2. Conservation of Momentum

So, starting with the last slide, where we left in the last lecture and the we derived the general form of Reynolds transport theorem here. So, this is the general form and now in the upcoming lecture and slides, what we are going to do is we will apply this Reynolds transport theorem for derivation of different conservation equations.

In this introductory statement, the speaker revisits the concept learned in the last lecture about Reynolds Transport Theorem, which provides a framework for deriving various conservation equations in fluid dynamics. The focus will shift towards applying this theorem specifically for motion conservation, emphasizing the practical use of these concepts in hydraulic engineering.

Think of the Reynolds Transport Theorem like a set of guidelines that help you track fluid movement as it shifts from one place to another. Just as you may track the flow of people in a busy store to understand how many enter and exit, this theorem helps us analyze and conserve quantities like momentum in fluid dynamics.

So, now we are moving from a system to a control volume. So, we will see how when we apply B system to mass, what is going to happen. We will see about linear momentum.

This chunk discusses a shift from analyzing entire systems to focusing on a control volume, which is a specific region in space through which fluid flows. In this context, the conservation of mass and linear momentum will be highlighted. The control volume allows for easier application of conservation laws by focusing only on the mass within that defined area, rather than the entire system.

Imagine you are observing the flow of a river through a section you’ve marked out with ropes on either side. By focusing only on that section, you can measure how much water flows in and out, making it easier to understand the dynamics of flow rather than trying to consider the entire river at once.

So, now conservation of mass, how can we apply what we have learned in Reynolds transport theorem to the conservation of mass. So, B is total amount of mass in the system, when we do the conservation of mass B which we saw in Reynolds transport theorem is the total amount of mass in the system.

This section explains the concept of mass conservation in relation to the Reynolds Transport Theorem. The total mass in a system is denoted as 'B' and is crucial for analyzing fluids. The conservation of mass assures that the amount of mass entering a control volume equals the amount exiting it over time.

Think about a bathtub. If you have the faucet running, the water level will rise until it reaches the overflow drain, at which point the amount of water entering equals the amount leaving. This is similar to how we analyze mass conservation in fluid flows.

So, what is this, we know that in conservation of mass the total mass of the system with respect to the time DM, DM /Dt = 0. So, we can simply write in the control volume sys... This equation here, tells us that mass leaving minus mass entering is minus rate of increase of mass in the control volume.

Here, the continuity equation is introduced, stating that the change in mass within a control volume over a period of time is zero. This equation shows that the mass flowing into the control volume minus the mass flowing out equals any accumulation of mass inside that volume, which is a foundational concept in fluid mechanics.

Imagine a busy subway station. The number of people entering and exiting the station can be tracked. If the number of people entering matches those exiting, then the overall number of people in the station remains constant, just as the continuity equation indicates for mass in fluid flow.

So, now we are going to see, so, unit vector n cap here is normal to the surface... So, if the mass in the control volume is constant...

In discussing momentum conservation, the text illustrates that the unit vector normal to the surface helps in calculating the momentum at the control surfaces. If the mass remains constant within the control volume, it implies that the outgoing momentum must balance the incoming momentum, which is fundamental to understanding how forces act on moving fluids.

Picture a soccer game. When the ball is kicked, the momentum changes as the ball enters the goal area versus when it rebounds. Here, just as the soccer ball exchanges momentum with the goal posts (or players), fluid momentum is conserved as it moves through different cross-sections of a control volume.

So, now the linear momentum equation. An example here is, if you see this figure here... So, here we can actually apply the momentum conservation linear momentum.

The linear momentum equation is introduced through an example where water is shot against a wall. Here, the momentum before hitting the wall is compared to the momentum after, demonstrating a change in momentum. This scenario serves to highlight the practical application of momentum conservation principles in real-life fluid mechanics.

Think of it like throwing a basketball against a wall. The ball has momentum as it approaches and comes to a stop upon hitting the wall. The change in momentum relates directly to the force applied to the wall, similar to the linear momentum discussion in fluid scenarios.

So, now let’s discuss some examples of mass conservation, the flow out of a reservoir... So, Q out will be simply - area of reservoir into d h / dt or we can simply write the rate of fall of height with respect to time is this - Q / area of the reservoir.

This chunk introduces practical applications of the conservation of mass and linear momentum through example problems. By calculating the flow of fluid from a reservoir, the relationship between flow rate, area, and height is highlighted, illustrating how these principles are intertwined and useful in engineering calculations.

Visualize a reservoir filled with water. If you have a spout at the bottom, and you know how fast water flows out, you can easily calculate how quickly the water level drops. This calculation is similar to how engineers use mass conservation principles to manage water resources.

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