3.1.2. Integral Approach

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.

The integral approach in fluid mechanics involves analyzing fluid flow using a defined control volume. A control volume is an imaginary box or area we choose within a fluid system to analyze forces and fluid behavior without needing to understand every detail inside the volume itself.

Imagine a swimming pool. Instead of analyzing every single drop of water, you can just look at a specific section of the pool (your control volume) to see how much water is flowing in and out, and how forces might be acting on the sides of the pool.

So if you look at these problems, we consider a control volumes and we try to look at the velocity components, what could be expected inflow, outflow, also the outflow here and what could be the velocity vectors and based on that we apply the mass considerations and the momentum equations to estimate the force.

When we analyze a control volume, we pay attention to the velocity of the fluid entering and exiting the volume. These velocities determine how much fluid is flowing in and out, and we apply mass conservation and momentum equations to calculate the forces acting on the control volume, such as pressure from the fluid.

Think of a hosepipe watering a garden. The amount of water that comes in (inflow) must equal the amount that comes out (outflow), unless it's absorbed by the soil. By measuring how hard the water is flowing in and out, we can determine the pressure and force on the garden’s soil.

But if you look at that, when you do these type 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.

In fluid mechanics, the integral approach allows us to treat the interior of the control volume as a 'black box', which means we do not need to know all the specific behaviors of the flow inside of it. We can focus on the mass inflows and outflows to derive equations that describe the overall behavior of the system.

Consider a factory assembly line. You might not need to know every detail of how each item is made (the black box), but you can still analyze how many items come in and out of the factory daily to measure efficiency.

But 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 differs significantly from the integral approach. Instead of treating the fluid flow as an aggregated whole within a control volume, the differential approach breaks down the flow into infinitesimally small points. This allows for detailed analysis at each point, taking into account local pressure, velocity, and density variations.

Imagine zooming in on a map. Instead of looking at a city as a whole, you would examine each street and building individually to understand traffic patterns. This detailed analysis helps in fine-tuning approaches for better traffic management.

If my dx is tending towards 0. The dy tendings towards 0 and the dz tending towards 0.

As we reduce the dimensions of our control volume down to near zero (dx, dy, dz → 0), we approach a point in space. In this scenario, we can derive differential equations instead of integral equations, focusing on the specific behaviors of the fluid at that point, leading to the formulation of conservation laws for mass and momentum.

Think of focusing a camera: as you zoom in closer, instead of capturing the entire scene, you start to see fine details—like each leaf on a tree. This enables you to capture the intricate interactions occurring at that singular point.

Basically, I will get the four basic equations. 4 basic partial differential equations.

From the differential analysis of fluid flow, we derive four fundamental partial differential equations: one for mass conservation and three for momentum conservation in three-dimensional space (x, y, z). These equations are interrelated, meaning changes in one affect the others, allowing us to understand the complete state of the fluid flow.

Think of a complex system like a classroom with students. How one student behaves (velocity) could affect the whole classroom dynamics (momentum), while the classroom must still adhere to its total capacity (mass conservation). Understanding these relationships helps in managing situations more effectively.

So the basic concept is that the for given control volumes the dimensions of the control volumes we have been reducing it.

In summary, the integral approach simplifies analysis by considering a large control volume and applying conservation principles, while the differential approach focuses on the local behavior of fluid flow at infinitesimally small points, providing detailed insights and the formation of precise equations governing the flow dynamics.

Consider a factory: the integral approach would analyze the total production output, while the differential approach would inspect each machine's output individually. Both are valuable for understanding the factory's efficiency.

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