4.5.1. Newtonian Fluids

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 section, we start by examining fluid flow between two parallel plates. The bottom plate has a velocity of zero, representing a stationary condition, while the top plate moves with velocity V. This setup creates a scenario where we examine how fluid behaves under sheer stress. Rather than focusing on molecular motion, we’re concentrating on the fluid as a cohesive unit, referred to as a 'fluid element'.

Imagine a layer of honey spread between two plates. The top plate is like a spoon moving through the honey, while the bottom plate remains still. The honey next to the moving spoon starts flowing, while the honey next to the bottom plate stays put, creating a motion in the fluid.

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.

Under the no-slip condition, the fluid's velocity is equal to the boundary plate it is in contact with. Therefore, at point A (the top plate), it moves with a velocity V, while at point B (the stationary plate), the velocity is 0. Between these two points, there’s a linear increase in velocity, meaning as you move from B to A, the fluid speed increases gradually.

Picture a traffic jam where cars at the front are moving quickly, while those at the back are stationary. The cars gradually speed up as they approach the moving traffic, creating a smooth transition in velocity just like the fluid between the plates.

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

As time progresses, the fluid elements start to deform due to the differences in velocity between adjacent layers. This results in angular deformation, where the shapes of these fluid elements become distorted over time. The angular changes are represented as an angle θ, indicating how much the fluid has adjusted in response to shear stress.

Think of a deck of cards being pushed at one end. The cards on the bottom do not move, but the cards at the top are pushed forward, causing the deck to bend and distort. This bending is similar to how the fluid elements deform under shear stress.

This part indicates 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 velocity gradient refers to the rate at which velocity changes in the fluid as you move along the vertical axis (y-direction) between the two plates. This gradient is constant and equates to the shear strain rate, which helps us quantify how 'quickly' the fluid elements are deforming under shear stress. Essentially, a steeper gradient indicates a greater variation in velocity, leading to higher shear strains.

Consider a slide at a playground: the steeper the slide (higher gradient), the faster kids slide down (higher velocity change). Similarly, in our parallel plates, where the gradient is steep, we experience greater stress and deformation in the fluid.

The shear stress is proportional to the shear strain rate and that is the Newton's laws of viscosity that shear strain is having a, this experimentally proved it.

Newton’s law of viscosity states that the shear stress in a fluid is proportional to the shear strain rate, represented mathematically. This fundamental principle shows that as shear stress increases, so does the strain rate, demonstrating the fluid's resistance to deformation. This law is key to understanding how fluids behave under various flow conditions and reinforces the concept of Newtonian fluids.

Think of spreading butter on bread: the harder you push the knife (increasing shear stress), the more the butter spreads (increasing strain rate). The relationship between your pressure and how easily the butter spreads illustrates Newton's law in action.

Now let us commit that how does the temperature effect on the coefficient of the viscosity. Now we have to look at the molecular levels okay.

Temperature significantly affects the viscosity of fluids. As temperature rises, the molecular motion within liquids increases, reducing the intermolecular forces that hold the molecules together. This weakening leads to lower viscosity, meaning fluids flow more easily at higher temperatures. Conversely, in gases, increased temperature increases molecular motion which can lead to higher viscosity due to more collisions between molecules.

A good example is cooking oil: when heated, it becomes thinner and easier to pour, showing how an increase in temperature decreases viscosity. Cold oil, in contrast, is thicker and harder to pour, demonstrating the effect of temperature on fluidity.

The fluids that follow this relationship, this constant the proportionality relationship between the shear stress and the velocity gradient, we call them Newtonian fluids.

Fluids are classified based on their response to shear stress. Newtonian fluids exhibit a constant viscosity, meaning their shear stress is directly proportional to shear rate. Examples include water and air. Non-Newtonian fluids, however, do not maintain this simple relationship as their viscosity can change with the rate of deformation. Examples include ketchup and toothpaste, which behave differently under varying shear conditions.

Think of running water (Newtonian fluid), which flows steadily and predictably when you turn on a tap. Now imagine squeezing a tube of toothpaste (Non-Newtonian fluid); initially, it resists flow until sufficient force is applied, showcasing how different fluids respond to stress differently.

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