4.2.2. Newton’s Laws of Viscosity

If I consider the same fluid flow through a parallel plate. One of the plates has velocity zero (at rest) and the other top plate is moving with a velocity V. As we consider this from a microscopic point of view, we focus on fluid elements defined by points A, B, C, D. At point A, we have velocity V (due to no-slip condition), while at point B, we have velocity zero. Therefore, the velocity varies linearly from B to A, with maximum velocity V.

This chunk introduces the basic setup of fluid flow between two parallel plates. One plate is stationary, while the other moves, creating a velocity gradient. The no-slip condition states that fluid in contact with a stationary plate must have zero velocity. As we move from the stationary plate (point B) to the moving plate (point A), the fluid's velocity increases linearly. Understanding this setup is crucial as it lays the foundation for discussing shear stress and viscosity.

Imagine spreading butter on bread. The knife (moving plate) moves across the butter (fluid), and the bottom layer of butter stays still due to the bread (stationary plate). The butter closest to the knife moves faster than the layer in contact with the bread, illustrating the linear velocity distribution.

After a short time, these fluid elements will deform, leading to angular deformations. At position B and points E, F and D, similar angular deformations occur, which we denote as θ. Over short time intervals Δt, the rate of angular deformation can be related to the flow characteristics.

As fluid moves between the plates, it doesn't just slide but also deforms angularly. This deformation is essential because it relates to how the fluid exhibits shear stress. The angle θ represents a small deformation related to the flow velocity and time. Analyzing these angular changes helps us understand how fluids respond to forces, which is critical in fluid mechanics.

Think of a rubber band. When you stretch it (apply a shear force), it deforms. After releasing it, the band changes shape, showing its angular deformation. Similarly, fluid elements experience deformation due to shear stress in a flow scenario.

If I consider the y-direction and the angular velocity distributions, I can calculate the velocities at these points (u) and the ratio of the distances. From the relationship established, we see that the velocity gradient corresponds to the rate of shear strain.

The velocity gradient is crucial in defining how shear strain rate is calculated. It measures how quickly the velocity changes over distance (y-direction). In fluid mechanics, this relationship signifies how the fluid reacts under stress. A higher velocity gradient implies that the fluid is more susceptible to deformation, relating directly to the shear stress applied.

Imagine you have a thick syrup. If you tilt the container, the syrup flows slowly at the bottom and faster on top. The difference in velocity is like the velocity gradient — the steeper the incline (or more force applied), the quicker the syrup flows.

The rate of change of shear strain rate is proportional to the shear stress applied. This relationship defines Newton's law of viscosity where shear stress is proportional to the shear strain rate, not to the absolute shear strain, marking a key distinction between fluid mechanics and solid mechanics.

Newton's law of viscosity establishes that shear stress (τ) is directly proportional to the shear strain rate (du/dy), where μ (dynamic viscosity) is the proportionality constant. This means when you apply shear stress to a fluid, its deformation rate increases proportionally, differing from solids, where stress relates to strain. This foundational principle governs fluid behavior under stress.

Think about honey compared to water. When you push each, honey flows slowly and requires more force (higher viscosity) than water. This difference in response under the same stress illustrates Newton’s viscosity law: the thicker the fluid, the more resistance it has to flow.

When discussing the effects of temperature on viscosity, it is observed that with increasing temperatures, the kinetic energy of the fluid molecules increases, leading to reduced viscosity for liquids due to weakened intermolecular forces. In gases, the trend is the opposite and viscosity tends to increase with temperature.

The relationship between temperature and viscosity is critical in understanding fluid behavior. For liquids, increased temperature reduces viscosity because molecules move more rapidly and are less held together by intermolecular forces. Conversely, in gases, increased temperature leads to increased molecular movement, thereby increasing viscosity as more collisions occur among gas molecules.

Consider cooking syrup. As you heat it, the syrup becomes thinner and easier to pour (lower viscosity). Conversely, imagine a gas at high temperatures like steam. As temperature rises, steam can become more difficult to compress, hence increasing its viscosity.

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