 What we're going to do now is we're going to take a look at a couple of definitions that are used for wall-bounded shear flows or even free shear flows, but typically we'll start with wall-bounded shear flows, turbulent or laminar boundary layer. The first one is the displacement thickness, and we saw the boundary layer thickness, but displacement thickness is, seems simple as boundary layer thickness, delta, but it has a star and that denotes the displacement thickness. We'll take a look at what that is defined as being. So the displacement thickness is defined as being the distance by which a solid boundary layer would need to be increased or displaced in order to provide the same mass deficit as the boundary layer is providing, and so let's take a look at what that is visually. So what we can see is that this is the mass deficit in this region here where we have lower velocity due to the presence of the boundary layer, and what we would need to do, we would need to increase the body thickness by this amount here, delta star, in order to match that mass deficit. So it's basically a thickening of a body due to the fact that we have a boundary layer present, and then outside of that we would just have the free stream velocity as noted here, and mathematically the displacement thickness can be expressed in the following manner. So that is mathematically how we can express it. Now if we're dealing with incompressible flow, what we're able to do is we're able to take the density out from the integral and we end up with the following for the displacement thickness. So that is the displacement thickness. It's a parameter that you'll sometimes come across when people are talking about boundary layers. Another one that is quite common for free shear flows or wall bounded shear flows like we're looking at here for the boundary layer, is that of momentum thickness. Momentum thickness is a quantification, essentially of the amount of viscous activity that is taking place within the shear flow, and it is a length scale, and this came out of the von Karman analysis, we already saw it, and momentum thickness theta, what it refers to is the following. So drawing out a picture again for the momentum thickness. So that in terms of a diagram is what the momentum thickness represents, it would be the amount that the boundary layer would need to move away from the wall in order to have the same amount of deficit as what occurs within the region of velocity deficit due to the boundary layer. And we can write this out in terms of an equation, very similar to displacement thickness in concept. So we have that, and then we can write out an equation for it, and that is momentum thickness. So for both displacement thickness and momentum thickness, you would need to know what the velocity profile was within your particular flow, either going in and doing a velocity survey or having some estimate that you're going to use. But this doesn't only apply to wall bounded flows like we're looking at here. So wall bounded shear flow is something like this, a boundary layer is a wall bounded, and the reason why we call it wall bounded is because there's a wall here. But you could also have a free shear flow. So an example would be where you have one velocity here, and then the velocity gets larger over here, and there's no wall, it's just out in the open. So where might you find that? If you have a splitter plate, for example, and you have a higher velocity here, and then a lower velocity down here, and the plate ends, the profile immediately downstream of the splitter plate would be an example of a free shear layer. And so you're going to get a profile like that. And the viscous diffusion will occur, and then eventually you'll get instabilities developing in this shear layer that will roll up and do all kinds of interesting things, and then eventually it becomes turbulent. But there are planar shear layers, jet flows have the same thing, the wake of a cylinder, when you look at the boundary layer, we'll be talking about that in a later segment. The boundary layer lifts off, and it starts to develop a free shear flow. And it's right in this region here, when you look at that velocity profile, we sometimes use the momentum thickness. So momentum thickness is used quite extensively within fluid mechanics to characterize essentially the thickness of the shear zone where viscous effects have become predominant. And that is displacement thickness and momentum thickness. So what we're going to do in the next lecture, we're going to move into Blasius solution that Blasius was a student of Prandtl's, and he came up with an expression for the velocity profile in a laminar boundary layer. And then what we'll do is we'll compare that to what Theodore von Karman did using his momentum integral techniques. That's where we're going, and we're looking at external viscous flows.