10.1.2. Rotation

I think you please remember these figures okay which is very interesting figures for a fluid mechanic student point of view that when you have this a flow passing over the plate, you can anticipate it that you will have a viscous effect zone which is called boundary layer formations, there is a zone where there is large gradient of velocity vectors, the velocity will start from 0 to a large gradient will be there.

When a fluid flows over a surface, like a flat plate, it doesn't move at a uniform speed. Near the plate, the velocity is zero because the fluid particles are stuck to the solid surface due to viscosity. As you move away from the surface into the fluid, the velocity increases until it reaches a maximum. This region where the velocity changes from zero to its maximum is known as the boundary layer. In this zone, viscous effects are significant and impact how the fluid flows. The changes in velocity create varying speeds and lead to rotational motion in fluid particles, especially within this boundary layer.

Imagine when you run your hand through water. Your hand moves quickly, but near your hand (close to your skin), the water is still and moves more slowly. This is similar to how the fluid behaves close to a surface, creating a layer of slower-moving fluid alongside faster-moving fluid further away.

Since, there is a large gradient of velocity variation is there as I said it earlier that velocity vector change from one point to other point, so the fluid particles will not go straight line, they will start rotating it, so that way this figure is retreating that when the fluid particles entered here, there are large velocity gradients are there, the turbulence is there, so the boundary layer formations are the zone where viscous effect dominates okay.

As fluid flows, variations in speed lead to changes in the direction of movement of fluid particles. Instead of flowing in parallel lines, particles experience rotational motion due to the differences in velocity across the boundary layer. This rotation contributes to the formation of vortices or eddies, where fluid particles spin around an axis. The presence of these eddies can significantly change the flow characteristics, indicating the complexity of fluid movement in real-world scenarios.

Picture a swirling drain in a sink: water moves in straight lines toward the drain, but as it approaches, it starts to spiral. This spiral motion is akin to the rotational effect experienced by fluid particles within the boundary layer.

So, then we tell it irrotational fluid, this is the outside of the boundary layers, you can see that fluid particles are moving it or the virtual fluid balls are moving it without any rotations but within the boundary layer formations, the small regions near to a surface, you will see there is a change of the velocity gradients; the drastic change of the velocity gradient and those the regions; a thin region is called boundary layer.

In regions outside the boundary layer, the fluid flows smoothly without rotation; this is referred to as an irrotational flow. The fluid particles exhibit translational movement, meaning they slide past each other in a uniform manner without spiraling. However, within the boundary layer, there is considerable change in velocity gradients, causing the rotational behavior in particle motion due to the viscosity of the fluid.

Think of a soccer ball rolling perfectly on a smooth field. The ball moves straight without any spinning. However, as it hits the grass at the edge of the field, the ball may wobble or rotate due to the uneven surface. This is similar to how fluids behave: smooth motion in irrotational regions and more chaotic behavior in boundary layers.

So that what we measure in terms of vorticity, so please do not have a very confusions between the vorticity and angular vector because vorticity is easy to define is a cross product between the delta and the V where is when you talk about angular rotations, we have half of that so, it is very easy the people who are not looking the angular rotations, they are looking it in terms of how the vorticity is playing it or vortex formations happening it, they talk about at the vorticity level not at the rotations level.

Vorticity is a measure of the local rotation within a fluid. It's calculated as the cross product of the velocity gradient and the velocity vector. Unlike simple angular momentum, vorticity provides insight into the rotational behaviors of fluid elements. Understanding vorticity is crucial for comprehending swirling flows and turbulent patterns of movement in fluids.

Consider the spinning motion of a whirlpool in water. The vorticity gives you a measure of how intense and how fast the water is rotating at any point in the whirlpool. Just as a whirlpool shows turbulence, vorticity helps predict the behavior of fluids in diverse conditions.

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