3.2. Types of Motion or Deformation

A fluid element can undergo the following 4 types of motion or deformation:

1. Translation
2. Rotation
3. Extensional strain (or dilation)
4. Shear strain.

In fluid mechanics, when we analyze how fluids behave, we categorize the movements or deformations they experience into four main types:

1. Translation: This is where the entire fluid element moves from one point to another in space without changing its shape.
2. Rotation: In this type of motion, a fluid element spins around an axis. For example, if you consider a small volume of water in a rotating bucket, the water will rotate along with the bucket.
3. Extensional Strain: This occurs when the fluid element expands or contracts. Imagine squeezing a balloon; as you apply pressure, the balloon's shape and volume alter.
4. Shear Strain: This refers to the deformation resulting from forces applied parallel to a surface, leading to a change in the shape of the fluid element without changing its volume. For example, when you stir cream into coffee, the cream undergoes shear as it is dragged along by the spoon.

Think of a swimming fish. As it glides through the water, the fish experiences translation as it moves forward. When it turns to the side or spins to evade predators, that’s a rotation. If the fish expands its body by pushing water away to gain speed, that’s extensional strain. Lastly, when it's pushing against the water to change direction quickly, it undergoes shear strain.

Consider a fluid element A, B, C, D moving in an xy plane. At time t, its positions are labeled, and at time t + dt, the positions are altered due to fluid motion showing transformations such as translation, rotation, etc.

To visualize how a fluid element deforms over time, consider a square fluid element formed by points A, B, C, and D in an xy plane. Initially, at time t, these points are in the corners of the square. As time progresses to t + dt, the movement of the fluid causes:

• Translation: The points may shift to new locations (for example, point B moves to a new location B').
• Rotation: The diagonal line connecting two of the points (BD) may rotate, indicating that different parts of the fluid element are moving at different velocities, leading to rotation.
• Dilation/Extensional Strain: As the fluid flows, certain sides of the element may stretch or compress due to the velocity differences, illustrating extensional strain.

Imagine a pizza being tossed in the air. As the chef spins the pizza, the outer edge stretches while the center may compress. This is akin to what happens to the fluid element as it transforms: it translates (as the pizza is lifted off the counter), rotates (due to the spinning motion), and undergoes strain (the dough stretching).

The rate of rotation of the fluid element is given by the angular velocities of the sides BC and BA, which have opposite senses.

When analyzing the rotation of a fluid element, we can look at the velocities of the sides of our defined fluid shape, specifically sides BC and BA. The rate of rotation considers how these vertices rotate around a central point. This means that one side may be rotating clockwise while the other is rotating counterclockwise, producing an opposing rate of rotation. Mathematically, we average the angular velocities measured at these respective sides to compute the overall rotational effect in the fluid element.

Think about how a door swings open. As the handle moves forward, the point near the hinges (side BA) moves in a different direction than the outer edge of the door (side BC). Therefore, if one half (BA) rotates a little clockwise and the other half (BC) rotates a little counterclockwise, the combination creates a complex rotational motion similar to the fluid rotation described.

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