4.1.3. Angular Deformations

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. Here what we consider that fluid element. That means, I consider some space of A, B, C, D, this is what the fluid element.

This chunk introduces the concept of fluid flow between two parallel plates. One plate is stationary while the other moves with a certain velocity (V). At the position of the no-slip condition, the fluid adjacent to the stationary plate has zero velocity, while the fluid closest to the moving plate has velocity V. The fluid is considered as a 'fluid element' defined by points A, B, C, and D, which allows for a clearer analysis of how velocity varies within the fluid layer.

Think of this scenario as a two-lane highway where one lane is blocked (stationary) and the other is flowing smoothly (moving). The cars (fluid particles) closest to the congested lane (stationary plate) hardly move, while those closer to the free lane (moving plate) can accelerate up to the speed limit (velocity V).

As you know it that 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.

In this section, we discuss the concept of linear velocity distribution within the fluid element. Given that point A is closest to the moving plate with velocity V while point B is adjacent to the stationary plate with a velocity of zero, we can infer that the velocity changes uniformly from point B to point A. This results in a linear gradient of velocities, an important concept when analyzing shear stress in fluid mechanics.

Imagine a water slide—at the top (point A), the water flows quickly, while at the bottom (point B), it slows down as it interacts with the ground. This gradual change in speed creates a 'ramp' of speed that allows the riders to experience a smooth descent rather than a sudden drop.

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

After a small amount of time ∆t, the fluid element will undergo a deformation due to the velocity gradient that exists between the two plates. This deformation results in angular displacements (θ) at points B, E, F, and D, showing how the originally rectangular shape of the fluid element is altered as fluid layers slide past one another. This concept is crucial to understanding how materials respond to shear forces.

Picture stretching a slinky: as you pull one end away from the other, the coils begin to lean and create angles rather than remaining neatly organized. This represents how a fluid element adjusts to external forces, changing shape over time in the flow.

At the ∆t time we have angular deformations of θ. Consider a very simple geometry of these problems. I can consider these distance of AE is V into delta t divide by l will be the 10 θ since θ is very small.

Here, we derive a relationship that relates the changes in deformation to time and geometry. We consider the distance between points A and E (where the angular deformations occur) to calculate how much the angle θ changes over the time interval ∆t. The assumption that θ is small allows us to use small angle approximations which simplify calculations of angular deformations in fluid mechanics.

Think of pushing a door open. If you push it slightly (like a small angle), you can easily predict how much more it will open based on how long you push it—much like calculating the angle change in a fluid element.

The shear strain rate and the velocity gradient is equal for these conditions.

In the context of fluid flow between parallel plates, we find that the shear strain rate is directly proportional to the velocity gradient. This means that how fast the fluid layers slide past one another (shear rate) is directly related to the difference in velocities over distance. This relationship is foundational in fluid mechanics as it describes how fluids behave under shear forces.

Imagine spreading butter on bread. The faster you glide the knife over the surface, the more 'shear' you are applying to the butter. As the speed increases, the layer of butter deforms more—reflecting how shear stress functions in a fluid.

The shear stress is a proportionality to the shear strain rate and that is the Newton's laws of viscosity that shear strain is a proportionality to the shear strain rate.

This chunk introduces Newton's law of viscosity, which states that the shear stress in a fluid is directly proportional to the shear strain rate. This relationship illustrates that as the fluid's layers move faster relative to each other, the resulting shear stress increases. This law is essential for understanding fluid dynamics and is often used in engineering applications to calculate the behavior of fluids under various conditions.

Think of honey. If you apply a gentle force (shear stress), it flows easily (shear strain), but if you try to push it harder (more shear stress), it resists more. This resistance is what Newton’s law of viscosity helps describe.

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