1.2. Fluid Kinematics

Fluid can be assumed to be made up of infinitesimal small particles tightly packed together. At any instant, a description of any fluid property, such as density, pressure, velocity, and acceleration, may be given as a function of the fluid's location.

In fluid mechanics, we treat fluids as being made up of tiny particles rather than considering them as a whole. This helps us understand how different properties, like density and velocity, change depending on where you are in the fluid. At any given moment, these properties can be described by their positions in space, meaning for any location in the fluid, we can know its density, pressure, and velocity.

Imagine a crowded concert where everyone is pushing and moving in different directions. To understand how the crowd is moving at any given moment, you'd look at small groups of people (like the infinitesimal particles of fluid) and see how they are positioned relative to each other. This way, you can describe the crowd's density (how close people are together), how fast they are moving (their velocity), and how pressure changes when everyone shifts.

This representation of fluid parameters as a function of spatial coordinates is termed as field representation of the flow. Velocity can be described with components: u (in x), v (in y), and w (in z). Each of these can be functions of x, y, z, and t.

When we talk about field representation, we are expressing the properties of fluid—like velocity—based on where you measure them (at specific x, y, z coordinates) and when (at specific times). The velocity can be broken into three parts: u for movement along the x-axis, v for the y-axis, and w for the z-axis. These components help us describe how fluid moves in three-dimensional space.

Think of a weather report that also tells you the wind speed and direction. The report may say the wind is blowing at 10 km/h from the north (u), 5 km/h from the east (v), and a slight upward gust (w). Each component tells you how the wind behaves in a three-dimensional space and allows for accurate predictions of its impact.

The velocity of the particle is the time rate of change of the position vector for that particle.

When we track a single fluid particle, its velocity can be understood as how quickly and in which direction that particle moves over time. Essentially, if we are looking at how the position of a particle changes, the velocity tells us the rate of that change. Mathematically, this is written as the derivative of the position vector with respect to time.

Imagine tracking a runner in a race. If you measure where the runner is every few seconds, you can determine their speed (velocity) at any moment by seeing how far ahead they've gone in that time. The same idea applies to tracking a fluid particle's journey through the air or water.

The two most famous descriptions of fluid flow are Eulerian and Lagrangian. The Eulerian approach focuses on properties at fixed points in space, whereas the Lagrangian approach tracks individual fluid particles as they move.

In the Eulerian method, we study how fluid properties like velocity and pressure change at specific spots in space as time passes. It's like watching a river from the bank—you're fixed in one location and observe how the water moves past you. In contrast, the Lagrangian method involves following specific water droplets as they travel along the river, noting how their properties change over time. This approach gives us a different perspective on fluid flow, focusing on the journey of individual particles.

Consider a soccer game; using the Eulerian method would be like watching from the sidelines and noting where players are positioned and how they move as the game progresses. Using the Lagrangian method would be akin to following a specific player throughout the match to see how they navigate and interact with others on the field.

Fluid flow can be complex, involving three-dimensional and time-dependent phenomena. Fluid flows are generally three-dimensional but can be simplified to two-dimensional or one-dimensional under certain conditions.

Real-life fluid flows typically have velocity components in three dimensions (x, y, and z). For instance, when considering how air moves around an airplane wing, we account for changes in all three dimensions over time. However, in some cases, one or two dimensions may have minimal effect. For example, if a fluid flows in a long, straight pipe, the motion might primarily be along the length of the pipe (1D), making analysis simpler.

Imagine a river flowing straight down a mountain. From a distance, you can think of its flow primarily in one direction (downward), simplifying your analysis. However, if you move closer, you see how it swirls and bubbles in various directions (3D), and waves can even splash vertically (z). Understanding these dimensional aspects helps in simplifying the complex behavior of fluid flows.

One classification of the flow is steady and unsteady. In steady flow, fluid flow conditions at any point do not change with time, while unsteady flow parameters change over time.

In steady flow, the properties such as velocity, pressure, and density remain constant at a specific location over time. If you were to measure them, they wouldn't show any change. Conversely, in unsteady flow, these properties fluctuate as time progresses. This fundamental distinction is crucial for analyzing how fluids behave under different conditions.

Think about the water flow from a tap. When you open it slowly and keep it at a steady rate, that's steady flow—the water rate and pressure remain consistent. However, if you were to turn it on and off quickly or change the flow intensity, that would be unsteady flow, with varying speeds and pressures affecting the water's behavior.

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