 All right, are you ready to be challenged? What we're now going to do is we are going to derive the differential equation of linear momentum. It might not sound like a lot, but in the next three segments we are going to go through the steps of deriving the Navier-Stokes equations, which are the most important equations in fluid mechanics. So where are we going to start? We're going to start with control volume analysis, and we're going to look at a little differential element and then we're going to work our way through and we'll eventually come up with the Navier-Stokes equations, as well as the Weyler equations that we can look at. But we'll begin by conservation of momentum, and this will be the control volume formulation. So if you recall, control volume formulation, we had surface forces, body forces, it was equated to a time rate of change of momentum within our control volume, and then we had momentum flux across our control surfaces. So that's conservation of momentum. Now what we're going to do, we're going to apply this, just like we did with continuity, to a little differential element. And we'll say the volume of that is dv, which is dx, dy, dz, the three dimensions of our little element. And consequently we can rewrite the conservation of momentum for that differential element. Okay, so that's what we get. What we're now going to do, we're going to take this and we're going to look at the x-momentum flux in the little differential element that we're interested in. So let me begin by writing a cube, drawing out a cube, and hopefully I'll put it in the middle this time. Okay, so there is our little differential element. And what I'm going to do now is I'm going to sketch out momentum flux into the back surface and out of the front surface. So going in the x-direction. So there we have our momentum flux, and I should have put here the size of the differential element. We have dx, dy, sorry, that should be dz, then in the vertical we have dy. So that's the size, dimensions of our differential element. And what we have is momentum flux into the back and out the front, and we use the Taylor series expansion to do that. And in doing that, what we're doing is we're assuming that we are expanding about the midpoint of our little differential element. So what we're now going to do, looking back at the momentum equation, we're going to play around with this term here, looking at the momentum flux into and out of our control volume. There's our control volume. We've written momentum flux in one direction, but we will have to expand that for all three directions. We're not going to do that. Well, we will, but I'll show you the result. So anyways, let's start with the momentum flux in the x-direction, and take a look at what that looks like. And again, just like all other control volume analysis that we've been doing, we have to be careful about the sign of what is flowing in and what is flowing out. So that's flowing out of the right-hand side, and then coming in on the left-hand side, we'll have a negative, just because the way the vectors work out with the dot product. Okay, and that's the area that it is flowing out over. And then we have plus y-direction, and then plus c-direction. So if we go in and we perform this for all three directions, and we add up terms, and we cancel out and do a bunch of different things like that, what we'll find is we get something that looks like the following, and that is then multiplied by dx, dy, dz, which is the volume of our differential element. So with that, what we can do is we can combine this with the first term on the right-hand side of the control volume formulation, and the left-hand side is going to be our forces. And we have our surface forces plus our body forces. And then on the right-hand side, I'm going to pull that volume term up to the front, and we get this. So we have that. Now what we're going to do, we're going to zoom in on this term here, the whole term here in brackets, and we're going to play around with that a little bit more, and we're going to use continuity. And it turns out that you can rewrite that term in the following manner. So what can we do with this? Well, it turns out that this here is continuity, and we know from continuity that is equal to zero. So that term disappears. What we are then left with for conservation of linear momentum is the following. And this last term here, this is the total acceleration of a fluid particle. If you remember the material or substantial derivative, that's kind of what that is there. So we have that resolved on the right-hand side. However, we don't know about our forces, and so we have to look at the surface forces and the body forces. Body forces are relatively easy. It's just mass and using the density and the gravitational constant. Surface forces are a little bit more challenging, embedded within there, is going to be pressure, which is not that big of a deal, but we also have viscous shear, and that is going to be a big deal. So what we need to do is come up with formulations for those using the differential analysis, and that's what we're going to do in the next segment. And once we have that resolved, then we're going to come back to this equation here, and we'll plug everything together and take a look at what we end up with. So let's proceed by figuring out the surface forces in the next segment.