 So, we're talking about control volume analysis and what we said in the last segment was that we needed to be able to recast the basic laws into a form that enables us to have mass crossing the boundaries. Before we can jump to having mass crossing the boundaries, let's take a look at the basic laws to begin with. So these are basic laws for a fixed mass system. So we have our basic laws applying to a fixed mass, so let's say we've got some chunk here and let's say that is mass M and it also occupies some volume V. So typically when we're dealing with the laws of physics, we refer to this as being our system, so that is an arbitrary quantity of mass and then conservation of mass would apply to that momentum, both linear and angular as well as energy. So let's take a look at those four equations and we'll express them in terms of this system that we're looking at. Beginning with conservation of mass, so we can write what conservation mass states is that the mass of the system does not change with respect to time. Now how do we define the mass of the system? We can do it this way. We can integrate little differential elements of mass across the entire system that we're looking at, so that chunk of fluid, or we can also do that as a volume integral where we integrate the volume of the entire system and if we're looking at mass, that would be rho, so the density that's mass per unit volume multiplied by some differential element dV. So that is how we could define mass. Let's take a look at the next one, which is conservation of linear momentum and this we will use in Newton's second law. So Newton's second law of conservation of linear momentum, it's important in fluid mechanics. This is where we determine forces, be it pressure, shear, but that helps us understand what is happening within a system or within our control volume. So here we have a vector formulation and it says that the forces are equal to the time rate of change of momentum within our system, where momentum is defined as P and this is linear momentum. And then P, again what I'm going to do is I'm going to write it out either as little chunks of mass dM or as dV, so the way that we would determine that is we could integrate the mass of our entire system. Instead of being just dM, it's going to be the velocity times dM and this could also be expressed as being a volume integral across the entire system and again the velocity and then the mass is replaced by rho dV. So that is Newton's second law of conservation of linear momentum, that's another one of the laws that we will be using. Next one is the conservation of angular momentum or moment of momentum and this would apply, for example if you had some rotating system and you want to be able to measure the torque on it, that's where we would be looking at moment of momentum. And so we have a balance between torsional and the time rate of change of the moment of momentum and that's across the system and we define that by H and again just like before we can integrate that over the mass of the system and here we would have r cross V dM or we can do that across the volume of the system and again that would be r cross V and dM would be replaced by rho dV. So that is the third basic law that we can have and the last one is the first law of thermodynamics or conservation of energy and here we have a relationship between heat transfer work and the internal energy of, in this case we're considering it to be a fixed mass system and quite often we look at the rate form so heat transfer across our boundary plus work across the boundary is equal to the time rate of change of energy within our system and I write out energy as being a capital E and energy can consist of again it's going to be an integral of either the mass in which case it would be E, little E is energy per unit mass or it can be a volume integral and little E is defined then as being U plus V squared over 2 plus GZ and this would be our internal energy second one is kinetic energy and then the last one is potential energy so that's the first law now those are the different laws we have mass linear momentum angular momentum and the first law of thermodynamics now let's go back and look at all of them and one thing to notice in our governing equations we always have this time rate of change derivative here we have another one for Newton's second law momentum momentum we again have time rate of change and finally for the first law of thermodynamics conservation of energy again we have this time rate of change term and what we will need to do is be able to find a way to be able to translate from the fixed mass reference frame which is what these equations are developed for and a control volume where we have mass crossing the boundary so let me just make a comment here about that so the governing equations that we've just looked at or the basic laws they are for systems of fixed mass and the problem is is that in fluid mechanics we don't want to have to follow a single particle because that would become very very complex and would not be very easy to do with analysis that we're going to be doing the other one is for most of the problems that we look at our region of interest is fixed it's stationary and the fluid is flowing into or around the objects or systems that we're considering and consequently the system approach is really not that good for fluid mechanics and so what we're going to do in the next segment we're going to go through a fairly long and laborious derivation but if you recall we had that d by dt of whatever of a system what we're going to try to do is recast that into a control volume approach to be able to express that for a control volume and that's going to take quite a bit of time to go through and derive that but we will have it and then once we have that we can then go back and apply all of our basic laws to analyze fluid systems we have conservation of mass linear momentum angular momentum and energy and those will be the ones that we will be analyzing in this course so that's a little bit of a background of the basic laws now let's move on and do this derivation and like I said it's going to be kind of a long one so brace yourself