8.2. Particle Levels and Time Derivative of Velocity

Now, let us look it at the particle levels, if I had to find out what is the acceleration of the fluid particles? Is nothing else is a time derivative of velocity of the particles, as you know it from class 10, 11th and 12th I am just doing this time derivative of the velocity; particle velocities with respect to time and that is what represents the accelerations, at the particles levels you will have these.

Acceleration is defined as the rate of change of velocity over time. When considering fluid particles, the acceleration can be determined by taking the time derivative of the particle's velocity. In simpler terms, if you know how the velocity of a particle changes as time passes, you can calculate the acceleration. This is similar to how you might have learned in previous classes about the concept of acceleration in physics.

Think of a car going from rest to speeding up. When you press the accelerator, the speed of the car increases. If you measure how fast your speed is changing at different moments (like 5 seconds, 10 seconds, etc.), that tells you how much you're accelerating. The same principle applies to fluid particles in motion.

Now, if you look at this the velocities has a variability in a positions and the time because of that though when you define a; derivative with respect to time you will have a local component okay, you will have a with a x particle directions, y particle directions and z particle direction.

The velocity of fluid particles can change based on their position in space (x, y, z coordinates) and over time. This means that when calculating acceleration, you need to consider how the velocity changes in different directions (left-right, up-down, forward-backward) and how it can vary with respect to time. Thus, when defining derivatives, you often break it down into components along these different axes.

Imagine a river. The flow speed might change in different parts of the river due to various factors like rocks or bends. If you were to measure how fast the water at one point flows compared to another point, you are essentially observing velocity variability based on position. This is similar to how we analyze changes in velocity for fluid particles.

This is nothing else if you are considering is a 2 variables like I just discussed you the Taylor series, if you remember it defining for the 2 variables in this case, I have a Taylor series of 4 variables the x, y, z and the t.

In mathematics, a Taylor series can be used to approximate functions. When dealing with fluid dynamics, you can extend this idea to account for multidimensional changes, such as in x, y, z coordinates and time (t). A Taylor series expansion allows us to express data (like velocity) in terms of derivatives at a single point, helping to analyze how these variables interact.

Consider a balloon being inflated (where changes occur in all directions). If you want to predict how the balloon's size changes with time based on how you inflate it at different rates, you can think of extending that logic using a Taylor series for your calculations—meaning that knowing how one part affects another can inform you about the whole balloon.

So, this is 2 components; one is local acceleration component and other is convective acceleration. In vector (()) (45:37), we can very simple way represented the local accelerations and the convective acceleration component.

Local acceleration refers to how the velocity of a fluid particle changes at a specific point in time and space, whereas convective acceleration takes into account the change in velocity due to the motion of the fluid particles themselves. In simple terms, local acceleration is about how fast the flow is changing in one location, and convective acceleration is about how flow properties change as you track the movement of the particles over space.

Imagine riding a wave in the ocean. As you move with the wave (convective acceleration), you are also noticing how the water around you changes in speed (local acceleration). These two effects work together to give you a complete picture of your experience in the wave's motion.

But many of the times people talk about material derivative, it is nothing else, it is that the change it is a special name of material derivative means, when you talk about a particles along the particles, you compute the derivative with respect to time is a total derivative or the material derivative or the particle derivatives is nothing else, we already proved it that these derivatives we can define as these functions that is what we have defined it, okay.

The material derivative is a way to describe how a fluid particle's properties change as it moves through space and time. It's an important concept in fluid mechanics because it combines the effects of local and convective changes. In simpler terms, it helps us understand the changes experienced by a fluid particle as it flows along.

Think of a floating leaf on a river. The material derivative would help describe how the leaf’s position and speed change as it travels down the river (convective changes), as well as any other changes it experiences (like the water having different speeds at different locations).

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