18.3. 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 approximation of extensive and intensive properties and as of now we have derived mass conservation equations. Today, I am going to derive conservation of linear momentum.

In fluid mechanics, flow can be classified as steady or unsteady. A steady flow means that the fluid's velocity at a given point does not change with time. In contrast, in an unsteady flow, the fluid velocity at a given point changes over time. The Reynolds transport theorem helps in analyzing these flows by allowing us to set up equations for control volumes that consider both mass and momentum conservation.

Think about a garden hose. If you have the water flowing steadily without changing the speed, that's like a steady flow. But if you were to turn the tap on and off, causing the water speed to change, that would be an example of unsteady flow.

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: with respect to time is it steady or unsteady.

The first step in applying the Reynolds transport theorem is to identify whether the flow is steady or unsteady. If the flow is steady, we can simplify our calculations considerably. In steady flow, properties such as fluid velocity and density do not change with time, allowing us to focus on surface integrals rather than time-dependent factors. This simplifies our equations significantly.

Imagine driving down a highway at a constant speed. That’s like steady flow; your speed remains unchanged. Now, think about driving through traffic, where you constantly start and stop. That's akin to unsteady flow as your speed varies frequently.

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.

In steady flow situations, the time derivatives of flow properties average out to zero because nothing changes as time progresses. This means the volume integral of the Reynolds transport theorem can also be simplified. In practical terms, this makes it easier to focus solely on the surface integrals, which directly relate to the flow of fluid across the boundary of the control volume.

If you think about filling a bathtub with water at a constant rate, the water level rises steadily without fluctuating. This is similar to a steady flow where the rate of water entering equals the rate of water in the tub, resulting in no change in water level over time.

Then, with respect to density change or the variations of the density, we divide it, flow is compressible or incompressible. So, we can have two types of approximations, steady compressible, steady incompressible.

Fluid flows can be further classified as compressible or incompressible based on how much the fluid’s density changes during flow. For incompressible flows (like water), density changes are negligible. For compressible flows (like gases), density can change significantly. Understanding whether the flow is compressible or not is crucial for applying appropriate equations and assumptions.

Think about a balloon. When you blow air into it, you compress the air, making it a compressible flow. Now, consider water flowing through a pipe. The density of the water remains almost constant, representing an incompressible flow.

So, when you have a steady incompressible flow, most of the case what we consider for flow devices or engineering applications, we can consider steady incompressible flow, then the problems become very simplified when you apply for Reynolds transport theorem.

In engineering applications, most flows we deal with can be approximated as steady and incompressible. This significantly simplifies calculations because we can assume constant density and streamline the equations we use to analyze the flow. This makes it easier to predict outcomes in systems such as piping, open channels, and many mechanical devices.

Consider a simple faucet. When you turn it on, the flow of water is steady and behaves as incompressible fluid. Engineers can predict how much water will flow out based on the size of the faucet and the pressure in the pipes without considering complex changes in density.

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