4.1. Angular Velocities and Rotation

So, 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.

In fluid mechanics, it's important to understand how fluid elements move and deform. In this case, we observe that point B moves to a new position (B') and the diagonal BD rotates. This rotation is described as anti-clockwise. Additionally, the fluid element experiences dilation, which refers to the change in volume or shape due to the motion and rotation. Essentially, when a fluid particle moves, it can both change position and alter its shape at the same time.

Imagine a lump of dough in your hands. When you push down on one side, it rotates and also gets stretched or flattened, similar to how the fluid element transforms as it moves.

So, we can say tan d alpha is equal to del v del x into dx dt divided by dx + del u del x dx dt.

This equation helps us relate the angular rotation (dα) of the fluid element to the velocity gradients in different directions. The term 'tan d alpha' is derived using basic trigonometry and describes the rate of change of angle due to velocity changes along the x-direction for a small time interval. This can be useful in understanding how fast a fluid particle is rotating in response to changes in its velocity as it moves.

Think of a car making a turn. The sharper the turn (like the dα angle getting larger), the faster the car must adjust its direction according to the speeds of the wheels on the inside versus the outside of the turn.

The rotation of the element is given by the arithmetic mean of the angular velocities of sides BC and BA.

To determine the overall rate of rotation of a fluid element, we calculate the average of the angular velocities on two sides of the element. This approach allows us to account for both the contributions to rotation from different sides of the fluid element and ensures the measurement reflects the combined effect of the rotations happening within the same timeframe.

Imagine you're swinging a ball on a string in a circle. The speed at which the ball moves along different points of the string can be different, but to understand the overall effect (or rotation), you would need to average those speeds to see how quickly the ball is being pulled around.

These 3 are convective derivative and this is local derivative, so this is substantial or material derivative.

In fluid dynamics, understanding different types of derivatives is essential for analyzing fluid motion. The convective derivative captures how properties change due to the movement of the fluid itself, while the local derivative shows how properties change at a specific point in space over time. Together, these derivatives form the substantial (or material) derivative, which combines both effects into one expression that describes the overall change in a fluid property as it moves.

Think of a river flowing past a rock. The water near the rock experiences different conditions as it moves compared to the water upstream. The local derivative could describe the water conditions right next to the rock, while the convective derivative describes how the rock’s presence alters the conditions as the entire river flows.

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