14.2. Reynolds Transport Theorem

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.

This chunk describes the fundamental principles of conservation in fluid mechanics. Specifically, it highlights three major conservation laws: conservation of mass, momentum, and energy. Each principle applies to fluid systems in a similar manner as they do in solid mechanics. The conservation of energy, for example, is crucial for determining how much work a fluid does while moving from one place to another, and how it can transfer energy into or out of a system.

Think of water flowing through a pipe. As it moves from a higher elevation to a lower one, it does work by turning a water wheel, generating energy. This visualizes the principle that as fluids move, they carry energy and momentum with them.

Now, let me define two types of properties that we have; one is called extensive property and the other is the intensive property.

This chunk introduces the distinction between extensive and intensive properties. Extensive properties depend on the size or mass of the system, such as mass, momentum, or energy. In contrast, intensive properties, such as temperature or pressure, remain constant regardless of the system's size. This distinction is crucial in understanding how different properties behave under varying conditions in fluid mechanics.

Consider a glass of water. The total mass and energy (extensive properties) depend on how much water is in the glass. However, the temperature (an intensive property) would be the same whether it’s a full glass or just a drop.

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 Reynolds transport theorem (RTT) connects the concepts of fluid motion and conservation laws at both system and control volume levels. It provides a mathematical framework to analyze how extensive properties change within a control volume as fluid enters and exits the system. By establishing this theorem, we can apply it to various problems in fluid mechanics.

Imagine a container filled with gas. As gas molecules flow in and out, the RTT helps us quantify how much gas is present inside at any given time, even as the influx and outflux vary. This is similar to a busy restaurant, where people continually enter and leave, yet you can still keep a count of how many customers are being served at any moment.

I can define this 3 into 3 different spaces, like the space defined by this part I can give it as I and this can be II and this can be used as the control volume space.

This segment discusses how, during a control volume analysis, the fluid domain can be divided into three distinct regions: the influx region, outflux region, and the overlapping region where fluid exists at both present time t and future time t + dt. This division assists in understanding how fluid properties change over time and how to analyze the net changes in these properties.

Think of it like a bathtub. The area where water is entering is the influx, where it is draining is the outflux, and the amount of water that remains after a certain time is the space that overlaps. This visualization makes it easier to understand how changes in input and output affect the overall water level.

Now, how to compute the outflux surface, okay? That means, assuming it is a three-dimensional control volume, and over that surface I want to integrate it...

This section deals with the mathematical computation of fluid flux through the surfaces of a control volume. By considering small surface elements and their associated normal vector and velocity, one can use calculus to integrate values over the control surface, which allows for the calculation of the total influx or outflux of the fluid.

Imagine a water slide at a park where kids are going down. Each section of the slide has a specific number of kids going down in a given time. By observing each section of the slide as a control volume, we can calculate how many kids are sliding down (outflux) versus how many are approaching the top (influx).

That is what is Reynolds transport theorem, and the basic physics you can understand.

In the conclusion of the RTT discussion, we frame the theorem succinctly: the rate of change of a property within a system is equal to the accumulation of that property in a control volume plus the net outflux through its boundary. This serves as a powerful tool for solving complex fluid dynamic problems by linking the behaviors of the system with control volume analysis.

Returning to our bathtub example, if we think of it in terms of how full the tub is (the system), we can account for the water that flows in from the faucet and what flows out from the drain (the control volume), giving us a complete picture of how the water level changes over time.

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