14.2.3. Derivation of Reynolds Transport Theorem

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...

This chunk introduces the fundamental principles of conservation that are vital in fluid mechanics such as the conservation of mass, momentum, and energy. These principles form the basis for analyzing any fluid system. In fluid dynamics, understanding how these quantities behave as fluid moves is crucial for solving real-world problems in engineering and science. The flow of fluid involves considerations of forces acting on the fluid particles, work done by or on the fluid, and heat transfers that happen as fluids interact with their environments.

Imagine a river flowing from a mountain to a lake. As the water flows, it carries momentum (due to its mass and velocity), does work by eroding the river banks (energy conservation), and can transfer heat to the surrounding air or ground. Understanding how the water conserves its energy, momentum, and mass helps in predicting how the river will behave under different conditions, such as during a heavy rain.

Now, let me define two types of properties that we have; one is called extensive property and the other is intensive property... Extensive property is proportional to the amount of mass. For example, mass conservation, momentum conservation, and energy conservation. Intensive properties, that means it is independent of mass...

In this chunk, we define extensive and intensive properties. Extensive properties are those that depend on the amount of fluid present (e.g., mass, volume, energy), meaning if you have more fluid, these properties increase. Conversely, intensive properties do not depend on the size or mass of the system (e.g., temperature, pressure, and density). Understanding these properties is crucial because it helps in applying the right equations in fluid mechanics problems.

Think of extensive properties like the total amount of frosting on a cake—a larger cake (more mass) means more frosting (extensive property). In contrast, the frosting's sweetness is an intensive property, which does not change regardless of the cake's size. Whether you have a small cupcake or a large wedding cake, the sweetness remains constant.

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... So, first, what we are considering is a non-deforming control volume.

This segment outlines the beginning of the derivation for the Reynolds Transport Theorem (RTT). The RTT connects the conservation laws at a system level and a control volume level, essential for analyzing fluid motion. The control volume is typically a fixed or non-deforming volume through which fluid flows in and out. Understanding how the fluid's total property changes both at the system level and control volume level is crucial for solving fluid dynamics problems.

Consider a water tank connected to a pipe. The water flowing in and out represents the control volume. RTT helps us understand how the volume of water changes in this tank based on the flow rates into and out of the tank. By understanding the total volume (system) versus the volume inside the tank (control volume), we can predict how long it takes to fill up or empty the tank.

...I can define there will be a control surface defined by A, D, C, and F... the streamlines are indicating how the influx is coming into the control volume and going out from the control volume, through this surface.

Here, we discuss the definition of a control volume, which is the specific region in space where we analyze the fluid motion. The control surface is the boundary that defines this control volume. Understanding the influx (fluid entering) and outflux (fluid exiting) is essential for applying the conservation principles and helps in deriving the equations governing fluid flow.

Continuing with our water tank example, the walls of the tank represent the control surface. The water entering through the pipe is the influx, while the water leaving through a drain is the outflux. By observing the flow in this fixed volume, we can analyze how the water levels change over time.

Now, if I look at simple definitions... dB B (t + dt) - B (t) / dt.

In this step of the derivation process, we introduce 'dB', which represents the change in an extensive property (like mass, momentum, or energy) per unit time within the system. The change is calculated over a time interval 'dt'. This mathematical relationship sets the stage for how we will relate the properties at the system level with those at the control volume level.

Think about filling a bathtub. If you turn on the faucet and track the change in water level over a minute, that change can be expressed as dB per time. Just like measuring how much water fills the tub as time passes gives you an insight into the change in the water level, understanding dB gives insight into how mass or energy changes in our control volume over time.

Now, how to compute the outflux surface... that means how much of influx is coming into the control volume or going out from control volume.

Here, we discuss how to calculate the influx and outflux of fluid through the control surface. This involves considering the velocity vector and the area vector at each point of the surface. The calculations define how much fluid passes through per unit time based on the geometry of the control volume and the velocity of the fluid, helping to fully characterize the fluid behavior through the control volume.

Let's say we are measuring how fast water is flowing out of a garden hose. By knowing the diameter of the hose and the speed of the water, we can calculate the total volume of water flowing out per minute. This is analogous to calculating outflux across the control surface.

Now, coming to the third part. The first two terms we can write it as earlier you will have the influx and outflux...

This part brings together the various elements of the derivation and expresses the Reynolds Transport Theorem mathematically. The theorem essentially states that the rate of change of a property within a control volume is equal to the net influx of that property into the control volume, which combines all previously discussed concepts of influx, outflux, and time rate changes.

Imagine you are filling a swimming pool with water from a hose while some water is also leaking out through a crack. The RTT helps you determine whether the water level will increase, stay the same, or decrease based solely on the rates of water flowing in and leaking out.

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