17.1.5. Application of Mass Conservation Equation

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.

The concept of incompressible flow arises when the Mach number (a measure of speed relative to the speed of sound) is less than 0.3. At this low Mach number, density variations in the fluid are so slight that they can be ignored. Consequently, we treat the fluid as incompressible, meaning that density remains constant throughout the flow. This simplification greatly simplifies our calculations and analyses in fluid mechanics.

Think of a water pipe. When water flows through it at a slow speed, changes in the water's density are negligible, much like how a baseball doesn't change size as it moves through the air at slow speeds.

Now, there, the scalar product of V and n and d A, okay, and density can come out. 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?

In fluid mechanics, mass flux refers to the mass of fluid passing through a unit area per unit time, while volumetric flux refers to the volume of fluid passing through a unit area per unit time. When we ignore density changes in incompressible flow, we can use volumetric flow rate, which simplifies our calculations: Q = A × V, where Q is the discharge, A is the cross-sectional area, and V is the average velocity.

Imagine a garden hose. The amount of water that flows out per second depends on the cross-sectional area of the hose and how fast the water is pushed through. Similarly, knowing the area of a pipe and the flow rate helps us understand how much water moves through it.

So, 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.

A control volume is a defined space through which fluid flows, helping us analyze mass flow in and out of that space. Understanding how velocity changes at different points within that volume is key to applying the mass conservation equation. If we know how fast fluid is moving and in what direction, we can calculate the mass entering and leaving the control volume, allowing us to determine the overall change in mass within the volume.

Imagine a classroom filled with students. If you know the entrance (where students come in) and exit (where students leave) and can monitor how many students walk in and out over time, you can calculate how many students are in the classroom at any given moment.

So, basically, when you apply the mass conservation equation your knowledge of velocity variation is important to you, how you are simplifying the velocity field on the control surface.

When applying the mass conservation equation, understanding how velocity changes across the control surface is crucial. If velocity is constant, the calculations simplify considerably. In real-world problems, however, velocity often varies, which requires a more nuanced approach, potentially involving integration to account for these variations.

Think of a river with varying speeds. In some areas, the water flows quickly, while in others, it flows slowly. Understanding these changes helps us estimate how much water is moving downstream at any point.

Whenever you solve the problem, you classify the problem. Once you classify problems it gives indirectly that these are the assumptions that are valid for this problem that we are solving.

Classifying the flow helps establish the assumptions we can safely make. Are we dealing with laminar or turbulent flow? Is the flow steady or unsteady? These classifications dictate how we apply the mass conservation equation and select the appropriate control volume, simplifying our calculations while ensuring accuracy.

Consider a chef preparing different dishes. Each dish might need a different cooking method (stir-frying, boiling, baking) based on its ingredients (vegetables, meat, desserts). Similarly, classifying fluid flow helps us determine the best method to analyze it.

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