15.2. Control Volume and Reynolds Transport Theorem

Let us come back to what we learned in the last class. If you remember it, I discussed very thoroughly what is the difference between systems and the control volume. Mostly in fluid mechanics we follow the control volume aspect. That is the reason we need a relationship between the system and the control volume. The Reynolds transport theorems establish the relationship of conversions of mass from system to control volume.

In fluid mechanics, the concept of 'control volume' is crucial. A control volume is a specific region in space through which fluid can flow. The main idea behind using control volumes is to analyze how mass and energy enter and leave this defined space. This is different from the concept of a 'system', which could be more rigid and not necessarily change in a fluid dynamic context. The Reynolds transport theorem helps us understand how quantities like mass are conserved as they pass in and out of the control volume.

Imagine a water tank with an inlet pipe and an outlet pipe. The tank itself acts as the control volume. Water flowing in through the inlet and out through the outlet represents the mass entering and exiting the control volume. By analyzing this tank, we can understand how much water (mass) is in the tank at any given time.

So, that is the strength of the Reynolds transport theorems and, as already I discussed, you can have three type of control volumes, fixed control volume, that means with reference to the space it remains at the same location. As the time goes, it does not move it. That is what fixed control volume is. It has definite control surfaces. The other one is moving control volume, like the ship movement in a sea or the river. The ship and the adjacent fluid can be considered as moving control volume. And we solved many of the problems with moving control volume, considering this control volume is moving with a particular velocity. Not only that, the shape of the control volume can change. That was the deformable control volume. The shape of the control volume can change with respect to time, then we call it deformable control volume. So, there are three types of control volume, fixed control volume, moving control volume, and deformable control volume.

There are three primary types of control volumes in fluid mechanics:

1. Fixed Control Volume: This volume does not change position over time. It is useful for analyzing flows around stationary objects, such as a building in a wind tunnel.
2. Moving Control Volume: In this case, the control volume moves with the fluid, which is common in applications like ships navigating through water. The control volume travels along with the fluid, allowing for analysis of dynamic interactions.
3. Deformable Control Volume: This type can change its shape over time, adjusting to fluid flow dynamics. An example of this could be the deformation of an airfoil as it moves through air. Understanding these types helps in choosing the right approach when applying the Reynolds transport theorem.

Think of a fixed control volume as a swimming pool that doesn't change size or position. A moving control volume would be like a diver performing flips in mid-air, where the diver and the surrounding air act as the control volume, moving dynamically together. A deformable control volume can be visualized as a balloon being squeezed or stretched while being inflated, changing shape as more air enters.

So, as I have given very briefly Reynolds transport theorems and moving and deformable control volume, conservation of mass we will derive for simple case to complex problems, how we can write conservation of mass. Then, we will talk about some examples, real life examples we will take and how the appropriate choosing of control volume and the control surface can solve many of our problems.

The conservation of mass is a fundamental principle in fluid mechanics, describing how mass cannot be created or destroyed in a closed system. When applying this to control volumes, the Reynolds transport theorem allows us to express mass conservation as an equation that states: the rate of change of mass within a control volume is equal to the mass inflow minus the mass outflow. This relationship can be used to set up equations for both simple and complex fluid flow situations, helping to model real-life scenarios such as pipeline flow, aerodynamics, or even natural phenomena.

Consider a bucket being filled with water while simultaneously draining water from a hole at the bottom. By applying conservation of mass, we can calculate how fast the bucket fills or empties. If water flows in at 2 liters per minute through the top and out at 1 liter per minute through the drain, we can determine the net inflow to see how long it takes to fill the bucket.

Now, let us go to very interesting applications. As you know it Indian Space Research Organisation launched the satellite which is called MOM programmes, that means Mars Orbiter Mission programme. But if you look at this, it is the fluid flow problem. If we look at the trajectory which is started from the earth, and if you look at the trajectory part here, how this trajectory goes on changing, which was launched on December 1, 2013, okay?

The Mars Orbiter Mission (MOM) is an excellent example of advanced fluid mechanics in action, particularly in the areas of aerodynamics and trajectory optimization. When launching a satellite, engineers must carefully consider factors like atmospheric drag, propulsion forces, and the controlling of the satellite’s path as it travels from Earth to Mars. Fluid mechanics principles are essential in calculating the optimal force applications and trajectory paths to successfully navigate the satellite through varying conditions of the atmosphere and space.

Think of launching a ball into the air. Just like how you would need to angle the throw to ensure it reaches its target, engineers must calculate the exact angle and speed for the rocket to escape Earth's gravity and travel to another planet. With MOM, every calculation is so precise that it safely reached Mars after executing multiple maneuvers, resembling a perfectly thrown ball landing where intended.

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