mass, rho times volume. So that's why it has triple integral over here. So the only part that is unfamiliar to me now is V dot n. But before that, I want to show something from the previous segment because I'm still using the same notes. Look here. This actually, the last term that I have in the general form of the RTT is two terms: exit minus inlet. Remember this. I'll refer back to this momentarily. Okay, let's go down and discuss them. And I draw this for a nozzle, right? It was a stationary nozzle, it was non-deforming. Except the area was getting smaller, right? And I said that let's pick the control volume like this in this particular case. Now let's discuss V dot n. What will happen? So my goal is to look at V dotted with n. So do you remember the dot product from your calculus? That's the first thing I want to ask. If no, you may wanna revisit it. It's not a biggie. It's easy. It's like this: the magnitude of the velocity vector. Okay. And the magnitude of the n vector. What is the magnitude of n vector? That's called the unit vector. So this is one, so that drops out. Times cosine of the angle between the vectors. That are dot producting. Let's take a look at the velocities and normals. Let's assume that the flow is in this direction, from left to right by convenience. Okay. And look in here. So this will be the velocity at the inlet, and this will be velocity at the exit. Okay. And let's look here. What will be the velocity over here and over here? Well, I can give you two options. The first option is, as there is no slip, the velocity there will be zero. Okay. And the second option is, well, maybe there's a slip. I'm getting into these tricky situations in real life where there may be slip. But then let's say that this is V slip, and I'm gonna call this V is equal to 0. So in one arbitrarily assigned, in this one there will be some slip. In this one I said zero. But you will find out the result will be independent whether there is a slip or not. That’s the final answer for a dot product will be the same. Okay, next is normals. Normals are always perpendicular to the surface and pointing from the out. Okay. And let's write over here what are the cosine of some angles to help us out. Cosine of zero, cosine of Pi over 2, which is 90 degrees, and cosine of Pi. That's right over here because this will come in handy. Cosine of zero, you may know this already, this is one. Cosine of 90 degrees or Pi over 2 will be zero. Cosine of Pi will be minus one. Okay, let's stop at this. What is the angle? Now, I'm looking at the definition of this. It says velocity, the magnitude of velocity, times the cosine of the angle between these two vectors. Looked here, this n is pointing that way, and the velocity is pointing that way. What is the angle between them? It is zero. Okay, so that's write over here. For exit, I got myself a plus V. So let's look at the inlet. What did I get, indeed? Velocity is this way, normal is that way. So what is the angle between them? Let's plot it. Sometimes it gets tricky. This is the velocity, and from the same point, that is the normal. But on this angle that I have here, it's gonna be 180 degrees or Pi, right? And what is the cosine of Pi or 180? It's minus 1. So from here I get myself, if this is an inlet, then I get myself a negative V. Okay, now let's look at the V dot n for these two surfaces, the surface and the surface. Okay, so let's first look at this. V is equal to zero. This is easier. What do you think is gonna get? If this is zero, the resulting dot product will be zero. Okay, so let's write it over here. If no slip, neither inlet nor exit, I got myself 0. Okay, so the dot product V dot n, so let’s write over here, V dotted with n is what I'm investigating over here. Okay, if slip, so let's look at the case when there's a slip. It's like the slip is like this, right? And the normal is that way. Well, what will be the angle between the V and the n? That will be actually 90 degrees, right? So I can simply wrote and put this over here. This is 90 degrees, right? Because V is this way and n is that way. What is cosine of 90? It's zero, right? So then, okay, let's write it in here. If slip, neither inlet nor exit, I get a zero. Now let's look at this case. Whether there's a slip or no slip, I still get a zero. So I don't have to distinguish anymore whether there's a slip or no slip. Okay, so let me rephrase this whole thing. V dotted with n will be: if this is an exit, it will be plus V. If this is the inlet, minus V. Neither inlet nor exit, it would be zero. So let's assess what's happening over here. First question I'm gonna ask is, did I even define the coordinate axis? No. Did I make the x aligned with the positive x? So this whole result is based on me selecting x that way. No, I didn't even say that. What happens if x is this way? Same. I'll get this exact same. So this is independent of the coordinate axis. What happens if I flip this? Let's say that the flow now is going from here to here. What will happen? Well, this will be inlet, that will be negative V. This will be the exit, this will be positive V in that particular case. But this whole thing doesn't change. I said the same: exit is plus V, inlet is negative V. If I don't have any inlet or exit, it is zero. So let's go back to the over here and let's just write this here. So note here. This is plus V, negative V, and zero. So let's call those exit, inlet, otherwise. Okay, so let me go back to the general form of the not integral, but the general form. Do you see something over here? Do you see a plus sign in front of exit, negative sign in front of the inlet? Yep. Negative sign in front of the inlet, positive sign in front of the exit. So that's why these equations are the same. So it's a pretty brilliant way of representing this as plus V or negative V. Look at it. This is b times Rho times either plus or negative or zero V A. b Rho V A. Let me go up. What was B dot? b Rho V A. Again, I'm writing the same thing. This is also important. If there is no exit or an inlet anywhere, I'm not even touching that for the second term because I'm gonna have zero. The whole thing is multiplied by zero. So this completes fluid kinematics module. In the next module, now I'm gonna apply this purple color to Reynolds Transport Theorem and the integral form conservation of mass in the very next module. Okay, so now it will be making much more sense from the real-life perspective. So hang on with me. I am aware that this is on the conceptual side of things. We will address this very soon. Okay.