18.2.1. Previous Discussion on Conservation of Mass

Welcome all of you to fluid mechanics course. Today, I am going to deliver lecture on conservation of momentum. As you know, in the last class we discussed about conservation of mass. Also, we have solved few problems based on the conservation of mass.

In the starting of the lecture, the professor sets the context for the current lesson on conservation of momentum by referencing the previous class where the conservation of mass was discussed. This implies that understanding mass conservation is crucial for grasping momentum conservation, as both are fundamental principles in fluid mechanics.

Imagine a water balloon being thrown. Just before it hits the ground, the mass of the water inside is conserved (it doesn't disappear). However, when it bursts upon impact, the momentum also changes, in that the direction and speed of the splashing water molecules become very different. This analogy illustrates how mass conservation sets the groundwork for understanding momentum.

Basically, we have been following the Reynolds transport theorem as a basic concept to apply the setup of the system into physical equations to the control volume level. And, then, at the control column level, we have approximation of extensive and intensive properties and as of now we have derived mass conservation equations.

The Reynolds transport theorem is a vital theorem in fluid mechanics that enables the transition from a control mass perspective to a control volume perspective. It helps in formulating equations that describe the conservation of mass for a fluid that is flowing in and out of a volume of interest, which allows for more practical applications in engineering.

Think of the Reynolds transport theorem as a way to track water going in and out of a bathtub. If you measure how much water you poured in and how much is draining out, you can maintain the balance to understand how full the tub is at any time, just as the theorem helps quantify mass in a flow system.

we have two basic assumptions, that is, with respect to time is it a steady or unsteady. So, the steady we do the approximations of many fluid flow problems which are steady problems. Then, with respect to density change or the variations of the density, we divide it, flow is compressible or incompressible.

When applying conservation of mass, it's crucial to determine whether the fluid conditions are steady or unsteady. A steady flow means that the fluid's velocity and other properties don’t change over time, while unsteady flow means they do. Additionally, fluids can sometimes compress (like gases) or remain incompressible (like liquids), which impacts how we apply mass conservation equations.

Consider traffic on a highway: in a steady state, cars continue to fill lanes without speed changes (like water in a pipe), but if a road construction starts, the flow becomes unsteady. Similarly, if some cars are compact and others are larger trucks, this mimics compressible vs. incompressible flow encounters.

So, when you have the steady assumptions, you can remember that the component of Reynolds transport theorem of time, differentiate components become 0 or the volume integral component part of the Reynolds transport theorem becomes 0.

In situations where steady flow conditions are assumed, certain terms in the Reynolds transport theorem can be simplified to zero, making calculations more manageable. This allows engineers to focus on surface integrals, which are necessary for practical applications without the complexity of changing properties over time.

This can be related to how a factory runs efficiently when all machines work at a constant speed without interruptions, leading to predictability. When everything is steady, calculations are simplified just like understanding the production flow becomes easier.

Thus, the problems become too simple as compared to if you have compressible flow. So, when you have a steady incompressible flow, most of the cases what we consider for flow devices or engineering applications, we can consider steady incompressible flow, then the problems become very simplified when you apply for Reynolds transport theorem.

In many engineering scenarios, the steady incompressible flow condition is a common assumption because it greatly simplifies the mathematics involved. By considering the density of the fluid constant, it reduces the number of variables that need to be accounted for in calculations.

Imagine using a flexible straw to drink juice. If the juice doesn't change and flows steadily, it's easier to drink and calculate how much you've had, similar to working with constant density. However, if bubbles appear, and foam starts, the dynamics of flow become more complicated, like considering compressible flows.

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