14.1.1. Conservation of Momentum

The conservation of momentum which is required for you to know it, how the fluid particles are moving it, what could be the force exerting on that, what could be the velocity. Similar way, we can understand the energy conservation which plays a major role for us when the fluid comes from one location to other locations, how much of work is done by the fluid or into the fluid.

This chunk introduces the concept of conservation of momentum, emphasizing its importance in understanding fluid motion. It suggests that knowing how fluid particles move, the forces at play, and the velocities involved are essential for grasping momentum conservation. Furthermore, it connects momentum conservation with energy conservation, hinting at how these principles interplay during fluid movement.

Think of a river carrying leaves downstream. The leaves represent fluid particles, and their speed and the force of the water determine how quickly and effectively they move. The energy of the flowing water helps carry these leaves, demonstrating both momentum and energy conservation!

Similar way, whether there is heat transfers happening which you can feel it, if there is a temperature gradient there will be heat transfer either to the surrendering of the systems or into the systems or out of the system, that is what we can do.

This chunk explains how heat transfer is connected to energy conservation. It highlights that when there is a temperature difference (gradient), heat will naturally flow in a direction towards equilibrium, either into a system or out of it. This principle mirrors the concepts in momentum conservation and illustrates the interconnectedness of different conservation laws in physics.

Consider a hot cup of coffee in a cool room. The heat from the coffee warms the surrounding air — this is heat transfer due to a temperature gradient. Similarly, as the coffee cools, it demonstrates energy conservation, as the heat is not lost but transferred to the environment.

So, to summarise this, that means, we all know that there are three energy conservation principles that we follow in solid mechanics when you consider as a system: conservation of mass, conservation of linear momentum which is Newton’s law, and the conservation of energy which is the first law of thermodynamics.

In this chunk, the speaker summarizes three fundamental conservation principles: conservation of mass, linear momentum, and energy. The conservation of mass states that mass cannot be created or destroyed in a closed system. The conservation of linear momentum, based on Newton's laws, indicates that the total momentum in a closed system remains constant unless acted upon by an external force. Lastly, the conservation of energy aligns with the first law of thermodynamics, emphasizing the transformation rather than the creation of energy.

Imagine a perfectly balanced see-saw with two kids on either side. As one goes down, the other goes up without loss of total system energy, representing momentum and energy conservation perfectly in motion. Mass conservation can be visualized with a pizza; cutting it into slices doesn't change the total amount of pizza — it simply redistributes it.

Now, let me define two types of properties that we have; one is called extensive property and the other is the intensive property. The extensive property which is considered as proportional to the amount of mass.

This chunk defines two key types of properties in thermodynamics: extensive and intensive. Extensive properties depend on the mass of the system; examples include mass, volume, and total energy. Conversely, intensive properties do not depend on mass; examples include density and temperature. Understanding these properties helps in analyzing systems under conservation laws.

Think of a balloon. As you blow more air into it, both the volume and the mass increase, showcasing extensive properties. However, the temperature of the air inside remains the same regardless of how much air you add, demonstrating intensive properties.

Now, we will go to derive Reynolds transport theorem. The derivation of the Reynolds transport theorem are available in almost all the fluid mechanics books.

This chunk transitions to more advanced concepts by introducing the Reynolds transport theorem (RTT). This theorem establishes a relationship between conservation laws at the system level and the control volume level. It simplifies the analysis of fluid flows, making it easier to apply laws of conservation in practical scenarios.

Consider a large lake as a control volume. If you pour in a bucket of water (a system), RTT helps explain how the water's movement affects not only the lake but also the waves, temperature, and movement of fish, illustrating how different conservation principles interact in larger bodies of water.

The idea for me is to introduce the Reynolds transport theorem so that you can easily understand it. But the step wise derivations, if you are not understanding it, I could suggest you to follow any of the fluid mechanics books...

In this chunk, the focus is on introducing how to derive the Reynolds transport theorem. The speaker acknowledges that while the derivation can be complex, textbooks provide a structured approach to understanding it. The emphasis is on recognizing the significance of RTT in fluid dynamics as a fundamental tool to connect system behavior with control volume analysis.

Imagine you are trying to explain how a swimming pool's water level changes when someone dives in. The RTT helps you communicate the relationship between the swimmer's volume (the system) and the overall water level change (the control volume) — a practical demonstration of this theorem in action.

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