1.2.2. Field Representation of the Flow

So, first, we will talk about the velocity field, what actually is this velocity field. You know it from beforehand but too thorough to brush up your concepts will go through it once again. So, it is the actually the fluids can be assumed to be made up of infinitesimal small particles and they are tightly packed together. And this is implied by the continuum assumption. So, infinitesimal particles of fluids are tightly packed together. Thus, at any instant in time, a description of any fluid property, such as density, pressure, velocity and acceleration, may be given as a function of the fluids location. So, that is the basic assumption that is the basic thing that we are going to start with that these properties such as density, pressure, velocity and acceleration can be described as a function of fluids location.

This chunk introduces the velocity field, presenting the idea that fluids consist of tiny particles closely packed together, which is referred to as the continuum assumption. This means that we can describe physical properties of fluids—like density, pressure, velocity, and acceleration—based on their location in space at any given moment.

Imagine a crowd of people in a concert. Each person represents a tiny particle in a fluid. Even though you see a large group, each person can be understood individually based on their position and what they are doing at any moment.

So this representation of fluid parameters as function of a spatial coordinates is termed as field representation of the flow, spatial coordinate. So, that is, with respect to the space. So, in terms of x, y and z. For example, the velocity here can be written as u, u is a fluid velocity or the fluid speed in direction, is x axis plus v in direction, is y axis and w in direction and each of u, v and w can be function of x, y, z and t, each of those, this one, this one, and this one here.

This chunk explains how fluid parameters are represented in terms of spatial coordinates. It emphasizes that if we know the position in space (x, y, z), we can describe fluid velocity using components u (x-direction), v (y-direction), and w (z-direction). Each of these components can change depending on both the location and time.

Think of a weather map displaying different temperatures across regions. Just like the temperature varies based on location (latitude and longitude), the speed and direction of fluid flow vary depending on where you are in the space at any given moment.

By definition, the velocity of the particle is the time rate of change of the position vector for that particle. So, the velocity is the time rate of change of the position vector for that particular particle, this you already know.

This chunk defines velocity specifically for a particle as the rate at which its position changes over time. This emphasizes that velocity is not just a speed but also includes the direction of that movement, fundamentally connecting motion with time.

Consider a car moving along a road. Its velocity tells you how fast it is moving and in which direction. If it travels from point A to point B in a certain amount of time, the velocity can be calculated based on the change in position divided by the time taken.

So, by writing the velocity of all the particles, we can obtain the field description of the velocity. So, if we write the velocity of all the particles in the system, we can obtain the field description of the velocity vector v as this V = V (x, y, z, t) using the equation described in the previous slide. Since, the velocity is a vector, it has both a direction and the magnitude, this is the basic concept that you know from before.

This chunk discusses how by determining the velocity of each individual particle in a fluid, we can create a comprehensive field description of velocity as a function of position and time. Velocity being a vector implies that it has both size (magnitude) and direction, which we need to consider when analyzing fluid flow.

Think of a school of fish swimming in the ocean, where each fish is a particle. If we can figure out how fast and in what direction each fish is swimming, we can describe the overall movement of the school (the velocity field) in terms of both speed and direction.

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