14.3.1. Defining Influx and Outflux

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.

This chunk explains the foundational principles of conservation that govern fluid mechanics. The three key conservation laws are: 1. Conservation of Mass: This principle states that mass cannot be created or destroyed in a closed system. 2. Conservation of Linear Momentum: According to Newton’s laws of motion, the quantity of motion of a body is directly proportional to the net force acting on it and is in the direction of that force. 3. Conservation of Energy: This principle states that the total energy of an isolated system remains constant. Understanding these principles is crucial as they serve as the framework for analyzing fluid flows in engineering.

Think about a closed water bottle. No matter how you shake it or move it, the amount of water inside remains constant (conservation of mass). When you squeeze it, the water pushes against the sides, demonstrating how the motion of the water (momentum) is affected by the force you apply. Lastly, if you heat the water, it can turn into steam, showing how energy can change forms but the total energy remains the same.

Let us understand extensive and intensive properties. The extensive property is proportional to the amount of mass; as mass increases, extensive properties increase. Conversely, intensive properties are independent of mass.

This chunk helps in distinguishing between two types of properties in fluid mechanics. Extensive properties, such as mass, momentum, and energy, vary with the quantity of substance in the system—more mass means more extensive property. Intensive properties, such as temperature or pressure, do not depend on the amount of substance and remain constant regardless of the system's size. Understanding this distinction allows engineers to choose the appropriate formulas and approaches in fluid analysis.

Imagine a glass of water. The total volume of water in the glass (an extensive property) increases if you add more water. However, the temperature of the water (an intensive property) may remain the same regardless of the volume of water you're measuring; it’s the same temperature whether you have a cup or a bucket.

Let me define the Reynolds transport theorem which establishes the relationship between the conservation law at the system level and the conservation at the control volume level.

The Reynolds transport theorem is a fundamental equation in fluid mechanics that relates the change in a property of a control volume to the property entering or leaving that volume. It bridges the gap between the system approach (where we consider a fixed amount of fluid) and the control volume approach (where we consider a volume through which fluid flows). This theorem is vital for deriving equations governing fluid motion, such as the equations of motion for fluids.

Consider a swimming pool where you’re measuring how much water is added or removed. The Reynolds transport theorem allows you to account for the water volume (the system) as it flows in and out of the pool (the control volume). The focus is on understanding both how much water is entering or leaving and how that affects the overall amount of water in the pool.

I will consider that as the system for me at time t. That means I consider whatever fluid particles are there within this control volume that is what the system is.

In this chunk, we learn about the concept of a control volume. A control volume is a fixed or moving volume in space through which fluid can flow. When analyzing fluid behavior, we focus on what happens inside this volume at various times (t and t+dt). This helps establish how fluid properties change as they flow in and out of the control volume over time.

Imagine a water slide at a water park. As kids slide down, they are like fluid particles, moving in and out of a designated zone (the control volume). At any given moment, you can examine how many kids are on the slide and how many have just exited, helping you understand the flow of children through that area.

The time rate of change of B in the system as a definition t + dt minus the system at the t level. If we just do a rearrangement of this equation, we will understand it at the control volume level.

Here, the chunk describes the mathematical framework needed to analyze changes in properties over a control volume over time. By calculating the difference in a property (B) at specific time intervals, we can establish how that property behaves as fluid flows in and out of the volume. This mathematical approach is essential for deriving equations related to fluid dynamics.

Think of a bank account where you track deposits and withdrawals. The difference between the account's balance at two different times (like t and t+dt) tells you the rate of change in your balance due to the money flowing in (deposits) and flowing out (withdrawals). In fluid mechanics, we’re doing something similar but with fluid properties instead of money.

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