18.2.2. Reynolds Transport Theorem

Basically, we have been following the Reynolds transport theorem as a basic concept to apply the setup of the system into physical equations to the control volume level. And then, at the control column level, we have an approximation of extensive and intensive properties, and as of now, we have derived mass conservation equations.

The Reynolds Transport Theorem is a fundamental principle used in fluid mechanics. It allows us to relate the macroscopic behavior of fluid systems described by control volumes to the flow of individual fluid particles. It provides a framework for converting between the total derivative of a quantity and the integral form, allowing us to apply conservation laws effectively for mass or momentum. This theorem helps us to derive equations that model physical phenomena in engineering, particularly in fluid dynamics.

Imagine you are trying to capture the flow of water in a river using a large bucket (the control volume). The Reynolds Transport Theorem helps you to know how much water is entering and leaving the bucket over time, allowing you to understand the behavior of the water flow around it.

As I told you, we discussed about the Reynolds transport theorem for conservation of mass, and when we apply this conservation of mass to the Reynolds transport theorem, we have two basic assumptions, that is, with respect to time is it a steady or unsteady.

In fluid mechanics, flows can be categorized as 'steady' or 'unsteady'. Steady flow implies that properties at any point in the fluid do not change over time. In contrast, in unsteady flow, these properties do change. When analyzing a system using the Reynolds Transport Theorem, identifying whether a flow is steady or unsteady helps in selecting the right mathematical approach; for steady flow, specific simplifications can be made, leading to easier calculations.

Consider a garden hose. If you keep the nozzle open until the hose is empty, that would be steady flow—the amount of water flowing out remains constant. If you periodically cover and uncover the nozzle, changing the flow rate, that's unsteady flow— the water flow varies over time.

Then, with respect to density change or the variations of the density, we divide it, flow is compressible or incompressible.

Fluid flows can also be categorized based on the compressibility of the fluid. Compressible flow occurs when changes in density are significant, notably in gases at high speeds, whereas incompressible flow assumes constant density, applicable primarily to liquids under many practical conditions. Understanding whether a flow is compressible or incompressible affects how equations are formulated, as this greatly influences the behavior of the fluid as it moves through a system.

Imagine blowing up a balloon. The air inside the balloon is compressible; as you blow more air in, the density of the air changes because it gets packed into a smaller volume. In contrast, think about a water balloon; when you fill it with water, the water's density hardly changes regardless of how much more water you add—this is analogous to incompressible flow.

So, when you have the steady assumptions, you can remember that the component of Reynolds transport theorem of time, differentiate components become 0 or the volume integral component part of the Reynolds transport theorem becomes 0.

Under the assumption of steady and incompressible flow, the time-based components of the Reynolds Transport Theorem simplify significantly. The volume integral of changes over time becomes zero, which means that we only need to consider the boundary conditions (surface integrals). This greatly simplifies our calculations and leads to straightforward relationships between velocities and areas.

This is like focusing only on the amount of water flowing out of a stationary water tank through a pipe, ignoring any changes in flow over time; you only care about how much water flows out at a given moment, making it easier to measure and calculate.

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