3.2.2. Microscopic Analysis

If you look it if fluid is composed of the molecules in constants, motions and collisions okay. Then you can just think it that the fluid flow what is happening it that is what the representations of million number of molecules are in motions, also they are colliding each other.

Fluids consist of a large number of molecules that move constantly and collide. This motion and interaction among molecules define the behavior of fluids. When imagining a fluid, think of millions of tiny particles (molecules) moving around, bumping into one another as they flow. This dynamic nature is essential for understanding how fluids behave under various conditions, such as when they are at rest or in motion.

Imagine a crowded dance floor where everyone is moving freely. Each person bumping into others represents the molecules in a fluid. Just like people can change their movement patterns based on the actions of those around them, fluid molecules interact with each other, affecting how the fluid flows.

That is what is the average distance traveled by a molecule before collisions. That is what is called the mean free path.

The mean free path is a concept that describes how far a molecule travels on average between collisions with other molecules. For example, in oxygen gas at normal conditions, a typical molecule may travel about 6.3 × 10^-8 meters before colliding with another molecule. This indicates that there is a significant amount of empty space between molecules even though they are packed closely together.

Think of this like cars on a highway. If you imagine each car as a molecule, the mean free path would be the average distance one car travels before encountering another car. On a busy highway, cars are continuously changing lanes and bumping into each other, just like molecules in a fluid.

These properties what is happening it at the molecules levels that is what is reflect us very interesting concept what let us consider very basic fluid properties is the fluid density, okay? That means that is what is the mass per unit volumes.

Fluid density is defined as the mass of the fluid per unit volume. Since fluids consist of a vast number of molecules that are constantly moving and colliding, the density can fluctuate, especially when looking at small sample volumes. In a very tiny volume, the number of molecules can vary significantly, which results in changes in the measured density. This fluctuation is more pronounced in small volumes because the random motions of individual molecules have a more significant impact.

Consider a balloon filled with air. If you take a tiny sample from the balloon, sometimes you'll get a lot of air molecules, and other times you won't get many. If you average the density across the entire balloon, it remains constant, but in small samples, variations can occur.

That means, we call the macroscopic uncertainty. That sampling volume is too large that like if you consider this is the room is a sampling volume for me see in that case what will happen it that the density variations within this room that what will play the rules that will not be a constant value.

Microscopic uncertainty occurs when the sampling volume is too small, leading to significant variation in density due to the random behavior of individual molecules. Conversely, macroscopic uncertainty happens when the sampling volume becomes too large, potentially encompassing multiple regions with different densities. For accurate analysis, the sampling volume needs to be sufficiently large so that the variations average out and represent a uniform density.

Imagine trying to guess the average height in a school by measuring just a handful of students. If you only measured kids from kindergarten, you might think everyone is short! But if you measure all grades together, you get a better average. Similarly, small samples of fluid may not give a good average density, while large samples can provide a more accurate representation.

So we define as a fluid as continuity. So similar way as the densities having a continuous function we can define is the pressure is also a continuous functions of the space and the time.

Defining a fluid as a continuum means that, within a specified range of observations, the properties of the fluid (like density and pressure) can be treated as continuous functions. This allows us to use calculus to analyze fluid properties and fluid flow. For a fluid to be treated as a continuum, the sampling volume must be large enough to balance out the microscopic randomness of individual molecule interactions.

Think of a river: while individual water molecules are constantly moving and interacting, if you took a large enough section of the river, the water appears to flow uniformly. This allows engineers to calculate factors like flow speed and pressure as if the water behaves continuously, even though it's made of countless individual molecules.

--