9.6.3. Shear Strain Rate

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, okay. So, rate of increase in length for unit length that means, let you have a the initially, you have a the fluid element which is starting from P to Q okay, at the point P you have the velocity $v_p$ but at the Q, we should have a based on the Taylor series, you can have this velocity $v_q$.

The shear strain rate is defined as the rate at which deformations occur in a material (in this case, fluid). It is specifically the change in length (more accurately, the change in relative displacement) per unit length caused by the velocity differences across points in the fluid. For example, if two points P and Q are in a fluid flow and they move apart because of differing velocities, we can say that the rate of strain is determined by how quickly they stretch apart relative to how far apart they were initially. This is important in fluid mechanics as it helps us understand how fluids deform under flow conditions.

Imagine a group of students holding hands in a line as they run forward. If the students at the front run faster than those at the back, they'll start to stretch apart, illustrating how one part of the group experiences more strain than another. This stretching represents the shear strain rate as some parts 'move' away from others while still being connected.

At the dt times, these P point will be moved to Ṕ, the Q point will move to Q́, the distance between these 2 will the different, so that is the reasons they will be linear strain. Distance travelled by this P and Q will be the different as the velocities are the different at the P and Q, if the velocity difference between the PQ not there, then you will may not have a linear strain, will not have a linear strain.

As the fluid flows, the velocities at different points will often vary. This causes points P and Q (which were originally at the same distance apart) to now be farther apart after a given time interval (dt). This differential movement means the fluid element experiences stretching, known as linear strain. If both points P and Q move at the same rate, then there is no strain because they maintain their relative positions.

Think of a rubber band. If you pull both ends at the same rate, it simply stretches uniformly. However, if you pull one end faster than the other, the part of the band closer to the faster end stretches more than the part nearest the slower end, creating strain. This illustrates how variations in velocity can lead to differential strain in a fluid.

Since, there is a velocity variation, the fluid element which connected as a virtual fluid balls from P to Q; Q moves more as compared to the P, so you have a stretching of fluid virtual fluid balls, stretching of the virtual fluid balls that means, they will stretch it, there will be the same dimensions but there will be stretching it, so they are will be elongated as the width will be reduces, the length will be change it.

In incompressible flow, despite the fact that the density remains constant, the fluid still undergoes deformation. As certain parts of the fluid accelerate faster, the overall dimensions of the fluid elements may change. It leads to an elongation in the direction of flow while possibly compressing in transverse directions. Understanding these changes is crucial for predicting fluid behavior.

Visualize a balloon filled with water. If you push on one side of the balloon faster than the other, the water nearest your push moves quicker and stretches out, while the areas a bit farther from the push compress slightly. The result is the fluid dynamically adjusting and changing shape, though the total volume remains consistent, which reflects how incompressible flow behaves.

The shear strain rate is a half rate of decrease of angle between 2 initially perpendicular lines that intersect at the point. What it is saying that if you look at these figures that I have a figure like this, the fluid element at the t1 time at the fluid element at the t time. At the t time, this line is moving $\theta_1$ and $\theta_2$ angles.

The shear strain rate focuses on angular changes within a fluid element. It looks at how the angle between two lines (initially perpendicular) changes due to shear stress. As shear forces act on the fluid, these angles shift, contributing to the fluid's deformation. The shear strain rate quantifies this angular change over time, reflecting how quickly and in what manner the fluid becomes distorted.

Imagine a piece of clay resting on a table. If you push one side of the clay, the shape changes, but at the same time, the angle between two edges of the clay changes from the flat angle. Just like the clay, fluids experience changes in shape and angle when forces are applied, which is captured in the concept of shear strain rate.

So, as of now we have done it, we can get a shear strain rate in a Cartesian coordinate for different plane, xy plane, zx plane and yx plane and finally, we can write the shear strain rate tensor which will be linear strain formations and the formations of shear strain rate. So, it has a 9 component of stress strain rate tensors.

In fluid mechanics, we can represent how a fluid deforms using a strain rate tensor, which captures the variations in strain across different directions. This tensor has components for different planes: xy, xz, and yz planes. Each component represents how the fluid stretches or compresses in that particular direction, allowing us to mathematically describe the fluid's response to forces.

Envision a stretchy fabric that can be pulled in different directions. Depending on how you pull it, the fabric will stretch more along certain lines than others. Similarly, the strain rate tensor acts like a map of how fluid responds to different strain directions, providing a comprehensive tool for analyzing fluid behavior in response to forces.

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