3.1.4. Basic Concept of Control Volumes

We use as a control volume considers there is a dicks is mountain over a decks and we have the control volumes to just to estimate how much of force is acting on these dicks.

In fluid mechanics, a control volume is a predefined region in space where we analyze the behavior of fluids. When we consider a control volume, we focus on the forces acting on it to understand how fluid flows in and out. For example, if you have a flat surface or disk in a flowing fluid, we can analyze how much force the fluid applies to the disk.

Imagine a water faucet pouring water into a sink. The sink acts as a control volume. We can analyze the force of water hitting the bottom of the sink, the level of water rising in it, and how water flows out of the drain.

When you do these types of things, the interior part in the control volumes, we do not know anything about this. That means the interior part of this control volume we consider as a black box.

The term 'black box' refers to the idea that we do not analyze the internal characteristics of our control volume. We focus solely on how fluid enters and exits, without knowing the specific interactions or properties happening inside the volume. This simplification allows us to derive important equations without requiring detailed internal knowledge.

Consider a vending machine. When you insert money and make a selection, you don’t need to understand how the machine sorts and dispenses the items inside—it’s a 'black box' to you. You just care that your desired item comes out, similar to how we care about fluid entering and leaving a control volume.

If you look at the same problems if I go for a next levels where each point within the flow domains that is what is the flow domains.

The differential approach involves analyzing fluid flow at every individual point, rather than looking at the entire control volume as a whole. By understanding the velocity, pressure, and density at each point within the flow, we can derive more detailed insights about the fluid behavior compared to the integral approach, which gives us broader trends.

Think of a weather map. An integral approach might show average temperatures in a region, while a differential approach would analyze temperatures at every weather station in that region. This detailed approach provides a more accurate depiction of local weather conditions.

The basic idea comes here is that the my control volume which is have the dimensions of like if it is the x this is the y and this is the z coordinate.

In the fluid analysis, control volumes often have dimensions defined by x, y, and z coordinates. When we make these dimensions very small (dx, dy, dz approaching zero), we can derive differential equations that relate to mass and momentum in fluid flow. This means our control volume can represent an infinitely small point, allowing us to analyze fluid behavior at a more granular level.

Imagine a single grain of sand in a beach. While the beach as a whole may be a large stretch of land, analyzing just that grain gives insight into its composition. Similarly, by reducing control volume dimensions, we can analyze fundamental properties of the fluid at that scale.

So we will get 4 partial differential equations which will be the one will come from mass conservation equations, three vector form equations of linear momentum.

Let's discuss the four key equations derived from our analysis. The first is the mass conservation equation, which ensures that mass is neither created nor destroyed within the control volume. The other three equations relate to the conservation of momentum and encompass the three spatial dimensions (x, y, z). Together, these equations describe how the fluid behaves under various forces.

Consider traffic flow at a busy intersection. The mass conservation equation is like ensuring the total number of cars entering the intersection equals those leaving over a time period. The momentum equations can relate to how traffic changes direction and speed based on traffic signals and the movements of other vehicles.

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