 In this lecture segment, what we'll be doing is we'll be looking at the continuum definition of a fluid, and what we'll be doing in all of the analysis in this course is we will be considering fluid as a continuum, and so I'll give you an indication as to what continuum means throughout this lecture segment. So if we were to look at a fluid, and let's consider a gas, or you could think of a liquid as well, but typically what happens is that the molecules are not fixed, but they're free to float around, or in the case of a gas, they move around and they hit into one another. And so consequently, given that they're able to move with respect to one another, we have to consider what we call a continuum when we define properties of the fluid. So if we were to look at a gas, and if we're able to zoom in and look at the molecules themselves, we would refer to this as being a microscopic view of fluid mechanics. And there is a subject referred to as being the kinetic theory of gases, which helps us understand the behavior of a gas. And with kinetic theory of gases, you can get all kinds of properties such as viscosity and different things like that using only theory. So the kinetic theory of gases, which is a microscopic view of a gas, we would treat the molecules as being little balls, and each of these balls would have a characteristic diameter or size, so that might be ball three. And it's going to be moving at a characteristic velocity, so v3 in that case. Here we have another ball and let's say that's moving at v1. Another ball over here, we could say that that is moving at v2, and these balls are molecules of the gas, but it is being represented in a very simplistic model, but with the model you can then obtain quite a few things. So imagine that was our gas, and if we wanted to determine the density, if we were to say this is the area that we're going to use to determine the density, that would be different than if we were to do something like this. So what that shows us is that when we operate at the microscopic view, the size of the volume or sample that we're taking has an impact upon the properties and the density. And consequently that's why we get to what we call the continuum. So let's explore that a little bit more. So if you remember from any of your physics courses, density is mass per unit volume, or we can express that, and we always write density as row and fluid mechanics. We will say that in the limit as delta v, and delta v is going to be the sample volume, approaches what we call delta v star, and I'll define that in a moment. And if we take delta m, that's the mass in this delta v that we're looking at, that would be the definition of density. Now it turns out that this delta v star that I've indicated here, as that approaches a certain value, and that value turns out to be about 10 to the 9 millimeters cubed, as we approach that, that's when we start to be able to have the continuum approximation or assumption, where we assume that the properties then won't change much as we increase in our sample size. So that's kind of a rule of thumb or a value for all liquids and gases at atmospheric pressure. If you're dealing with gases at lower pressures, what will happen is the mean free path, that is the spacing between the collisions of those balls or the molecules we looked at in the kinetic theory, it becomes larger and larger, and consequently the volume that you would need to sample would become larger. So let's take a look at this with a diagram. So let's consider a density field, and so here we have some substance, and what I'm going to do is I'm going to draw a number of lines here, and these lines represent constant density lines. And we're talking about this delta V that we need in order to make our sample. So what I'm going to do, I'm going to draw in a little box, and that little box is our delta V, and that would be the sample that we're getting our mass from. Now if we were to plot density as a function of delta V, and so I'll do that over here on the side, so we have rho kilograms per meter cubed, and then delta V down here. If we were to plot that, what we would find is that when delta V is really, really small below our 10 to the minus 9 millimeters cubed, the density values are going to be fluctuating all over. But then what we'll find is as the density, or as the delta V gets larger and larger, you get to a point where you get to what would be a convergence, and then you have convergence in the density, and then eventually as delta V gets larger and larger, let's say it's a kilometer, two kilometers, three kilometers, four, we start getting macroscopic effects. And so what we want to do when we're dealing with the continuum approximation, we want to say that we're sampling in delta Vs that are larger than our 10 to the minus 9 millimeters cubed, and we also want to operate below where we start getting into the macroscopic uncertainty. And so that would be where, let's say you're going higher up in the atmosphere and the pressure drops, your density would change. So down here on the left, we have what is called microscopic uncertainty. And then over here on the right, we have what is referred to as being macroscopic uncertainty. So those are some aspects of the continuum approximation that we need to make when we're dealing with fluid mechanics. And in our analysis, what we'll do is we will make the continuum assumption, which simplifies matters, we don't have to worry about density changing. So that is the continuum assumption or the continuum approximation that we'll be using in fluid mechanics. What we're doing is that we're assuming that the sample size that we're using is such that we do not have any kind of variability. And so consequently, we're dealing for the most part in this zone here. And that's where our density has converged, and it does not vary with the delta Vs that we're looking at. So that would be the continuum assumption. And that is another one of the aspects that we need to know as a basis of studying fluid mechanics.