4.2.1. Shear Strain Rate Relation

If I consider the same fluid flow through a parallel plate, okay. One of the plate the velocity is zero which is at the rest conditions and other the top plate is moving with a velocity V. So as a microscopic point of view I am not talking about the molecular motions or exchange of the molecular motions or the momentum flux that what is resulting a shear stress.

This passage sets the stage for understanding fluid dynamics between two parallel plates. One plate is stationary (velocity = 0), while the other moves at velocity V. This setup creates a shear stress in the fluid between them due to the difference in motion between the two plates.

Imagine spreading butter on bread. The knife represents the moving top plate, while the bread represents the stationary bottom plate. The butter (fluid) moves along with the knife, and the friction between the knife and the butter creates shear stress.

At the A point we will have the velocity V as the no-slip conditions. At the B point we will have velocity zero. So velocity assuming it, it will have a linear velocity variation from B to A. That linear velocity from zero to V as the y increases, the velocity will increase it and maximum velocity will be the V.

In this chunk, we explore the velocity profile in the fluid between the two plates. According to the no-slip condition, the fluid at the stationary plate has a velocity of 0, and at the moving plate (A), the fluid travels at velocity V. This creates a linear gradient of velocity throughout the fluid, which means that as we move from the stationary plate to the moving plate, the velocity gradually increases.

Think of a slow-moving escalator. The people at the bottom (the stationary plate) are not moving, while the people at the top (the moving plate) are moving at the speed of the escalator. As you move up the escalator, you gradually gain speed.

After ∆t time these fluid element will be deformed as a angular deformations will happen it that there will be angle will come it at the B and E, F and D.

During the fluid flow, the fluid elements experience angular deformation over a small time increment (∆t). This means that the shape of the fluid element changes, causing the angles at different points of the element to shift, indicating that the fluid is not rigid and can deform under shear stress.

Imagine stretching a rubber band. When you pull it, it deforms and changes shape. Similarly, the fluid elements deform in response to the movement of plates, which results in angular changes in their shape.

At the ∆t time we have a angular deformations of ... . I can consider these distance of AE is V into delta t divide by the l will be the 10 (angle) for since angle is very small, I can consider.

Here, we calculate the rate of change of angular deformation by relating distance (AE), velocity (V), and time (∆t). Since the angle of deformation is small, we can approximate this relationship to understand how quickly the fluid is deforming relative to the distance between the plates.

Think of a pencil being twisted while one end remains still. The degree of twist at the free end is proportional to how hard you twist and how far along the pencil you apply that twist.

So we can now consider it that the shear rate strain rate and the velocity gradient is equal for these conditions...

This point clarifies that under linear flow conditions, the shear strain rate (how quickly the shape changes due to shear) is directly related to the velocity gradient (the change in velocity over the distance between the plates). This illustrates a fundamental property of fluid mechanics.

Consider squeezing a sponge. The tighter you squeeze, the more the shape of the sponge deforms, directly related to how hard and how much you push down on it.

We are talking about the stress is proportionality to the strain rate. That means the change of the shear strain rate with respect to time, not the absolute value of the shear strain.

In this segment, it distinguishes between the concepts of stress in fluids versus solids. In fluids, shear stress is proportional to the shear strain rate rather than the absolute change in deformation. This highlights how fluids behave differently under stress compared to solid materials.

Think of cooking pasta. When you stir the water (fluid), it starts swirling faster without visibly changing its shape because the motion (strain rate) increases with your stirring effort (stress) without changing the size of the pot.

That is what it happen this shear strain is a proportionality to the shear strain rate. Again I can emphasize to that shear strain rate not the shear strain.

This emphasizes the critical understanding of Newton's law of viscosity which states that shear stress in a fluid is directly proportional to the rate of shear strain, thus establishing a key relationship in fluid mechanics.

Consider a person trying to push a cart on a smooth floor versus a sandy surface. The cart moves easier and faster on smooth floors (low viscosity) compared to sandy surfaces (high viscosity), demonstrating how the resistance (stress) changes with different types of materials (strain rates).

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