The conservation of momentum which is required for you to know it, how the fluid particles are moving it, what could be the force exerting on that, what could be the velocity. Similar way, we can understand the energy conservation which plays a major role for us when the fluid comes from one location to other locations, how much of work is done by the fluid or into the fluid.

Similar way, whether there is heat transfers happening which you can feel it, if there is a temperature gradient there will be heat transfer either to the surrendering of the systems or into the systems or out of the system, that is what we can do. So, to summarise this, that means, we all know that there are three energy conservation principles that we follow in solid mechanics when you consider as a system.

Same concept also we can use at the system levels to solve the problems, conservation of mass, conservation of linear momentum which is Newton’s law, and the conservation of energy which is the first law of thermodynamics.

Now, let me define two types of properties that we have; one is called extensive property and the other is the intensive property. The extensive property which is considered as proportional to the amount of mass. When you apply extensive properties, that means, you are the properties which are proportional to the amount of mass. That means, as mass increases you will have extensive properties going to increase.

Mass decreases, the extensive property decreases. It is proportional to the mass. For example, as we discussed it in three basic laws, we talked about mass conservation, momentum conservation, and energy conservation. So, m will be the mass conservation part, momentum and energy conservation. But when you look at the intensive properties, that means it is independent of mass, that means, which is denoted as specific values like velocity or specific energy. For energy conservation the extensive property will be the one, but in the case of the momentum, but intensive property will be the velocity vectors.

Now, we will go to derive Reynolds transport theorem. The derivation of the Reynolds transport theorem are available in almost all the fluid mechanics books. The idea for me is to introduce the Reynolds transport theorem so that you can easily understand it. But the step wise derivations, if you are not understanding it, I could suggest you to follow any of the fluid mechanics books, F.M. White, Cengel Cimbala, or any other advanced fluid mechanics books, you can see the derivations of Reynolds transport theorem. Only the symbol of representation of extensive properties, intensive properties, either B or b or are used in different books in different forms.

Now, coming to the third part. The first two terms we can write it as earlier you will have the influx and outflux. That part can be written as integrals of influx and outflux and if you combine it next what you are getting? This is total cross section, okay? That means net outflux of mass, momentum, or the energy flux going through the system, either influx or the outflux but as you integrate it it represents the net outflux that is going through this control surface.

Let us consider a steady incompressible flow, that means the flow does not change with time, the density changes in this case of the steady compressible flow. If that is the condition, your major part of this becomes 0, because if d by dt becomes 0.

With these things let me summarise today’s lecture. We discussed about system and control volume. We talked about fixed and deformable and moving control volume concept. The more important thing is that we derived the Reynolds transport theorem which can be used for fixed control volume, deformable control volume, moving control volume. And we also demonstrated the use of the simplification of the steady problems.