9.4.2.2. Rotations

Now, let us go for the translations which is very easy concept okay, which is the velocity factor which is responsible for shifting a fluid particle from A location to B locations, it depends upon the velocity components like in this case, you have a small u, v and w is a scalar velocity component in x, y, z directions respectively.

Fluid motion can be understood through the concept of translations. The movement of fluid particles occurs when they shift from one point, A, to another point, B. This movement relies on the fluid's velocity components in different directions: u in the x-direction, v in the y-direction, and w in the z-direction. Each of these components tells us how fast and in which direction the fluid is moving along that particular axis.

Imagine a swimmer in a pool. If the swimmer moves from one end of the pool (point A) to the other end (point B), we can describe their movement in terms of how fast they move forward (along the x-direction), how much they shift sideways (y-direction), and how deep they go into the water (z-direction). Just like the swimmer, fluid particles move through a medium based on their velocities in these three dimensions.

Now, coming to the rate of rotations, which is called angular velocity okay, what could be the angular velocity of the fluid point. See if I take it this is the fluid element okay, it will have a this the velocity variations that means, if this is the u velocity is here at this point, you will have a velocity perpendicular to it.

Angular velocity refers to how fast a fluid element is rotating about a certain point. When we consider a small portion of fluid (a fluid element), the velocities at different points within this element can vary. For instance, at one point there might be a velocity 'u,' while at another point, there might be a velocity 'v' in a perpendicular direction. This difference in velocities creates a rotational motion within the fluid element, indicating that it is turning or rotating around the point of interest.

Think of a rotating merry-go-round. Kids sitting on the edge experience different speeds based on their position. The kids at the edge move faster along the circular path compared to kids near the center. Similarly, in a rotating fluid element, different parts of the fluid can experience different velocities, leading to a rotation of the entire fluid mass, just like how the merry-go-round spins.

If you want to have very detailed derivations writing from the Taylor series, find out in the distance travels that computing all these component of the velocity at each points, you can follow of any of my wave lectures which is there in NPTEL’s.

To calculate the angular rotations within a fluid element, we can use mathematical concepts like Taylor series to understand how velocity changes at various points. This involves examining how far the velocities at different locations cause the fluid element to rotate. By considering these velocity differences and applying derivatives, we can mathematically represent the rotation, allowing us to quantify it.

Think of how a small wind-up toy car rotates when you let it go. Each part of the car spins differently based on the spring's release. If we measured the angles and distances of each part's movement, we could calculate how fast the entire car would spin as time goes on, similar to how we use calculations in fluid mechanics to determine rotations based on velocities.

Now, coming to the very simple thing is called linear strain rate that means, what is the strain rate; rate of increase the length per unit length, this is very simple definition.

Linear strain rate is a measure of how much a fluid element's length changes relative to its original length. If a fluid element stretches or compresses, this change can be quantified as a strain rate. Specifically, if the length of the segment of fluid increases due to different velocities acting on its ends, this change results in a positive strain. Conversely, if it decreases, then you'll have a negative strain.

Consider a rubber band. When you pull on a rubber band, its length increases. The rate at which it stretches can be likened to the linear strain rate. If you measure how much longer it gets each second while you pull, you can determine the linear strain rate for that rubber band. In fluid dynamics, a similar stretching occurs when velocities vary across a fluid element, leading to changes in its length.

So, let us look at another component is called the vorticity or the rotationality, which basically we are representing as a vorticity vectors, the basically we are representing as a vorticity vectors which actually the measures of rotations of a fluid particle.

Vorticity is a measure of the local rotation in a fluid. It is represented mathematically as a vector that indicates how much and in what direction fluid particles are rotating. The vorticity vector can be thought of as twice the angular velocity of that fluid particle, providing insight into how the fluid is swirling around a point. This property is essential in analyzing flows, particularly in turbulent conditions where rotations are significant.

Imagine a whirlpool in a pond. Water at the center spins around in a circular motion, creating a vortex. If you could measure how quickly and in which direction the water is spinning, you'd be measuring the vorticity of that whirlpool. Just as the whirlpool shows us the rotation of water, vorticity in fluid mechanics helps us understand the swirling motions in all types of fluid flows, from oceans to air currents.

--