4.5. Newtonian and Non-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. 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.

In this chunk, we explore fluid flowing between two parallel plates. One plate is stationary (velocity = 0), while the other moves with a velocity V. This creates a scenario where the velocity of the fluid is not uniform. At point A (near the moving plate), the fluid has a high velocity, while at point B (near the stationary plate), the velocity is zero due to the no-slip condition. The flow in between results in a linear increase in fluid velocity as we move from point B to point A.

Imagine spreading butter on a piece of bread with a knife. The knife is like the moving plate, and the butter represents the fluid. The butter close to the knife moves quickly, while the butter farther from the knife stays still until you apply more pressure. This helps visualize how fluid layers slide over each other due to different velocities.

So next one is the velocity lesser than that or above one will go moving with the fluid is more than that. So if as a linear velocity distribution as you assume it as we got it some of the parallel flow conditions after ∆δt time these fluid element will be deformed as a angular deformations will happen it that there will be angle δ will come it at the B and E, F and D.

As the fluid continues to flow, after a time interval (∆t), the fluid element will deform. This deformation can result in a change in the shape of the fluid element, indicated by angular deformations at points B and D. The fluid's behavior can be analyzed using the concept of velocity gradient, which describes how velocity changes with distance across the gap between the two plates.

Consider rolling dough with a rolling pin. As you apply pressure, the dough doesn’t just slide; it stretches and deforms. Similarly, as the fluid flows, it deforms, demonstrating how the flow of different layers interacts.

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 . 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.

Newton's law of viscosity relates shear stress to the shear rate (velocity gradient). It states that shear stress (τ) is proportional to the velocity gradient, represented mathematically as τ = μ (du/dy), where μ is the dynamic viscosity. This indicates that in fluids, shear stress increases with the rate at which the velocity changes from one layer of fluid to another.

Think of honey pouring from a bottle. It flows slowly due to its high viscosity compared to water. If you were to apply a force (like shaking), the honey moves more easily. This resistance to flow at different rates illustrates how viscosity affects the behavior of fluids under stress.

Now we are coming to the exactly the solid mechanics concept the stress having the proportionality with the strain. 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.

In solid mechanics, stress is proportional to strain (the deformation of the material). However, in fluid mechanics, stress is proportional to the rate of strain (shear strain rate). This distinction shows how fluids respond differently to stress compared to solids; fluids can flow and change shape continuously, whereas solids have fixed shapes until a certain limit.

Imagine pushing on a rubber ball. It deforms under your pressure (strain), but if you stop pushing, it returns to its original shape. In contrast, when you apply pressure to water in a container, it simply moves and takes the shape of the container without returning to a ‘resting’ state.

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.

Temperature affects viscosity significantly. For liquids, as temperature rises, molecular motion increases, which reduces the cohesive forces between molecules. This decrease in interaction results in lower viscosity. Conversely, for gases, increasing temperature increases the random motion of molecules, which can lead to an increase in viscosity as the gas particles become more active.

Consider making ice tea. When you add ice to hot tea, the sugar dissolves more slowly due to higher viscosity at lower temperatures. Yet if you were to heat the tea, the sugar dissolves rapidly, showcasing how temperature changes a liquid’s viscosity.

So the fluids those follow this relationship, this constant the proportionality relationship between the shear stress and the velocity gradient, we call them Newtonian fluids, okay. Those fluid flow problems we call the Newtonian fluids. That means what the coefficient of viscosity which is a directly proportionality constant, that does not vary as the rate of deformations are, or the time is changing.

Newtonian fluids are defined by their consistent viscosity that does not change, regardless of the shear rate. Common examples include water and air, where the relationship between shear stress and flow remains linear. Non-Newtonian fluids (like ketchup or toothpaste) don’t have a constant relationship; their viscosity can change with the applied stress or rate of deformation.

Think of stirring maple syrup versus water. Maple syrup is thick (non-Newtonian) and requires more energy to stir; its flow changes under different forces, while water (Newtonian) flows smoothly, maintaining its viscosity regardless of the force applied.

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