4.4.1. Sutherland Correlation

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.

In this discussion, we begin with fluid flowing between two parallel plates. One plate is stationary (at rest), represented as having a velocity of zero, while the other plate is moving with a velocity (V). This setup allows us to consider the concept of shear stress not at the molecular level but at the level of fluid elements, which are defined spaces within the fluid where we analyze characteristics like velocity and deformation.

Imagine a thick syrup being spread on a cutting board. One side of the board is stationary, while you drag a spatula (the other plate) across the syrup. The syrup closest to the cutting board has no movement (like the stationary plate), while the syrup at the top moves with the spatula. The flow of syrup from one side to the other illustrates how fluid elements behave under the influence of movement.

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.

Due to the no-slip condition in fluid dynamics, the velocity at the stationary plate (point B) is zero, while as you move towards the moving plate (point A), the velocity gradually increases to V. This results in a linear change in velocity throughout the fluid layer, which can be visualized as a gradient. Essentially, as you move up from the stationary plate (B) towards the moving plate (A), the speed of the fluid increases proportionally.

Think about a slide at a playground. If the bottom of the slide is against the ground (no height or velocity) and the top of the slide is where the kids gain their maximum velocity as they go down. As they slide down, they start from a stop and gradually build speed; similarly, in the fluid flow, you go from zero velocity at the lower plate to maximum velocity at the top plate.

Now if I consider the at the y distributions and the angular velocity distributions if I consider it I will get it the u, the velocity at these points and ratio of the distance as a linear velocity distributions concept, we can get it, √V/y.

In fluid dynamics, we define shear stress and shear strain rate. The velocity gradient, which is the change in velocity per unit distance (here, represented as V/y), illustrates how fast the fluid's velocity is shifting as we move away from the stationary plate. This is essential for understanding how shear stress develops in the fluid, as shear stress is directly proportional to how quickly the velocity is changing at a given point in the fluid.

Consider frosting a cake with a spatula. When you apply pressure and drag the spatula across the frosting (the fluid), the movement creates different levels of velocity at different heights of the frosting. The bottom layer (the closest to the cake) doesn't move, while the top is spread faster. This unequal distribution creates a shear effect.

We remember this is what is indicating us is that the velocity gradient. 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 shear deformations per unit time. So we can now consider it that the shear rate strain rate and the velocity gradient is equal for these conditions.

The relationship established indicates that the velocity gradient in the fluid (the rate at which velocity changes with vertical position) is directly tied to the shear strain rate. This connection allows us to use Newton's law of viscosity, which states that shear stress (force per area) is proportional to the shear strain rate, giving us a framework for calculating how fluids behave under stress.

Think about honey and how it behaves when you pour it. If you pour it slowly, it flows steadily and smoothly. As you increase the speed at which you pour, the resistance (or shear stress) increases due to the honey's viscosity. The faster you try to pour, the stickier it feels. This is the shear strain rate in action!

Now let us commit that how how does the temperature effect on the coefficient of the viscosity. Now we have to look at the molecular levels okay. So if you look at this molecular levels, when you talk about the liquids, they will have a molecular bonding forces between two molecules okay. But that is much weaker when talk about the gases.

Viscosity is also affected by temperature. As temperature increases, the energy of the molecules increases, causing them to move more rapidly and reducing the intermolecular forces holding them together. This means that typically, the viscosity of liquids decreases as temperature rises. In contrast, gases, which already have weaker intermolecular forces, can experience increased viscosity with rising temperatures due to greater molecular motion and collision frequency.

Think about how syrup flows at room temperature—it is thick and slow-flowing. However, when heated, it becomes much thinner and flows easily. On the contrary, if you heat up some air (like when blowing up a balloon), it expands and can lead to an increase in speed, thereby affecting how it behaves under certain conditions.

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