4.5.2. 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.

This chunk discusses the concept of fluid flow between two parallel plates where one plate is stationary, and the other is moving. It sets the stage for understanding shear stress by defining a 'fluid element' as the space between these plates. At one point (Point B), the fluid has zero velocity, while at the other point (Point A), the fluid has a velocity 'V'. This arrangement allows us to visualize how velocity changes across the gap between the plates.

Think of a sandwich with peanut butter where one slice of bread is held in place while the other is being spread with a knife. The peanut butter moves with the knife, creating a flow while the bottom slice remains stationary. Here, the peanut butter is the fluid, and the slices of bread represent the plates.

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. That linear velocity from zero to V as the y increases, the velocity will increase it and maximum velocity will be the V.

The velocity profile of the fluid is described as linear between the two points (A and B) due to the no-slip condition, meaning the fluid at Point B (the stationary plate) has zero velocity, while at Point A (the moving plate), it reaches the maximum velocity 'V'. Hence, as we move from the stationary plate towards the moving plate, the velocity of the fluid increases in a linear manner.

Imagine riding a slide. At the starting point (the top), you are at rest (zero velocity), but as you slide down (towards the bottom), you pick up speed. The increase in speed is consistent, similar to how fluid velocity increases linearly between the two plates.

After ∆t time these fluid element will be deformed as angular deformations will happen it that there will be angle θ will come it at the B and E, F and D. That is a new positions of the fluid element after ∆t time. 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 tan θ.

This chunk discusses how, after a certain time (∆t), the fluid elements experience angular deformation due to the shear caused by the movement of the top plate. The deformation results in an angle (θ) at points where the fluid has been displaced. The calculation of this angle involves the fluid's velocity and the distance between the plates.

Consider a piece of clay that you are stretching. If you pull the top of the clay in one direction while keeping the bottom stationary, the shape of the clay changes. The change in shape represents angular deformation similar to how fluid elements behave between the plates.

The shear strain rate and the velocity gradient are equal for these conditions. The shear rate strain rate is defined as ∂u/∂t dy. These parallel flow conditions when you have velocity is linear distributions, the shear strain rate and the velocity gradient they are equal.

In non-Newtonian fluid mechanics, it's essential to understand the relationship between shear strain rate and velocity gradient. Under constant conditions, when fluid flows between the plates, the rate at which the fluid deforms (shear strain rate) is equal to the rate of change of velocity across the distance (velocity gradient). This indicates how quickly the velocity is changing as you move away from the stationary plate.

Picture spreading butter on toast. The rate at which the butter spreads (how quickly it deform) is directly related to how fast you are moving the knife (velocity gradient). As you push the knife more quickly, the butter strains and spreads more, showing a similar relationship to fluid flow.

The stress having the proportionality with the strain. Here, we are talking about the stress is proportionality to the strain rate, indicating a basic distinction between fluid and solid mechanics.

This part encapsulates Newton's laws of viscosity, where shear stress in fluids is directly proportional to the shear strain rate. This is a critical difference from solid mechanics, where stress is proportional to strain. In fluids, as the rate of deformation increases, so does shear stress, evidencing a dynamic relationship in non-Newtonian fluids.

Think about a sponge versus a rock. When you press hard on a sponge (fluid), it compresses easily (strain), and if you press faster (strain rate), it also resists more (stress) compared to a rock that just fractures under pressure. This illustrates the differences in behavior under stress for fluids and solids.

We define the non-Newtonian fluids based on their behavior: shear thickening fluids, which require more force as deformation rate increases, and shear thinning fluids, which become easier to deform under increasing shear.

Non-Newtonian fluids can behave differently under stress compared to Newtonian fluids. Shear thickening fluids become more viscous when stirred or mixed faster, requiring more effort to move. Conversely, shear thinning fluids are easier to stir as you increase the flow rate or movement. Understanding these behaviors is essential in various applications where flow characteristics change with force.

Consider cornstarch mixed with water, often referred to as oobleck. When you gently press or stir it, it flows easily (shear thinning), but if you push it hard, it becomes solid-like (shear thickening). This behavior perfectly illustrates the unique properties of non-Newtonian fluids.

How does the temperature effect on the coefficient of the viscosity? The viscosity generally decreases with increasing temperatures for liquids due to reduced intermolecular forces.

Temperature has a significant impact on fluid viscosity. As temperature increases, the kinetic energy of fluid molecules increases, meaning they move around more and the forces holding them together weaken. For liquids, this results in decreased viscosity as they flow more easily. In contrast, gases may exhibit increased viscosity with higher temperatures due to increased molecular collisions.

Imagine how honey behaves. When it's cold, honey flows slowly, but if you heat it up, it pours out much more readily. The decrease in viscosity with temperature makes it easier for the honey to flow, showcasing how temperature influences fluid behavior.

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