17.2.3. Applying Reynolds Transport Theorem

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 dynamics, the Reynolds transport theorem helps us analyze fluid motion in a simplified manner. When the Mach number is less than 0.3, both gases and liquids can typically be treated as incompressible fluids because the density of the fluid does not vary significantly with pressure changes. This assumption simplifies calculations since we can treat density as a constant.

Think of a water balloon. If you squeeze it gently (just like low Mach number flow), it barely changes volume. But if you were to squeeze a balloon filled with air hard, the air gets compressed. At low speeds, like the water balloon, the fluid behaves in a stable, incompressible manner.

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 incompressibility in flow with a Mach number below 0.3, we can treat density as a constant value throughout the system. This significantly simplifies our calculations and leads to straightforward equations to work with in fluid dynamics. Essentially, rather than tracking varying densities, we focus on volume flow rates.

Consider water flowing through a garden hose. Whether the water is pouring fast or slow, its density doesn’t change. In the same way, for low-speed flows (Mac < 0.3), we treat fluids like water in the hose as having a constant density.

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, we can talk about volumetric flux.

In incompressible flow scenarios, the concept of mass flux (mass flow per unit area) can be simplified to volumetric flux (volume flow per unit area). This means instead of worrying about how density affects mass flow, we can simply calculate how much volume flows through a specific area within a given time period.

Think about how we measure water flowing from a tap. We often say liters per minute without needing to calculate density because water’s density remains constant. This simplifies how we determine how much water is available by just looking at the volumetric flow.

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 pipe flow, the velocity of fluid is not constant across the entire cross-section. It tends to be highest in the center and decreases to zero at the pipe walls due to friction. This creates a velocity profile, where knowing how this velocity changes across the surface is crucial for solving fluid dynamic problems.

Imagine driving a car in a tunnel. The car in the center can speed along, while the slower vehicles near the walls can only crawl due to friction with the tunnel. This analogy helps visualize how fluid flow behaves in a pipe.

So, the velocity field knowledge is required for mass conservation equations.

To apply conservation of mass laws in fluid dynamics, understanding how velocity varies in a flow field is critical. This information allows us to analyze how much fluid is entering, exiting, or changing within a control volume, ensuring our calculations are accurate and relevant to the flow situation.

Think of a river delta, where water splits and flows in various channels. Knowing how fast the water moves helps us understand where it will deposit sediment and how the delta will change over time, similar to how we analyze flow in pipes.

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