15. Fluid Mechanics

Welcome all of you to this fluid mechanics course. Today, we are going to plan about conversion of mass. As you could remember it, then, in the last class we discussed about Reynolds transport theorem. So, the same Reynolds transport theorem will be used to derive mass conservation equation which is an important equation for any fluid flow problems.

This introductory statement sets the stage for the importance of mass conservation in fluid mechanics. It references the Reynolds transport theorem, which is a fundamental principle for understanding how mass is transferred within a fluid system. The focus today will be on deriving the conservation of mass equation, essential for analyzing fluid flow problems.

Think of a swimming pool. When you dive into the water, you displace a certain volume of water. The concept of mass conservation helps to explain that the water you displace must go somewhere—it can’t just disappear. Similarly, in fluid mechanics, mass conservation ensures that mass is neither created nor destroyed as fluids flow around.

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, it’s crucial to differentiate between a ‘system’ (where we can analyze a fixed amount of fluid) and a ‘control volume’ (a specified region in space through which fluid can flow). The Reynolds transport theorem provides a mathematical framework that relates changes in the system and the control volume, especially in terms of mass flow.

Consider a water bottle being emptied into a glass. The water in the bottle is a system, but as it flows into the glass, that glass represents the control volume. The Reynolds transport theorem helps us understand how mass (water) is moving from the bottle (system) through the ‘control surface’ (the bottle opening) into the glass.

So, there are three types of control volume: fixed control volume, moving control volume, and deformable control volume.

Control volumes in fluid mechanics can be categorized into three types: fixed, where the boundary is stationary; moving, where the control volume itself is in motion (like a boat in a river); and deformable, where the shape of the control volume changes over time (like a balloon being inflated). Each type presents unique challenges in analyzing fluid flow.

Imagine a moving car (moving control volume) pushing air out of the way. The air around the car’s exterior forms a 'fixed control volume'. Now, think about a balloon being blown up—this balloon represents a 'deformable control volume' as it expands and changes shape as more air is pumped in.

Today, we will talk about how we can derive the conservation of mass. As I said earlier, when you have the control volume, through the control surface there is mass influx coming into the control volume, going out from the control volume.

To derive the conservation of mass equation, we begin by considering the mass entering and leaving the control volume. The principle states that the mass inside a control volume changes based on the net mass flow through its boundaries. This leads to an important relationship: the change in mass within the control volume equals the mass inflow minus the mass outflow.

Think of a bathtub filled with water. If you let the water flow out through the drain while leaving the tap running, the conservation of mass principle helps you understand how the water level changes. The amount of water entering the tub minus the amount flowing out through the drain gives you the change in the water volume in the tub over time.

Now, let us go to very interesting applications. As you know it Indian Space Research Organisation launched the satellite which is called MOM programmes...

Fluid mechanics isn't just theoretical; it has practical applications like the design and function of spacecraft. For instance, the Mars Orbiter Mission (MOM) required fluid mechanics principles to calculate trajectories and handle the drag experienced during its launch and travel through atmosphere. This highlights the relevance of the principles learned in fluid mechanics.

Launching a rocket into space can be compared to a high-speed car driving through a flooded road. In both cases, fluid mechanics helps predict how to minimize drag and maximize efficiency, whether for the water around the tires or the air around the rocket.

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