9.4.2.3. Deformations

If you come back to our basic fluid mechanics okay, so as in the last class I talked about 2 types of descriptions; one is Eulerian frame of descriptions, another is Lagrangian frame of description and that the descriptions we try to visualize as form of virtual fluid balls, okay. So, today I will talk about that I have the fluid element which is you know in any of fluid mechanics book. They talk about the fluid element which is representing a certain space of the fluid particles which is much larger scale than the molecules levels or it is not that bigger scale to represent the flow process.

In fluid mechanics, we can understand fluid behavior using two distinct frames of reference: Eulerian and Lagrangian. The Eulerian approach focuses on specific locations in space, observing how fluids flow past these points over time. Conversely, the Lagrangian perspective tracks individual 'fluid elements' or 'virtual fluid balls' as they move through the fluid. This means that instead of just looking at motion at one point, we can visualize the movement and transformations of these elements as they interact with their surroundings. You can think of these virtual fluid balls as being larger than individual molecules but smaller than sections of fluid, helping us analyze the fluid's properties effectively.

Imagine you are tracking a ball in a crowded stadium (Lagrangian approach) versus watching the crowd from a fixed spot (Eulerian approach). When you focus on the ball (Lagrangian), you can see its journey and how it interacts with people around it. In contrast, if you're at your fixed spot (Eulerian), you notice the patterns of movement of many people as they flow past your view.

So, when you have balls then it can have go through any type of motions and the differences like it can be translations like variations that you have a ball, it can be translated, it can go in a u direction or v direction. Or it can have a displacement in a x direction, y direction, z direction and if the displacement is caused because of the velocity component...

Fluid elements can undergo two main types of movements: translation and rotation. Translation refers to the movement of the fluid element from one place to another, which can occur in any direction based on the velocity components of the fluid (u, v, w). For example, if a fluid element moves from point A to point B in three-dimensional space, we can calculate its new position based on its velocity at the initial time. Rotation, on the other hand, involves the fluid element spinning around an axis. This rotational motion can be clockwise or counterclockwise and depends on the velocity gradients within the fluid. Understanding these movements helps us analyze how fluids behave in different scenarios.

Think of water in a rotating carousel (rotation) and water flowing from a faucet (translation). As water flows from the faucet, it travels in a predictable path, moving from a higher to a lower point (translation). Meanwhile, the water in the carousel swirls around due to the rotation of the carousel, demonstrating how water can also spin around an axis.

the fluid element can go for a rotation, so that means we can compute what could be this angular velocity, if it is going through a rotation; the clockwise rotations or anti-clockwise rotations. So, if you try to understand that these rotations all depends upon the velocity fields, the velocity of the field also talk about how the translations motion will happen from point A to point B...

When fluid elements rotate, they can experience angular velocity which tells us how fast and in what direction they are rotating. This rotation can be influenced by the velocity distribution around them. Additionally, fluid elements can deform when they are subjected to forces; this can happen in the form of linear strain or shear strain. Linear strain occurs when the fluid element is stretched or compressed, altering its dimensions, while shear strain occurs when adjacent particles slide past each other, changing the shape but not the volume. Understanding these deformations is crucial because it helps us describe fluid behavior under various flow conditions.

Imagine stretching a rubber band (linear strain) or twisting a piece of clay (shear strain). As you stretch the rubber band, it becomes longer; as you twist the clay, its shape changes even though the amount of material stays the same. Similarly, fluid elements experience deformations as they flow, which affects their dynamics in a fluid system.

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