4.5.3. Apparent Viscosity in 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 in 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. 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 analyze how a fluid behaves between two parallel plates, where one is stationary and the other is moving. The point A on the moving plate has a velocity V, while point B on the stationary plate has a velocity of 0. As we move from B to A, the fluid experiences a linear change in velocity—starting from 0 at the stationary plate and increasing to V at the moving plate. This linear velocity profile is important as it sets up the conditions for studying shear stress and viscosity in fluids.

Imagine spreading butter on bread. When you press down the knife (like the stationary plate), the butter starts at rest and gradually slides (like the fluid) towards the edge of the bread as you pull the knife. The butter closest to the knife moves slowly, and the butter further away moves faster, creating a gradient from still (0 m/s at the knife) to moving (the outer edge of the butter).

If it that the conditions now if you look it that if a point which is the y distance from the plate which is at the rest conditions you will have velocity U. okay. So as a linear proportionally we can find out what is the velocity of that one. 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 ∆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.

As the fluid continues to flow, the difference in velocity between the fluid layers results in angular deformations of the fluid element. This means that after a small time interval (∆t), the initially rectangular fluid element becomes distorted. The angle θ represents the degree of deformation at points B, E, F, and D in the element. This illustrates how varying fluid velocities can cause not just a change in speed, but also a change in the shape of the fluid area over time, which is crucial in understanding non-Newtonian flow behavior.

Think of stretching a rubber band. When you pull on one end, the rubber band doesn’t just move but also changes shape—similar to how the fluid elements deform as they flow. The velocity difference between fluid layers can be imagined as different amounts of pull on different parts of the rubber band, creating distortion.

The angular deformations at this point is ∂θ/∂x. Similar way the ∂θ at this point. So consider a very simple geometry of these problems. I can consider these distance of AE is V into delta t divided by l will be the 10 ∂θ since ∂θ is very small, I can consider d∆θ/dt . So that is what we will consider.

The angular deformations generated can be described mathematically with shear strain rates, which are defined as the change in angle (angular deformation) per unit length of the fluid element over time. The relationship between the distance the fluid travels and the time taken further illustrates how the fluid’s internal structure behaves as forces are applied. Here, the rate of angular deformation of the fluid is essential for determining the apparent viscosity.

Imagine holding a balloon filled with water. When you squeeze it slightly, the water inside occupies slightly different angles at its sides. The way the water shifts relative to the squeezing force helps us understand how the internal fluid behavior changes—similar to how we analyze shear strain rate in non-Newtonian fluids.

If I differentiate it, I will get this part, 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 strain rate and the velocity gradient is equal for these conditions.

This discussion leads us into the critical concept of apparent viscosity in non-Newtonian fluids. The velocity gradient (how velocity changes with height) is equivalent to the shear strain rate (how much the fluid deforms over time). In non-Newtonian fluids, the apparent viscosity changes with the speed of deformation, unlike Newtonian fluids where viscosity remains constant regardless of the speed of strain. This concept is key in fluid mechanics, particularly in applications involving complex fluids like polymer solutions and other mixtures.

Consider cornstarch mixed with water, often referred to as a non-Newtonian fluid. When you press or stir it (high deformation rate), it feels solid; but when you apply a gentle force, it flows easily. Its viscosity appears to change based on how hard you’re stirring it, illustrating apparent viscosity effectively.

So we can now define as shear stress is proportional to shear strain rate, and that is the Newton's laws of viscosity that shear strain is having a, this experimentally proved it. That is what it happens this shear strain is a proportionality to the shear strain rate. Again I can emphasize that shear strain rate is not the shear strain.

The relationship of shear stress to shear strain rate forms the basis of Newton’s law of viscosity for Newtonian fluids, where the viscosity remains constant. However, for non-Newtonian fluids, this relationship becomes more complex, and the viscosity varies with strain rate. This distinction is what characterizes how we categorize different fluids and understand their behaviors in applications.

Imagine pushing a heavy object across a carpet versus a smooth floor. You can view the carpet as a non-Newtonian fluid, offering varying resistance based on how fast you push (strain rate), while the smooth floor remains consistently easy to push across, akin to a Newtonian fluid.

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