18.2. Introduction to Conservation of Momentum

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 this introduction, the significance of conservation of momentum is established, linking it to previous knowledge on conservation of mass. The speaker indicates that conservation laws are fundamental in understanding fluid mechanics and will build upon concepts from earlier classes, particularly focusing on how these laws relate to fluid flow.

Think of conservation laws like a bank account. Just as you cannot spend more money than you have without going into debt, in physics, you cannot lose mass or momentum without it accounting to other systems. Conservation of mass is like ensuring you always account for every penny spent or received in your fluid system.

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.

The Reynolds Transport Theorem is crucial in fluid mechanics as it simplifies the analysis of fluid systems by relating the behavior of a fluid within a control volume to the physical laws governing it. The theorem bridges the gap between the system (the fluid’s internal behavior) and the surroundings (the flow as it interacts with the environment).

Imagine trying to keep track of the water flowing in and out of a bathtub. The Reynolds Transport Theorem is like a set of rules that help you understand what's happening to the water level based on how much is coming in from the faucet and draining out of the tub.

As I told you, we discussed about the Reynolds transport theorem for conservation of mass and when we apply this conservation of mass to the Reynolds transport theorem, we have two basic assumptions, that is, with respect to time is it a steady or unsteady.

Here, the lecturer emphasizes the importance of recognizing whether a system is in steady or unsteady flow, which determines how conservation laws can be applied. Steady flow means that the properties of the fluid at any point do not change over time, while unsteady flow means they do. This distinction is vital when analyzing fluid behavior and applies equally to momentum conservation.

Consider a traffic flow at a busy intersection. If every car passes through at the same rate (steady flow), predicting traffic patterns is easier. But during rush hour (unsteady flow), the influx and outflux of cars changes constantly, making predictions much more complex.

Then, with respect to density change or the variations of the density, we divide it, flow is compressible or incompressible.

The nature of fluid flow can significantly affect the behavior of momentum. Compressible flows involve significant changes in density (like gases), while incompressible flows assume density remains constant (like water). Understanding this classification is necessary because it dictates how equations and conservation laws are formulated and solved.

Think of a balloon. If you squeeze it (compressible flow), changes occur in its density and volume. But when water flows from a sink (incompressible flow), its density remains the same, simplifying the analysis of its flow behavior.

So, we can have two types of approximations, steady compressible, steady incompressible.

This chunk discusses the approximations that make solving fluid flow problems easier. By assuming steady conditions (where fluid properties don't change with time), and considering whether the fluid is compressible or incompressible, engineers can simplify equations and make predictions regarding fluid behavior more manageable.

Think of a diving board. If someone jumps off and you observe their motion steadily as they dive down, you're assuming steady conditions. If they were to change form drastically mid-dive, like going from a simple dive to a belly flop (similar to shifting fluid density), you'd need to account for those changes, complicating your predictions.

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 steady flow conditions, certain mathematical terms in fluid equations simplify to zero, which makes calculations easier. This directs focus on the surface integrals, disregarding changes over time, allowing for more straightforward problem-solving in fluid dynamics.

Imagine tracking a train on a specific track. If the train moves at a constant speed (steady), you can forecast its position at any given time without needing to consider sudden stops or speed changes, simplifying your calculations.

So, if you look at this project which was initiated or commissioned early in 1950s and 60s, generating and installing hydro power projects about 1300 megawatt power.

The speaker highlights real-world applications of fluid mechanics principles in the design of large-scale hydropower projects, which require a sound understanding of conservation principles to ensure efficiency and effectiveness in energy generation.

It's akin to building a large dam to harness river water for electricity. Engineers apply fluid mechanics theories to ensure that water flows optimally through turbines, generating maximum energy just like a bank efficiently manages deposits to maximize profit.

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