17.1.1. Simplifications and Assumptions

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.

In fluid mechanics, the flow can often be simplified by assuming it to be incompressible when the Mach number (a dimensionless number that compares the flow speed to the speed of sound) is less than 0.3. In this context, for fluids with a Mach number below this threshold, density variations are minor relative to other factors, allowing us to treat the flow as incompressible. This simplifies the mathematical modeling since we can consider the density as constant.

Think about water flowing slowly in a pipe. Even if there's some slight change in pressure or temperature, the density of water doesn't change significantly because the flow speed is much lower than the speed of sound in water. This slow flow is analogous to the Mach number being less than 0.3, making it a good example of an incompressible flow.

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.

This chunk further emphasizes the significance of treating the flow as incompressible when the Mach number is less than 0.3. When we assume density to be constant, it greatly simplifies calculations and the formulation of equations within fluid dynamics. This assumption allows engineers to focus on other variables without worrying about fluctuations in density, simplifying analyses and predictions.

Consider a tranquil lake where the water flow is very slow. As a result, the density of the water remains essentially unchanged. For engineers designing structures near this lake, they can treat the water's density as a constant when evaluating the forces acting on the dam walls, since they are dealing with incompressible flow.

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 height, volumetric flux, okay? So, please do not confuse, this is a different equation.

The distinction between mass flux (mass per unit time) and volumetric flux (volume per unit time) is critical in fluid dynamics. In cases where the density can be treated as constant, volumetric flux becomes a more useful parameter since it relates flow directly to this specific area and velocity. The equations become more straightforward, allowing easier calculation of how much fluid passes through a given surface in a given time.

Imagine a garden hose. As you increase the velocity of water flowing out of the hose without changing the diameter, you are effectively increasing the volumetric flux. The volume of water coming out per second can be easily understood just by considering how fast the water is shooting out and the size of the hose. This principle is similar to what engineers consider when calculating fluid flow in pipelines.

So, the velocity field knowledge is required for mass conservation equations. So, that way, whenever you take fluid mechanics problems, first you think what could be approximate velocity field, what could be the velocity direction.

To effectively apply mass conservation equations in fluid mechanics, understanding the velocity field in which the fluid flows is crucial. This knowledge allows us to gauge how fluid enters and leaves a control volume—defined as a specified region through which fluid flows. By analyzing the velocity at the surfaces of this control volume, it becomes possible to derive mass flow rates and ensure that mass conservation principles are appropriately applied.

Think about a busy intersection with traffic. To understand how cars are flowing through different directions, you would need to know their speeds at various points (the velocity field) as they enter and exit the intersection. Similarly, in fluid mechanics, knowing how quickly fluid moves at the boundaries of a control volume helps engineers design systems that accurately manage fluid flows, ensuring safety and stability.

If I have a pipe flow like this, you can anticipate it. As we discussed earlier, the velocity will be 0 near the wall, velocity will be maximum at the center and so there will be velocity distribution.

In typical pipe flow, the velocity distribution is not uniform. The fluid moves fastest in the center of the pipe due to lower resistance from the pipe walls, whereas the fluid near the walls experiences significant drag, resulting in a velocity of zero at the wall. Understanding this distribution plays a crucial role in designing fluid systems, as engineers use this data to determine pressure drops and flow rates in piping systems.

When stirring a large pot of soup, you’ll notice that the liquid in the center moves around faster than the soup near the surface or touching the sides of the pot. This is similar to what happens in a pipe; the center has the most motion, while the sides are held back, which must be considered during fluid flow calculations.

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