14.2.4. Control Volume and Influx/Outflux Regions

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.

In understanding fluid mechanics, we need to look at the conservation of momentum and energy. Conservation of momentum refers to how the motion of fluid particles is influenced by the forces acting on them, including their velocity. On the other hand, energy conservation helps us understand how much work fluid can do as it moves from one place to another. This involves tracking changes in energy as fluid flows, such as when it is compressed or expanded.

Think about riding a bicycle. As you pedal (apply a force), you accelerate (momentum conservation) due to your effort. If you're going downhill, the gravitational energy (energy conservation) can help you go faster without having to pedal as hard because the energy is transforming in the system.

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.

Heat transfer is another vital aspect to consider. When there is a difference in temperature (a temperature gradient), heat will always flow from the hotter area to the cooler area until equilibrium is reached. This can mean heat moving into the system, out of the system, or being exchanged between systems.

Imagine a cup of hot coffee in a cold room. The heat from the coffee moves into the air around it, causing the coffee to cool down and the air to warm up. Eventually, the temperature of the coffee and the surrounding air approaches equilibrium.

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. When you apply extensive properties, that means you are the properties which are proportional to the amount of mass.

Extensive properties are characteristics that depend on the amount of substance present. Examples include mass, volume, and total energy. For instance, doubling the amount of water will double its mass and volume. Intensive properties, however, are independent of the amount of substance. These include temperature and density, which remain constant regardless of how much of the substance is present.

Think of making soup. If you double the ingredients (extensive property), you double the volume of the soup. However, the temperature of the soup, whether it's hot or cold, does not depend on the total amount but rather the heat applied to it.

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 links the rate of change of a property within a control volume to the flux of that property across the control surface. It's fundamental for analyzing fluid motion in control volumes, providing a clear method to account for how properties like mass, momentum, and energy are conserved.

Imagine a water tank with a hose filling it at one end and a drain at the other. The Reynolds transport theorem helps you understand how the water level changes inside the tank by considering how much water flows in through the hose (influx) and how much drains out (outflux).

Let us understand the Reynolds transport theorem which establish the relationship between the conservation law at the system level and the conservation at the control volume level.

A control volume is a fixed or moving region in space through which fluid flows. This concept allows us to analyze physical phenomena and derive important equations for mass, momentum, and energy conservation. It establishes a boundary around a system to facilitate easier calculations and understanding of fluid motion.

Think of a car moving through air. The car itself can be seen as a control volume because it interacts with the air around it. By understanding how air flows into and out of this control volume (the car's boundaries), we can solve problems related to aerodynamics.

Let me 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.

The control volume is often divided into regions based on fluid movement. These can be defined as influx (where fluid enters), outflux (where fluid exits), and a common region (where fluid exists at both times t and t+dt). This classification helps in understanding how properties change within the control volume over time.

Imagine pouring water into a glass. The area where water enters the glass is the influx region, while the area where it might overflow is the outflux region. Understanding these regions helps you predict how full the glass will get over time.

--