17.3.2. Flow Classification

Now, let me come in to the, if the flow is incompressible. Again, we have a lot of simplifications. Again, I can repeat it. The flow systems when you have mac number less than 0.3, okay, whether it is gas, whether it is a liquid or any flow system, if you think that the within the flow system the flow becomes less than the mac number less than 0.3, then there will be density variation, but that variation of density is much much negligible comparing to other components. So, we can assume the flow is incompressible nature.

In fluid mechanics, an incompressible flow is one where the density of the fluid remains constant. This typically applies to both liquids and gases when their flow speeds are low, specifically when the Mach number, which is a dimensionless quantity used to represent the speed of a flow relative to the speed of sound, is less than 0.3. When this condition is met, any changes in density due to variations in flow speed are minor enough that they can be ignored. Thus, for calculations involving such flows, we can simplify our models by treating the fluid's density as a constant.

Think of water flowing through a garden hose. When you open the tap slightly, the flow is relatively slow, and the density of the water doesn't change significantly as it moves through the hose. So, for practical purposes in this scenario, we treat the water as incompressible.

Again, I am going to summarise that. When you have any flow systems, mac number is less than 0.3, so we can use flow as incomprehensible flow, density does not vary significantly. So, density becomes constant, as density becomes constant, as you know it, it is very simplified problem what we are going to solve. So, density varies negligible, as the density variation is not significant and beta equal to 1...

In flow systems where density is treated as a constant, we can simplify our analysis significantly. For instance, mass flow rate (mass entering or leaving a control volume) can be calculated using a volumetric flow rate, which is simply the product of the flow speed (velocity) and the cross-sectional area through which the fluid flows. This relationship becomes especially useful because it enables engineers to calculate how much fluid is flowing without worrying about density variations, as long as the conditions specified (Mach number) hold.

Imagine filling a bathtub with water. The rate at which the water rises (volumetric flow) can simply be calculated using the opening size (area) and how fast the water flows from the tap (velocity). Here, the water's density remains constant, so we only need to know these two factors to know how quickly the bathtub will fill.

So, what I am looking at is again summarised here. To solve this mass conservation equation I should have knowledge on velocity field. I should know how the velocity varies or I should know whether the velocity is a constant or the velocity varies.

To apply the mass conservation principles effectively, it's crucial to understand the flow's velocity distribution across the control volume. When dealing with incompressible flows, knowing whether the velocity is uniform or varies across the flow path helps determine how we can calculate inflows and outflows. For example, in a pipe, the water flow velocity is typically highest in the center and zero at the walls due to friction. This non-uniform distribution affects how we compute the overall flow rates through the pipe.

Visualize a river. In the center, the water flows rapidly, while the edges are slower due to contact with the riverbank. If you want to know how quickly the water is moving to gauge flood risk, you must understand where the fastest flow occurs – typically in the center – rather than just averaging across the entire width of the river.

If you know the velocity variations on this control surface, then I can solve the problem. So, basically, when you apply the mass conservation equation your knowledge of velocity variation is important to you.

When solving problems in fluid mechanics, particularly regarding conservation of mass, understanding the average velocity across a defined section is vital. If the flow is not uniform, we can calculate the average velocity by integrating the velocity profile across the section of interest. This average velocity allows us to compute how much fluid passes through a specified area, which is important for applications like designing ducts or channels.

Consider how you might measure average speed on a busy road. Instead of monitoring every single vehicle at different speeds, you might take a few measurements at various points, then calculate an overall average. Similarly, in fluid flows, even if water flows at varying speeds, finding an average lets us make accurate calculations.

Again, please remember we are still doing the mass conservation equation. The volumetric form has come in because you have taken out the density.

Though we work with average velocities and volumetric flow, it’s important to remember that these calculations are rooted in the conservation of mass principle. By treating density as constant, we can extract it from our equations, leading to a focus on volume-related calculations, which simplifies our approach to understanding fluid flows in engineering applications. However, we still fundamentally abide by the mass conservation law.

Think about pouring milk into a glass. You pour until it’s full, and while you’re aware of the volume of milk you’re using, you don’t consider that its weight (mass) is also part of the equation. You simply ensure the glass can hold the volume of liquid you are pouring in.

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