3.3. Derivation of Strain Rates

So, we are going to derive the strain rates, so one thing I would like to take your attention to, is this figure. So, we consider a fluid element A, B, C, D, so we consider this element A, B, C, D and that moves in an xy plane, the position of this element at times t, this is position at time t and its position at time t + dt is shown like this.

In fluid mechanics, we consider a small element of fluid, represented as a quadrilateral (A, B, C, D) moving through space. Initially, it is at position 't', and after a very small time interval 'dt', it moves to a new position. The study of this motion helps us understand how the fluid's properties, like velocity and strain, change over time. These changes are crucial for developing the equations that describe fluid flow.

Imagine a rubber sheet being stretched. As you pull at the corners, the positions of the points on the sheet change smoothly. Just like that rubber sheet, the fluid element transforms as it moves, and analyzing this transformation gives insights into how the fluid behaves under forces.

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.

As the fluid element moves, three types of deformations can be observed: translation (movement of points), rotation (the fluid element turns), and dilation (change in size or volume). These changes happen simultaneously and are important for calculating the strain rates, which helps us understand how internal forces within the fluid develop.

Think of a group of friends standing in a circle holding hands. If one friend walks a few steps forward (translation), the circle may stretch (dilation) and twist as everyone adjusts to keep holding hands (rotation). Each of these actions contributes to the overall motion of the group.

So we can see from the figure that I showed you, 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.

To find out how the fluid element is rotating, we use the tangent of the angle (d α) created due to the motion. By calculating the change in velocity in the x direction (del v del x) and combining it with the distance moved (dx), we can find this angular deformation. This mathematical relationship allows us to quantify the rotation experienced by the fluid element as it deforms.

Imagine a spinning top on a table. By measuring how far the top has moved across the table and the angle it has turned, you can determine its angular speed. Similarly, by measuring velocities and distances in fluid motion, we understand how fluids twist and turn as they flow.

So, the rotation of the element is given by the arithmetic mean of the angular velocities of sides BC and BA. Further, the rotations d alpha and d beta are in opposite sense.

In fluid dynamics, the rotation of a fluid element can be calculated as an average of how the edges are rotating. By considering that one side of the element rotates in one direction and the opposite side rotates in the other direction, we identify the net rotational effect. This understanding is essential for calculating the strain rates accurately in the fluid environment as they directly impact fluid behavior.

Consider a bicycle wheel. As you ride, the tire rotates while parts of the wheel experience different speeds and directions depending on where they are. Analyzing the rotation of the wheel helps engineers design better bikes by understanding how to apply force efficiently.

Similarly for x and y directions, simply we write sigma x by dt is equal to 1/2, so we have found out for z, same way we can find out for del w by del x, sorry, del w by del y - del v by del z.

The strain rates in different directions (x, y, and z) can be derived using the relationships we established previously. They help quantify how fast the fluid is distorting or changing shape in any direction. This information is essential for engineers to analyze stresses and failures in fluid systems and structures.

Think of a balloon being squeezed from the sides. Depending on how much pressure is applied, different points on the surface of the balloon will stretch or compress more than others. By examining the strain at various points, one can understand the strength and limits of the balloon material.

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