4. Fluid Flow Through Parallel Plates

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.

This chunk introduces the scenario of fluid flow between two parallel plates. One plate is stationary (with zero velocity), while the other moves at a velocity 'V'. The term 'fluid element' refers to a volume of fluid we will analyze in this context. This setup is essential to understand how fluids behave under shear stress, which occurs when layers of fluid slide past each other.

Think of it like two slices of bread with jam in between: if you spread the jam by pushing the top slice, the jam (fluid) gets sheared as the slices slide over each other, with one slice being stationary and the other moving.

That means, I consider some space of A, B, C, D, this is what the fluid element. That means, whatever the fluids are there that reasons, I am defined as a fluid element. 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 part, we define a fluid element between two points A and B. Because of the no-slip condition (where fluid adheres to the surfaces), the fluid at point A (the moving plate) moves at velocity V, while at point B (the stationary plate), the fluid is stationary with zero velocity. Consequently, the velocity of the fluid varies linearly between these two points. This linear relationship is crucial for understanding the gradient of velocity in fluids.

Imagine a tapered slide; the top of the slide (point A) is where kids start sliding down (moving quickly), and the bottom (point B) is where they stop (no movement). The speed of the kids changes smoothly from fast to slow as they slide down.

If as a linear velocity distribution as you assume it as we got it some of the parallel flow conditions 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.

After a time interval (∆t), the fluid element will experience deformation due to the velocity gradient. This means that the fluid element will no longer be rectangular but will instead become skewed, forming angles (θ) at certain points. This demonstrates how the motion of one plate influences the entire fluid element, leading to internal stresses and strains.

Think of stretching a rubber band. As you pull it on one end, the rubber band gets thinner and longer, creating angular distortions in the shape. Similarly, as one plate moves, it stretches and distorts the fluid element.

So that is what will consider. Just rearranging that we can see that the rate of change of angular deformations of this fluid element is given by ∂θ/∂t = v / l.

Here, we examine the rate of angular deformations in the fluid element. The equation derived shows that this rate depends on the constant velocity (v) and the thickness (l) of the plates. This relationship helps establish a quantitative measure of how shear stress affects the deformation rate within the fluid.

Consider turning a steering wheel in a car. The speed at which you turn the wheel affects how quickly the wheels of the car turn. Similarly, how quickly the fluid deforms relates to the speed of the moving plate and the distance between the plates.

If I consider this is the y directions how the velocity varies along the y directions that is the constant which is equal to the shear strain rate.

The velocity gradient reflects how the fluid's velocity changes perpendicular to the direction of flow (along the y-axis). This gradient correlates directly with the shear strain rate, which measures how quickly the shape of the fluid changes over time. Understanding these concepts is vital in fluid mechanics as they lay the foundation for how shear stress is calculated.

Imagine a stack of pancakes where each pancake is sliding on top of the one below it. The way the top pancake's speed varies compared to the stationary ones below it gives us insight into the shear rate just like how the velocity gradient shows changes in fluid flow.

But here, 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.

This section introduces Newton's law of viscosity, which states that shear stress is directly proportional to the shear strain rate. In fluids, it’s important to focus on how quickly the fluid deforms (strain rate) rather than the total deformation. This principle distinguishes fluid behavior from solid materials, where stress correlates with deformation directly.

Imagine pulling taffy; the force you apply and how fast the taffy stretches reflects the viscosity. The faster you pull, the more resistance you feel, akin to the shear strain rate.

The shear strain is having a proportionality to the shear strain rate. Again I can emphasis that shear strain rate not the shear strain.

This emphasizes the difference between shear strain (geometry of deformation) and shear strain rate (speed of deformation). In fluid dynamics, we often focus on the rate at which the deformation occurs because it directly impacts the fluid's behavior under shear stress.

Think of a balloon being inflated; the rate at which the air is pumped affects how fast the balloon expands, rather than just how much larger it becomes overall.

We can define as τ = μ (∂u/∂y).

This formula represents the relationship between shear stress (τ), dynamic viscosity (μ), and the velocity gradient (∂u/∂y). It summarizes how the fluid flows with respect to the force applied, establishing a direct relationship that is foundational in fluid mechanics.

Think of pushing a thick syrup; the harder you push (more shear stress), the faster it flows. The viscosity indicates how resistant the syrup is to the push.

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