14.2.5. Time Rate of Change in System and Control Volume

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.

In fluid mechanics, we can understand two key principles: the conservation of momentum and the conservation of energy. Conservation of momentum deals with how the motion of fluid particles is affected by the forces acting on them. It tells us that the momentum of a fluid system will be constant unless acted upon by external forces. Similarly, the conservation of energy states that as fluid moves from one place to another, it can do work on its surroundings or gain work itself due to the energy transferred within the fluid.

Think of momentum conservation like a game of pool. When the cue ball hits another ball, it transfers momentum to that ball, causing it to move. Without any obstacles (like frictional forces), the total momentum before and after the hit remains the same. In energy conservation, imagine a water slide; as the water flows down, it loses potential energy but gains kinetic energy, illustrating how energy moves from one form to another.

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.

Extensive properties are dependent on the amount of substance present. This means that if you have a larger quantity of a fluid, its extensive properties, like mass, momentum, and energy, will also increase. For instance, the total mass of water in a tank is the sum of the water in it; if you add more water, the total mass increases. On the other hand, intensive properties are independent of the amount of substance. They remain constant regardless of how much of the fluid you have. Examples include temperature and pressure.

Imagine a pure gold necklace. Its value (an extensive property) increases with its weight; more gold means a higher value. However, the purity of the gold (an intensive property) remains the same no matter how much you have. This concept applies to fluids where temperature or pressure does not change even if the volume of fluid changes.

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. As I discussed, there is a system and control volume.

Reynolds Transport Theorem (RTT) provides a linkage between the behavior of fluids in a system (fixed mass) and within a control volume (open mass). It allows us to calculate how properties like mass or energy change over time, connecting the accumulation of these properties within the control volume to their inflow and outflow across the control surface. The theorem essentially states that the rate of change of a property within a fixed system equals the change in the property within a control volume plus the net flux out of the control volume.

Consider a bathtub. If you fill it with water while simultaneously draining some out, RTT allows you to account for how much water is in the tub at any moment. The water that flows in adds to the total, and the water that drains out reduces it, which helps us understand how much water is accumulating in the tub over time.

Now, let us come to the derivations which I will highlight as I say it while derivation, which are the major components, not line by line. So, first, what we are considering is a non-deforming control volume...

To derive the Reynolds Transport Theorem, we first define our non-deforming control volume and observe how the fluid enters and exits through its surface. We analyze how the extensive properties like mass, momentum, and energy are distributed and how they change over time within this control volume. By applying the principle of conservation for each property, we formulate the mathematical relationships needed to express the changes of these properties within the control volume in terms of influx and outflux.

Think of a classroom where students enter and leave. The total number of students at any time can change based on who walks in and out. When we derive RTT, we're essentially computing how the number of students changes over time based on those entering (influx) and those leaving (outflux), just like we compute changes in fluid properties.

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