4.4. Dynamic Viscosity Correlations

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. 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.

This chunk describes a fluid flowing between two parallel plates. One plate does not move (zero velocity), and the other moves at velocity V. The fluid elements between these plates behave differently based on their position. For instance, the fluid at the moving plate (point A) has a velocity of V, while fluid at the stationary plate (point B) remains at zero velocity. This creates a velocity gradient between the two points, crucial for understanding shear stress in the fluid.

Imagine a row of children on a slide. The child at the top is sliding down quickly (point A), while the child at the very bottom (point B) is at a complete standstill. The space between them represents varying speeds and interactions in the fluid.

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. 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.

The concept of linear velocity distribution is introduced. As the distance from the stationary plate increases (moving vertically up through the fluid), the velocity of the fluid also increases linearly from zero to V. This relationship can be visualized as a linear graph where the y-axis represents fluid velocity and the x-axis represents distance from the stationary plate.

Think of a highway where cars gradually speed up as they approach the entrance ramp. The closer cars are to the ramp (point A), the faster they're moving, while cars waiting at a red light (point B) aren't moving at all. The gradual speed increase is similar to how fluid velocity changes between two plates.

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. That is a new positions of the fluid element after ∆t time. And the angular deformations at this point is θ.

As time progresses, the initially positioned fluid elements (A, B, C, D) will undergo deformation because of the motion of the upper plate. This deformation can create angular displacements (θ), leading to a change in the orientation of the fluid elements. This aspect highlights how fluid elements do not just slide; they also twist and turn under shear stress.

Picture a soft plastic sheet being pulled at one end. The closer the top edge (the moving end) gets to the fixed bottom edge, the more it bends and twists. Similarly, the fluid elements adjust their positions based on the shear forces acting on them.

The shear rate strain rate and the velocity gradient is equal for these conditions. Then the shear rate is ∂u/∂y. These parallel flow conditions when you have velocity is linear distributions the shear strain rate and the velocity gradient they are equal.

In this section, the relationship between shear rate, strain rate, and velocity gradients in fluid flow is discussed. Under linear flow conditions, the shear rate (how quickly the fluid is deforming) is directly proportional to the velocity gradient (difference in velocity over a distance). This relationship is key to understanding the behavior of fluids under shear.

Imagine squeezing a tube of toothpaste. The faster you push (the shear rate), the more the toothpaste moves. The change in motion relative to how far you push it (the velocity gradient) helps us determine how much force is needed to squeeze the paste out.

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.

Newton’s law of viscosity states that shear stress is proportional to the shear strain rate over time, which means that stress in a fluid due to shear is directly related to how quickly it deforms. This equation implies that fluids behave differently than solids — the relationship is not based on static deformation.

Think of how honey pours compared to water. If you gently tilt the honey container, it flows slowly (high viscosity), whereas water rushes out quickly (low viscosity). The stress applied (tilting the container) leads to different strain responses due to the fluids' viscosities.

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. 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. So gas is at more random motions as compared to the molecules are more random motion as compared to the liquids.

Temperature affects viscosity significantly at the molecular level. In liquids, attractions between molecules (bonding forces) are stronger than in gases. As temperature increases, the energy given to molecules results in greater motion, disrupting these bonds for liquids (lowering viscosity), while in gases, it increases randomness (affecting viscosity differently). Understanding this relationship helps predict fluid behavior under different thermal conditions.

Consider how syrup thickens in the fridge but flows freely at room temperature. As the syrup warms, molecular motion increases, making it less viscous and easier to pour. Conversely, cold temperatures can significantly decrease fluid flow, making it sluggish and thick.

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