3.4. Transformation of Fluid Element

Starting with a new, you know, so if there is a fluid element can undergo the following 4 types of motion or deformation. What are those? Of course, it can do translation; translation is moving from one point to the other, it can do rotation; it can do extensional strain or also dilation and shear strain.

A fluid element can move in four distinct ways: translation, rotation, extensional strain, and shear strain. Translation refers to the movement of the entire fluid element from one position to another without any change in shape. Rotation involves the fluid element spinning around an axis, causing different parts of it to move in circular paths. Extensional strain refers to the change in size or volume of the fluid element, typically when it is stretched or compressed. Finally, shear strain refers to the angular deformation when forces are applied parallel to the surface, causing layers of the fluid element to slide past one another.

Think of a balloon filled with water as a fluid element. When you push one side, the balloon expands and translates (moves) in the opposite direction; when you twist it, it rotates; pulling it from both ends causes extensional strain, and pushing laterally creates shear strain, as the layers of water slide over each other.

So, we are going to derive the strain rates, so one thing I would like to take your attention to, is this figure. So we consider a fluid element A, B, C, D, so we consider this element A, B, C, D and that moves in an xy plane, the position of this element at times t, this is position at time t and its position at time t + dt is shown like this.

In studying fluid dynamics, we visualize a specific fluid element defined by four points: A, B, C, and D, which moves in the xy-plane over time. At time 't', this element is in a certain configuration. After a small time interval 'dt', the position of these points changes due to the fluid motion. By observing these changes, we can analyze how the fluid element transforms and derive the strain rates over time.

Imagine a small piece of clay being gently pushed in a two-dimensional plane. At one moment, it has a certain shape and size. As you push, the corners of the clay shift position after a moment, showing how the material (or fluid in our case) transforms in response to the applied force.

With reference to the above figure, we can observe the following transformations, I mean, of transformation or deformation in the fluid element within the time interval of t + dt. See, we can see that B has translated from B to B dash, we can also see that this diagonal BD has rotated anti-clockwise to B dash, D dash, you see that, and we can also see the dilation or the extensional strain of the element.

Observing the fluid element over the time interval from 't' to 't + dt', we can identify three specific changes: point B has moved to a new position (B dash), indicating translation; the diagonal line connecting points B and D has rotated, signaling a rotational motion; and finally, the structure of the element has changed in dimension, evidencing dilation or extensional strain. By analyzing these observations, we can quantify the fluid's response to forces acting on it.

Think of a rubber band being pulled. As you stretch it, different parts of the band translate to new positions (translation), it may twist at certain points when you pull from different angles (rotation), and you can see the band elongating (dilation) as the forces are applied.

--