 We're now going to take a look at the topic of control volume analysis. And if you're called from one of the earlier lectures, we talked about different methods that you can do analysis in fluid mechanics. This is a fairly simple and quick technique. So control volume analysis has the characteristics that it's a quick technique for doing analysis of problems involving fluid flow. We'll be able to get quantitative results out of this, such as forces that might be acting on objects that are immersed in a fluid flow or forces upon a wall. And this is essentially a large scale analysis. So typically with control volume analysis, you're not going to be able to determine specifically what the velocity profile might be on the inlet or exit to a flow field. However, what it is that gives us these large macro scale values such as drag or lift based on having measurements of velocity coming into and leaving our control volume. And in a way, if you think back or if you've taken a course in thermodynamics, in thermodynamics, we have fixed mass and we also have the open system. And in certain respects, what we're doing now, the basic equations are always described for a fixed mass. And we'll be talking about that in the next segment. But what we're doing is formulating our calculations to be able to allow us to do open systems. So we have mass crossing the boundary. And if you're called from thermodynamics quite often, we use control volumes. We use control volumes and fluid mechanics as well. However, in thermal, we mainly focus on the energy equation. Here we can look at the momentum equation, linear, angular. We can look at mass conservation. We also look at energy. So we kind of expand the portfolio of things we look at when we're doing fluid mechanic analysis. But it is very similar to what we do in thermodynamics. So to begin with, what is a control volume? And so I want to take a quick look at what a control volume is before we get into the details of deriving the equations for the control volume. So let's take a look at an example control volume. And sometimes what you'll see me do is something like this. That is an abbreviation for control volume, CV, with a line through it. So let's take a look at something that we can all relate to. Let's say we have a jet. So we have a flow, it could be a garden hose nozzle or something like that. But it would accelerate the fluid coming through it. And so let's say this is all solid. And let's say we have a uniform velocity profile coming in, chances are you wouldn't. But you may condition the flow to give you something like that. So that's V in. And then on the outlet, because the area is contracting, let's say this is a liquid. So it's incompressible. The velocity is going to increase and that would be V out. So if we were to do control volume analysis of this, where you sketch your control volume is quite often important for doing the calculations. But an example for this, if you were to do control volume analysis, maybe you would decide to put the control volume around the outside of the block itself. And so that's what we would have for the control volume. And you'll see then we would do something like this to denote that is the control volume. Another thing is that we have surfaces where mass is crossing the boundary. So we have one on the inlet and one on the exit. And sometimes we'll write control surface one and maybe control surface two. And I just didn't know what the fact that that's where the mass is crossing the boundary. So with a control volume, fluid can cross the boundaries, but our governing equations are for fixed chunks of mass. And consequently, we need a way to be able to relate that. So the idea with the control volume is that you only are worried about the boundaries, the inlets and the exits. And we don't have to keep track of all of the particles of fluid going through it. However, this does increase complexity because, as I mentioned, our governing equations are for fixed mass. Let's take a look at that. So I mentioned before, a control volume is not a system. And so by a system, that usually means fixed mass, just like we talked about in thermodynamics. And so here we have mass crossing the boundary. And so we need a way to be able to express the equations, the governing equations that we're going to use. And those will be conservation of mass, momentum and energy. We need to be able to express those in a form where we can allow mass to cross the boundaries. And so what we're going to do in this lecture segment, we're going to go through a derivation of a way to enable us to be able to recast conservation of mass, momentum and energy for a control volume formulation. It's going to be kind of long and kind of theoretical. So bear with me, but we're going to walk our way through and come up with the equations or get in the position where we'll have the equations that we can do control volume analysis.