15.2.2. Derivation of Conservation of Mass

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.

In this introduction, the speaker sets the stage for discussing the conservation of mass in fluid mechanics. The conservation of mass states that mass cannot be created or destroyed in a closed system, and it emphasizes the importance of the Reynolds transport theorem, which provides a relationship between the system, its properties, and the control volume used for analysis.

Think of a balloon filled with air. If no air enters or leaves the balloon, the total amount of air (mass) inside remains constant, demonstrating conservation of mass.

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.

This chunk introduces the concept of control volumes, which are an essential part of fluid mechanics. A control volume is a specific region in space through which fluid can flow. The relationship established by the Reynolds transport theorem connects how mass flows in and out of this volume. Understanding this relationship is crucial for analyzing real-world fluid systems.

Imagine a river as a control volume. The water that flows in and out of a specific section of the river represents the mass moving through this control volume, which helps us track water quantity changes.

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

This segment categorizes control volumes into three types: Fixed control volumes remain stationary; moving control volumes move along with the substance, like a moving boat; and deformable control volumes can change shape over time, like a balloon being squeezed. Understanding these types helps in choosing the appropriate model for fluid flow analysis.

Consider a water bottle (fixed), a speedboat (moving), and a flexible bag of water (deformable). Each demonstrates how the fluid behaves differently based on the type of control volume.

When you have the control volume, through the control surface there is mass influx coming into the control volume... Today, I will focus on mass conservations only.

This chunk explains the concept of mass inflow and outflow through the control surfaces of the control volume. The net mass entering the control volume must balance the mass leaving to maintain conservation of mass. This is crucial for analyzing fluid flow problems effectively.

Think of a bathtub: if you turn on the faucet (influx) but forget to drain the water, the water level will keep rising. To keep the level constant (conserving mass), the inflow must equal the outflow.

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.

This section discusses how to derive the conservation of mass equation by analyzing the mass entering and leaving the control volume. The fundamental principle is that the change in mass within the control volume over time equals the difference between mass inflow and outflow.

Consider a factory that produces candy: the amount of candy produced (mass allocated to the control volume) must equal the candy packaged and sent out (mass leaving), ensuring the factory manages its production effectively.

Today, we will talk about how we can derive the conservation of mass... we will talk about some examples, real life examples...

This chunk emphasizes the practical applications of conservation of mass in real-world scenarios and how theoretical principles can be applied to solve complex fluid mechanics problems. It mentions the importance of choosing the right control volume carefully for accurate analysis.

In civil engineering, when constructing bridges or dams, engineers must consider how water flows around these structures. By applying the principle of mass conservation, they can ensure that their designs safely accommodate water flow.

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