2.4. Application of Reynolds Transport Theorem

So, to start fluid dynamics, one of the most important theorems that generally is not taught in fluid mechanics is Reynolds transport theorem. So, what we are going to do, we are going to derive this Reynolds transport theorem in a little bit more detail.

In this chunk, the speaker introduces the Reynolds Transport Theorem (RTT) as a crucial concept in fluid dynamics, which is often overlooked in basic fluid mechanics courses. The theorem helps in understanding how physical quantities like mass, momentum, and energy change as they move through a fluid system. The speaker plans to provide a detailed derivation of the theorem to clarify its importance and application.

Think of the Reynolds Transport Theorem like tracking a group of runners in a 5k race. Just as we can measure how many runners pass a certain point at different times, the RTT helps track how properties like mass or momentum flow through a fluid system over time.

All physical laws are stated in terms of various physical parameters. So, if B represents any of these, these parameters can be velocity, acceleration, mass, temperature, momentum, anything etc.

The fundamental aspect of RTT is that it relates extensive and intensive properties of a fluid. Extensive properties, like mass or total momentum, depend on the size of the system, while intensive properties, like velocity or temperature, are per unit mass and remain constant regardless of the system's size. The distinction between these properties is crucial for proper application of various fluid dynamics laws.

Consider a large jug of water (extensive) versus a single drop of water (intensive). The total weight of the water (extensive property) changes with the amount you have, but the temperature of each drop (intensive property) remains the same regardless of the amount.

The amount of an extensive property that the system possesses at a given instant B, sys, can be determined by adding up the amount associated with each fluid particle in the system.

To calculate the total amount of extensive properties in a system (B_sys), we sum the contributions from each individual fluid particle. This sum can become an integral when considering a continuous fluid system where we can express the total amount as an integration over all particles of the system, effectively accounting for varying densities and properties.

This process is similar to calculating total calories in a dish by adding calories from each individual ingredient. Just as you would sum the contributions of each ingredient to get the total caloric value, in fluid mechanics, we sum the contributions of each fluid particle to get the total extensive property.

Most of the laws governing fluid motion involve the time rate of change of an extensive property of a fluid system, the rate at which the momentum of a system changes with time, for example.

The laws governing fluid motion often involve how quickly an extensive property changes over time, like how momentum builds up in a flowing river. The time rate of change of properties needs to be expressed in terms of a control volume instead of just a system. This requires obtaining a new formula that accounts for both the changes within the fluid and those crossing the boundaries of a defined control volume, which can be thought of as a 'snapshot' of the system.

Imagine observing a crowded train station. If you count the number of people entering and exiting over time, you're studying the flow of people—a similar concept to studying the rate of change in fluid properties within a specific area, or control volume, while considering those entering and leaving the space.

To formulate the laws into a control volume approach we must obtain an expression for time rate of change of an extensive property within a control volume B cv and not within a system.

The control volume approach involves analyzing how extensive properties change not just within a fixed mass of fluid but across a defined volume through which fluid flows. This is crucial in engineering applications since systems often have inputs and outputs. The RR states that we can derive relationships for the rates of change of these properties by considering the fluid entering and leaving the control volume rather than merely analyzing the system itself.

Think of a factory assembly line. Instead of just looking at the parts in one machine (system), look at how parts enter and exit the entire assembly line (control volume). Tracking items at that scale gives a more comprehensive understanding of the overall process and efficiency.

Now we are going to derive Reynolds transport theorem. We consider the control volume to be that stationary volume within the pipe or duct between section 1 and 2.

The derivation of the Reynolds Transport Theorem begins by defining a control volume, usually a fixed region, and observing how fluid properties change as fluid flows in and out. The speaker describes the system's boundaries and how they shift over time as the fluid moves, leading to the formulation of the relationships needed to find how extensive properties change over time, accounting for the inflow and outflow of the control volume.

Consider water flowing through a park fountain. As you measure the amount of water that comes in and out of the fountain (control volume) at different times, you're applying the same principles in fluid dynamics as the RTT accuses that setup as it flows through a defined area.

This is the general form of Reynolds transport theorem for a fixed non-deforming control volume.

The final general form of the Reynolds Transport Theorem provides a comprehensive relationship to calculate the rate of change of an extensive property within a control volume in relation to what flows in and out. This allows for application to various problems in fluid dynamics, enhancing the predictive power of fluid motion analysis.

Imagine pouring milk into a glass while some spills over. The RTT helps you quantify what’s happening—the amount of milk entering the glass (rate in) versus the amount spilling out (rate out), allowing you to understand how much remains in the glass over time.

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