14.3.2. Integration Over Control Surfaces

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, if there is a temperature gradient there will be heat transfer either to the surrendering of the systems or into the systems or out of the system, that is what we can do.

In fluid mechanics, understanding the conservation principles is crucial. These principles dictate how momentum and energy behave as fluid particles move within a system. Momentum conservation means that the total momentum of a fluid particle moving from one place to another is constant unless acted upon by an external force. On the other hand, energy conservation focuses on the work done by or on the fluid, highlighting how energy is transferred and transformed as fluid moves. Additionally, energy changes can also indicate heat transfers, which occur when there are temperature differences between the fluid and its surroundings.

Think of a flowing river. As water moves from a higher elevation to a lower one, gravity exerts a force on it, conserving momentum as the current flows. The energy of the moving water can also do work, such as turning a water wheel or eroding riverbanks. The interaction with the surrounding environment, like heat exchanged with the air, can be understood through these conservation laws.

Now, let me define two types of properties that we have; one is called extensive property and the other is the intensive property. The extensive property which is considered as proportional to the amount of mass. Extensive properties are proportional to the amount of mass, meaning, as mass increases, the extensive properties increase, and vice versa. When looking at intensive properties, they are independent of mass.

Extensive properties depend on the amount of substance present in a system. For instance, mass, momentum, and energy are extensive properties because they increase when more substance is added. In contrast, intensive properties do not change regardless of the amount of substance. Examples include temperature and density, which remain consistent even if the mass of the substance increases. This distinction is crucial in fluid mechanics as it helps in analyzing different properties of fluids under various conditions.

Imagine a glass of milk. If you have half a glass, the temperature of that milk remains the same whether it’s a full glass or half. Temperature is an intensive property. However, if you pour in more milk, the total mass increases, which affects the extensive property of the milk’s volume.

Now, we will go to derive Reynolds transport theorem. The derivation of the Reynolds transport theorem is available in almost all the fluid mechanics books. The idea for me is to introduce the Reynolds transport theorem so that you can easily understand it.

The Reynolds transport theorem provides a framework for relating the behavior of a fluid system as a whole (system level) with a specific control volume that moves or deforms over time. It combines the principles of mass, momentum, and energy conservation with the changes that occur within a control volume. This theorem helps us analyze how fluid properties change as they flow in and out of defined boundaries, making it a fundamental concept in fluid mechanics.

Consider a bathtub where water is flowing in through the tap and out through the drain. The Reynolds transport theorem can be used to analyze changes in water level within the tub over time. It highlights how the amount of water in the tub (control volume) relates to the inflow and outflow at a system level.

First, what we are considering is a non-deforming control volume which I have drawn the streamlines representing the flow that is coming in and coming out. That is what is my control volume. Whatever the fluid particles present at time t will represent the control volume and the reason is these are fixed.

In deriving the Reynolds transport theorem, we start with a control volume that does not change shape (non-deforming). This control volume is like a designated area in space through which fluid flows. We define the boundaries of this area and examine how fluid properties are distributed and change over time. By analyzing how fluid enters and exits this space, we can formulate equations that describe the dynamics of the fluid and its properties.

Think about a water cup that has been marked with a line to indicate a fixed volume. The cup represents your control volume. You can observe water entering the cup from a faucet and leaving through a straw. By monitoring the amount of water that flows in and out, you can apply the concepts of the Reynolds transport theorem to analyze changes in water level.

Now, if I look at simple definitions, the calculus is that the time rate of change of B in the system as a definition t + dt minus the system at the t level. We are defining in terms of in or out.

When analyzing how fluid moves through a control volume, we need to account for both the fluid that enters the system (influx) and the fluid that exits (outflux). By observing the changes in these flows over a specific time period (from time t to t + dt), we can use calculus to quantify these rates of change. This involves looking at the difference in the property B between these two time points and integrating this across the control surface.

Consider a water tank being filled and drained. If you monitor how much water comes in through the inlet pipe and how much leaves through the outlet pipe in a minute, you can assess whether the tank is gaining, losing, or holding a constant volume of water. This concept of influx and outflux is crucial in determining the dynamics of the water level.

If I know the velocity and the unit vectors, if I know the b and if I know the density, how it varies, then we can find out what could be the net outflux that is coming out if we are doing surface integrals of these functions.

To compute the total change in property B within a control volume over time, we need to gather data on the velocity of the fluid, the density of the fluid, and a variety of other factors. By performing surface integrals over the control surface, we can calculate the net outflux or influx of fluid and how these factors influence the overall behavior of the system. This requires integrating across the entire surface area affected by the fluid flow.

Imagine a water hose attached to a sprinkler system. By measuring the speed of water coming out of the hose and the area of the sprinkler head, you can calculate how much water flows through in a given period. This involves understanding both the flow rate and the area of the exit points to comprehend how much water is applied to your yard.

Now, coming to the third part. The first two terms we can write it as earlier you will have the influx and outflux. If you combine these, you get the total change in B at the control volume level.

After establishing the influx and outflux components in the previous steps, we can combine these terms to express the total change of property B within our control volume. This is a critical step in applying the Reynolds transport theorem, as it allows us to relate how much of property B is accumulated in the control volume compared to what flows in and out. The theorem ultimately helps us create a comprehensive understanding of the behavior of fluid flows and their effects.

Think about a swimming pool that is being filled. If you monitor how much water comes in from the hose but also how much escapes through leaks, you can calculate the effective increase or decrease in the water level over time. This cumulative understanding of influx versus outflux is at the heart of managing water in the pool efficiently.

With these things let me summarise today’s lecture. We discussed about system and control volume. We talked about fixed and deformable and moving control volume concept. The more important thing is that we derived the Reynolds transport theorem which can be used for fixed control volume, deformable control volume, moving control volume.

The key points covered in this section include the importance of understanding systems and control volumes, the differences between fixed and deforming control volumes, and the derivation of the Reynolds transport theorem. This theorem is fundamental for analyzing fluid flow dynamics across various types of control volumes and forms the basis for many applications within fluid mechanics.

In summary, think of the principles discussed as tools in a toolbox for engineers. The understanding of control volumes and conservation laws provides them with the ability to tackle complex fluid behavior in real-world scenarios, much like knowing how to use specific tools for carpentry helps build a solid structure.

--