6.1.2. Momentum Influx and Outflux Components

So based on Reynolds transport theorems, we can write for linear momentum equations. We can write that the sum of the force is acting is equal to same path, change of is equal to since it is a one inflow and one outflow, so we will have...

In this step, we start by using the Reynolds transport theorem, which provides a framework to relate the rate of change of quantities to the flow into and out of a control volume. The theorem simplifies our linear momentum equations by stating that the net force acting on the system will equal the difference between the momentum entering and leaving the system. Because we are dealing with one inflow and one outflow, we can express this as the sum of forces equal to the total momentum influx minus the total momentum outflux.

Think of this scenario like a busy highway. If we consider a section of the highway as our control volume, the cars entering and leaving represent the influx and outflux of momentum. Just like the total number of cars in that section will reflect how many enter versus how many leave, the total momentum entering the control volume must balance with that leaving.

So that is what is outflux, mass outflux. Sorry, this is what momentum outflux and this is what momentum influx components. And here we can assume it the velocity 1 and being a steady flow again, this part becomes zero, as it is a steady flow conditions.

Next, we differentiate between the momentum influx and outflux components. The momentum influx relates to the mass and velocity of the fluid entering the control volume, while the outflux refers to the mass and velocity of the fluid leaving it. Since we assume a steady flow, any changes in momentum storage within the control volume are negligible, allowing us to focus solely on the momentum flowing in and out.

Imagine a water tank with a steady inflow and outflow of water. If you fill the tank at the same rate as water is being drained, the water level remains constant. Similarly, when fluid flows in and out at a steady rate, we don't need to worry about changes within the tank, allowing us to simplify our calculations.

So we will play along the x directions to get it. If I apply in the x direction, we will get it the sum of the force F x will be...

When calculating force components, we apply the linear momentum equations specifically along the x-direction. This involves computing the force based on the momentum flux components. The equation simplifies to show how the velocity, area, and density of the fluid relate to the overall force acting on the system. By substituting the values of mass flux and velocity into the force equation, we can determine the net force exerted in that direction.

Think of pushing a swing. The force you exert pushes the swing forward (much like the momentum in our equations). If you push harder (increasing your force), the swing moves faster and reaches higher - similarly, changing fluid velocity influences the overall forces calculated here.

Similar way if I apply in the y directions, I am not doing for the y directions detail equations. We will also get it the...

In calculating the resultant force, the process is similar for the y direction. Even though this section skips some detailed equations, we still recognize that each directional force contributes to the total force acting on the system. By summing up the x and y components, we can derive the overall or resultant force acting through the control volume.

Consider the force needed to lift a box: you can think of it as needing to account for both the upward lift (y-direction) and any pushes or pulls from the side (x-direction). The overall effort you put in combines both of these efforts, just like we combine forces in our equation.

As we know this F x and F y value then we can compute resultant force which will be...

Here, we take our computed forces from the x and y directions and calculate the resultant force using vector addition. This is crucial because the resultant force gives us the total force acting on the control volume, enabling us to see the full impact of the momentum changes inside our system.

This is like finding the overall direction and magnitude of wind walking through an open field. While the wind may be pushing from different sides at different strengths, when you combine those effects, you understand how it ultimately pushes you as you walk.

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