 So you'll recall from the end of the last segment what we're doing is we're deriving the most exciting equation in mathematical physics that is known as the Navier-Stokes equations. And we came up with a term on the right hand side with our controlled volume analysis which can be seen here. And that's actually what we refer to as being the inertia term. Now what we need to do is handle the left hand side of the equation. And so we've got to look at the body forces and the viscous shear forces. So that's what we're going to proceed to do. Then we're going to, in the segments after that, we're going to put it all together and take a look at the wonder of the equations that we have created. So starting with body forces, and this one is pretty straightforward, gravity is what we have acting within our object. So the differential body force is just the mass multiplied by the gravitational vector. So dx dy dz for the volume multiplied by the density that gives us mass times the gravitational constant. So that one is done. That was easy. The next one is not going to be so easy. That's going to take a little bit more time. And that is dealing with the surface forces. And the surface forces and a fluid that we're looking at are going to consist of both viscous shear and pressure. So we're looking at hydrostatic pressure plus viscous stresses. And the viscous shear or stresses, we're going to put that as a shear stress tensor tij. And we'll get into that in a few moments. What we're going to begin with is a stress tensor for the fluid itself. And we're going to write this out as a matrix. And you'll notice on the diagonal, we will have the pressure. So those are the first elements. And I'll draw out a diagram showing what all of these are. Okay, so there are nine of these. And in terms of a schematic of how they apply, let's draw a little chunk of fluid. So a differential element. Now we had our stresses denoted by sigma ij. And what this means is stress in the j direction on a face is normal to the i axis. So let's take a look at what that might mean. Beginning with the face on the right here, we have a stress sigma xx. So that is acting on the x face in the x direction. And other ones acting on the x face, which is normal to the x coordinate, we would have sigma xy. Because that is acting on the x face, but it's in the y direction. And then we have sigma xz. Or z, because it is acting again on the x face, but in the z direction. Now the face on this side, and let me change pens. So over here, we have sigma zz. And then we have sigma zy on the z face in the y direction. And then we have sigma z face in the x direction. And then finally on the top, that would be sigma yy. Because it's on the face y in the y direction. And then we would have sigma on the y face in the x direction. So you're getting the hang of it. And then sigma on the y face, but in the z direction. So that's the convention that we use within fluid mechanics. And what we now need to do is we need to come up with a way to be able to express these. And the way that we're going to do that is, again, we're going to go back and use a Taylor series expansion. So this is where it's going to get a little messy. But let's work our way through that. And we're going to look at all of the stresses or surface forces in the x direction, just to simplify matters. But you would have to do this for all three directions. And so I'll begin again by drawing out a differential element. Now if we're looking at the forces that are acting in the x direction, what we're going to have, we're going to have a force here. And I'll put a normal force on the back going in that direction. And on the front surface here on the z surface, we'll have a force. And then on the back z surface, we'll have a force that way. On the bottom, I'll draw a force that way. And then on the top y surface, we'll have a force that way. So those are all of the forces that could be acting in the z direction on our cube of fluid. Now what we need to do is we need to expand those out. We're going to expand those out using a Taylor series expansion. So let me just draw it out because this is going to be kind of messy. And then we'll summarize things after we've done that. So those are the two components on the y surface. We have one at the bottom and one at the top. Now what we'll do, we'll expand out and express the ones on the z surface. So those are the stresses that are on the z surfaces in the x direction. And finally what we'll conclude with will be the x surfaces in the x direction. Okay. So those are the stresses that we have acting in the x direction. So what we can do, we can sum those up and express the differential force in the x direction. So that's what we get. And now what we're going to do, if you recall a couple of slides ago, we introduced sigma and we had this tensor. We're going to introduce sigma into the expressions that we just have come up with. So that's what we get for the x direction. We can go through the same exercise for the y and the z directions. And when we do that, we get similar expressions and I'll just write those out. Are you having fun yet? Okay. So we have those. What we need to now do is take all three of those components and plug them back into the momentum equation. We have the body force, we have the surface forces, and we have the right hand side of the equation. So we're starting to see this all stitched together. It's a vector equation. So there are three components. So there we go. Is it over yet? No, it's not. And the reason why it's not over yet is because what we need to do, this equation is good. We're getting there. But the question is, how do we relate tau to being a function of velocity? Tau means nothing to us. It's just a shear stress. And what we're after is we're after velocity. So what we need to do is find a way to be able to relate tau to velocity. Before we do that, however, we can do one really large simplification of this equation. And that is, if you assume it to be inviscid flow, we'll come back to this form. But we're going to take a little bit of a sidetrack right now. And we're going to imagine that we have a fluid that has no viscosity. And this is what people did in mathematical physics several hundred years ago to make this equation that we just derived simpler. And they assume that the flow was inviscid. And so what that means is that if it is an inviscid flow, there's no shear. And if there is no shear, looking back at this equation, all of the tau terms that are in here, knock out. And they are removed from the equation. So if you have that, you can say tau ij is equal to zero. And then this is the equation written in vector form. So that looks a lot simpler than the one that we just had with all of the tau's. And this equation is called the Euler equation. And it is a differential form of a governing equation for inviscid flow, flows without viscosity. And so it flows external to airfoils far away from the airfoil outside of the boundary layer. It will work. It's an equation a lot of people have used for a lot of different things. And incidentally, if you do the dot product of this equation along a streamline, you actually end up with the Bernoulli equation. We won't do that in this course, but that's kind of a little academic exercise for those that like math. You can play around with that and see if you come out with the Bernoulli equation. But what we're going to do now, and this will be in the next and final segment of this lecture, is we're going to return to this equation here. And we're going to look first and foremost for this relationship, enabling us to plug in for the tau's. And with that, we will have the Navier-Stokes equations. So let's pause now, and then we'll move into the next segment where we will do that for Newtonian fluid.