 Now that we are through fluid statics, we can start to talk about fluid dynamics, and fluid dynamics is not unlike any other dynamics course they would have taken, whether kinematic dynamics or thermodynamics. We have a skill set or a toolbox that we apply to a problem, and that really looks like a number of equations. Here that would be the conservation of mass, the conservation of linear momentum, the conservation of angular momentum, the conservation of energy, and the increase of entropy principle. By applying these five relationships to a fluid, we can have a pretty good idea of how it's going to behave, and we can quantify many aspects of it. To limit our scope a little bit, we are going to be ignoring angular momentum and the increase of entropy principle for now. We are looking at the conservation of mass, the conservation of linear momentum, and the conservation of energy, and before we start applying them, we should start to think about how it is most convenient to apply them to the situations we are likely to encounter in chapter 3. Chapter 3 is all about control volumes. So we should try to think about how best to write out our conservation of math, conservation of linear momentum, and conservation of energy equations for a control volume. To start, let's look at the conservation of mass. When we are in thermo, we quantified our conservation of mass by writing the change in mass of our control volume in terms of mass entering and mass exiting. Or if we were to divide all three terms by dt, we wrote the rate of change of mass of our control volume with respect to time in terms of the rate of entering mass and the rate of exiting mass. And that's true. That's a form of the conservation of mass, but it is most useful when you're talking about the entering mass and the exiting mass, specifically the majority of the time, and you don't care so much about our system. Here, we are shifting our focus a little bit. We are looking at the system primarily and the inlets and outlets are only considered to the degree to which they affect our system. So we are focusing on the system and including the entering and exiting effects in our integral. So instead of writing mass entering and mass exiting, we are going to write the mass of our system is the integral of dm across our mass of system, which isn't particularly helpful. But what is helpful is writing the mass as the integral of density with respect to volume. If density is constant, it comes out. But if density changes with respect to volume, then writing it as an integral across the volume will account for that. Similarly, with the conservation of momentum, we are primarily focused on force when we have encountered this in previous dynamics courses. And remember that force is defined as the rate of change of momentum with respect to time. And in the past, we had written that as mass times velocity. That is momentum as mass times velocity. At which point we were taking the derivative of mass times velocity with respect to time, which because of the product rule, we could write as the derivative of mass with respect to time times velocity plus the mass times the derivative of velocity with respect to time. Which, if the mass is constant, simplifies down to just the mass times the rate of change of velocity with respect to time, which is acceleration. So every time that you've applied f is equal to ma, what you're saying is the rate of change of momentum for a constant mass. Here, we're going to write the momentum of our system as the integral across the mass of velocity with respect to mass, or the integral across the volume of velocity times density with respect to volume. Note that both momentum and velocity are vectors here, as is the force. Then with our conservation of energy, like with the conservation of mass, we had to use the energy balance to represent our first law of thermodynamics application, saying that the energy change of our control volume is equal to the energy entering our control volume minus the energy exiting our control volume, which if we divide all three terms by dt, gives us de, dt of the control volume, in terms of the rate of energy entering and rate of energy exiting. But again, we are defining our scope as looking at the system. So we write the energy of our system as being equal to the integral across the mass of specific energy of our system with respect to m, which is also the integral across the volume of specific energy of our system times density with respect to volume. And the reason that I have written all three like this is because I can make a generalization. I can write that in terms of an extensive property and its accompanying intensive property. Here I'm using uppercase, beta, and lowercase beta, which I will refer to henceforth as b and beta, respectively, for the extensive and intensive property. So when I plug in a beta value of 1 and a b value of m, what I'm writing is the conservation of mass. When I plug in a beta value of velocity and a b value of momentum, what I'm writing is the conservation of linear momentum. When I plug in a beta value of specific energy of our system and a b value of total energy of our system, what I'm writing is the conservation of energy. So, making this generalization, I can quantify all three at the same time in terms of beta and b using the Reynolds transport theorem. The Reynolds transport theorem allows us to relate system properties to control volume properties. The left term is the total rate of change of any arbitrary extensive property, b, of the system, that's a Lagrangian perspective. The middle term is the time rate of change of the arbitrary extensive property within the control volume. Note the lowercase beta is the extensive property per unit mass, and rho dv is the mass element in the control volume, so the integral is the amount of b in the control volume, if that makes sense. The right term is the net rate of flow of b out of the control volume, where rho vda is mass flow rate out of the control volume through dA. That means beta rho vda is the b flow rate out through dA on a net basis. An important note here is that the velocity vector is measured relative to the control volume, so in situations where we have a moving control volume, we will have to deal with this, and we will do that later. The Reynolds transport theorem could be a video series in and of itself. Heck, there are core sequences in universities that just use heat transfer electricity, fluid mechanics, and thermodynamics as applications of the Reynolds transport theorem. By starting at the Reynolds transport theorem and simplifying it for the situation in front of us, we are going to be able to write out whichever equations make sense for the problem that we actually have. So we're coming back to the Reynolds transport theorem and simplifying it every time. Then there are simplifications that we can make for some common situations, but the logic here is that we are starting from the Reynolds transport theorem and simplifying. And depending on what values of beta and b we plug in, we will get out relative terms for the conservation of mass, conservation of momentum, and conservation of energy, respectively.