14. Introduction to Conservation Principles

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 concepts of momentum and energy conservation in fluid dynamics. Conservation of momentum refers to the principle that the total momentum of a fluid system remains constant unless acted upon by an external force. For instance, if a fluid is flowing through a pipe and experiences no external forces, its momentum before and after any point in the pipe remains constant. Energy conservation in this context looks at how energy is transferred via work and is critical in understanding how fluids move from one location to another, influencing both their velocity and the forces they create.

Imagine pushing a soccer ball. The moment you apply a force, you are changing the momentum of the ball. If you kick the ball straight down the field, it moves forward due to the momentum you provided. If no one else hits it, it keeps rolling until friction eventually slows it down, demonstrating momentum conservation. Likewise, when you pour water from one container to another, you are doing work on the fluid; the more force you use to lift the water, the more energy is transferred.

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 focuses on heat transfer, highlighting the effect of temperature gradients on fluid systems. A temperature gradient occurs when there’s a difference in temperature between two areas, leading to heat transfer from the hotter to the cooler area. In fluids, this principle helps us understand phenomena such as how hot air rises and cold air sinks, along with more complex processes in thermodynamics relevant to engineering and natural systems.

Think of a hot cup of coffee left on a table. The heat from the coffee (hot) transfers to the cooler air around it until they reach thermal equilibrium. If you put a cold spoon into the coffee, the spoon will absorb heat and feel warm while the coffee cools. This is an everyday example of heat transfer due to temperature differences.

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.

This chunk highlights three fundamental conservation principles relevant in mechanical systems, emphasizing that they also apply to fluid mechanics. These principles include the conservation of mass, which states that matter cannot be created or destroyed; conservation of linear momentum, based on Newton’s laws; and conservation of energy, reflecting the first law of thermodynamics. Understanding these principles allows for solving various fluid mechanics problems effectively.

Consider a water balloon. When you squeeze it, you can feel the pressure (momentum) change. The mass of water inside remains constant (mass conservation), and the energy used to squeeze it is transferred to the water, increasing its pressure. This reflects principles at play in fluids concerning energy conservation.

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 distinguishes between extensive and intensive properties. Extensive properties depend on the amount of material, like mass and volume (the more material you have, the greater these properties are). Conversely, intensive properties, such as temperature or density, remain constant regardless of the quantity of material present. This understanding is vital in thermodynamics and fluid mechanics as it helps in categorizing what properties change with the system size and which do not.

Imagine a glass of water. If you have a small glass (less water), the temperature of the water stays the same (intensive property), yet if you fill up a big pitcher with water, the total mass and volume increase (extensive properties), but the temperature stays unchanged regardless of the water amount.

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 introduces the Reynolds transport theorem, which serves as a bridge between the analysis at a system level (where the mass and properties are considered as a whole) with control volume analysis (which examines the flow across a defined volume). The theorem is widely used to analyze and solve problems related to fluid flow by relating changes in the system and control volumes. Understanding this theorem is crucial for studying fluid dynamics as it provides integral forms for the conservation laws mentioned earlier.

Think of a busy highway with cars (the system) compared to a particular section of that highway (the control volume). The flow of traffic through a specific section can be analyzed using the Reynolds transport theorem to see how many cars enter and exit that section, correlating the changes in vehicle count to the conditions of the entire highway.

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