17. Incompressible Flow

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.

Incompressible flow occurs in fluid dynamics when the density of the fluid does not change significantly. This usually happens when the Mach number, a dimensionless quantity used to represent the ratio of the speed of the object to the speed of sound in the surrounding medium, is less than 0.3. Under these conditions, we can assume that changes in density are negligible compared to other factors in the flow system. This simplification makes calculations and understanding fluid dynamics much easier.

Think of a slow-moving river. The water is flowing gently, and even when there is a slight change in speed, the density of the water doesn't change much. Now imagine a fast-moving stream or a jet where speed approaches sound. Here, density would play a more significant role, leading to the ideas of compressible flow. However, at lower speeds, like the river, we can simplify our calculations by treating the water as incompressible.

So, we can assume the flow is incompressible nature. 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.

When we assume that the flow is incompressible (as indicated by the Mach number being less than 0.3), we can treat the density as a constant. This greatly simplifies the equations we use in fluid mechanics because many equations, such as those governing mass conservation, rely on density variations. By treating density as constant, we can focus on the flow rate and velocity without having to constantly account for density changes.

Think of filling a balloon with water. As long as you are adding water slowly, the density remains relatively stable, and you can predict how much more water it can hold without worrying about changes in density. However, if you were to fill a balloon with air rapidly, the density of the air could affect what happens inside, making that situation more complex.

So, instead of the mass flux we are now talking about volumetric flux. That means, if you multiply the velocity into area, then what you get is unit volume per unit time, volumetric flux, okay?

In the context of incompressible flow, we often talk about volumetric flux instead of mass flux. Volumetric flux is calculated by multiplying the cross-sectional area through which the fluid is flowing by the fluid velocity. This means that volumetric flux tells us how much volume of fluid passes through a given area per unit of time, simplifying our calculations since we treat density as a constant.

Imagine a garden hose. If you take the cross-sectional area of the hose and multiply it by the speed at which water is flowing out, you'll get a measure of how much water is flowing out per second. This is similar to our volumetric flux, making it easy to assess how much water can reach your garden without worrying about weight or density.

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.

For effective application of mass conservation principles, understanding the velocity field is crucial. This includes knowing whether the velocity is constant across a given area or if it varies. In flows where the velocity is uniform, calculations become straightforward, whereas, in cases of varying velocity, we might need to apply integrals across surfaces to obtain accurate results.

Think of a crowded room where people are moving in different directions. If everyone walks at the same speed, it’s easy to compute how many people exit the room in a minute. However, if some jog while others walk slowly, understanding how fast each person moves in different areas becomes important for knowing how quickly the room empties.

The flow coming and going out. If this is simple in and out system, you can know this velocity distribution area, find out the discharge from inflow and outflow, equate it, then you can solve the problem.

In many practical applications like pipe flow, we encounter situations where the velocity distribution is not uniform. For instance, fluid velocity can be maximum in the center of a pipe and decreases toward the walls. Understanding and calculating this velocity distribution are necessary for determining the total discharge and solving the fluid flow problem effectively.

Picture water flowing through a wide pipe. The speed of the water at the center of the pipe is faster than near the walls due to friction. Adjusting for these variations allows engineers to calculate how much water flows through the pipe and how quickly it can fill a tank.

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