17.2.1. Problem Statement

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.

In fluid dynamics, the Mach number (or mac number) is a dimensionless quantity that represents the ratio of flow velocity to the speed of sound in that fluid. When the Mach number is less than 0.3, the flow can be considered incompressible. This means that the density of the fluid remains roughly constant despite changes in pressure and temperature, allowing us to simplify equations governing fluid motion.

Think of riding a bicycle through a thick fog. The speed of the bike is much slower compared to the speed of sound, allowing you to move through the air without dramatically affecting the air density around you. The air behaves like a dense liquid, allowing you to maintain a steady flow without feeling the pressure changes caused by compressibility.

We talked about mass flux is coming in or going out from the control volume or mass flux is changing within the control volume. That is the concept to that. But as it is simplified, in case the flow is incomprehensible, density is a constant, that density component comes out from Reynolds transport theorem.

Mass conservation in fluids states that the mass of fluid entering a control volume must equal the mass leaving plus any change within the control volume. For incompressible flows, since density is constant, we can use volumetric flux (volume flow rate per area) instead of mass flow rates, simplifying our calculations.

Imagine a water pipe with a steady flow. Water enters the pipe at one end and exits at the other. Since the water's density doesn't change, we can easily calculate how much water is passing through any section of the pipe without worrying about compressibility effects. It’s like counting how many balloons fill a room without factoring in that they might be squished as more balloons are added.

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 dealing with fluid dynamics problems, understanding how velocity changes across a fluid's surface (called the control surface) is crucial. Knowing whether the velocity is uniform or varies allows us to apply conservation laws correctly. In more complex flows, velocity distribution can significantly affect calculated outcomes.

Consider a busy highway where all cars are moving at different speeds. If you want to know how long it takes for a car to travel through a traffic jam, you need to observe how the speeds of surrounding cars change. Similarly, in fluid flow, understanding how velocity varies helps in accurately predicting the flow's behavior across different conditions.

First, 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 a problem in fluid mechanics helps identify which equations and assumptions apply to a given flow type, such as whether the flow is steady or unsteady and whether it can be treated as incompressible or compressible. This is essential before proceeding to apply mathematical models and equations accurately.

Think of organizing a library. If you classify books by genre (fiction, non-fiction, mystery, etc.), it becomes easier to find what you need. In fluid problems, classifying flows helps streamline the problem-solving process, guiding you to relevant equations and appropriate approaches.

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.

Understanding how velocity varies within a control volume is critical for accurately applying the mass conservation equation. If velocity remains constant, calculations can become straightforward, but if it varies, more complex integration may be necessary for determining the overall flow behavior.

Imagine measuring the flow of water through a funnel. If the water flows steadily (constant velocity), it’s easy to calculate how fast the liquid drains. However, if the flow rate changes (like during a burst of water), you need to take multiple measurements to understand how quickly the funnel can empty, highlighting the complexity of variable flow rates.

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