1.3.1. Newton's Law of Viscosity

Newton's laws of viscosities establish the relationship between shear stress and the velocity gradient.

Newton's law of viscosity describes how a fluid's viscosity operates. It tells us that the force (shear stress) required to move one layer of fluid over another is proportional to the rate at which the layers slide past each other (velocity gradient). The law can be mathematically expressed as:

$\tau = \mu \frac{du}{dy}$

Where:

• $\tau$ (tau) is the shear stress,
• $\mu$ (mu) is the dynamic viscosity,
• ${du}/{dy}$ is the velocity gradient, which measures how quickly the speed of the fluid changes with respect to distance. This relationship is fundamental when analyzing fluid behavior under different conditions, especially in engineering applications.

Imagine spreading honey on a piece of toast. If you apply a small force, the honey moves slowly due to its high viscosity. If you try to spread water with the same effort, it flows much faster because it has a lower viscosity. This behavior illustrates Newton's law of viscosity: different fluids have different responses to applied shear stress based on their viscosity.

Dynamic viscosity (μ) is a measure of a fluid's resistance to flow or deformation.

Dynamic viscosity is an essential property of fluids that quantifies how resistant a fluid is to flow when subjected to shear stress. It signifies how thick or thin a fluid is. A high dynamic viscosity means the fluid is thick, such as oil or honey, while a low value indicates a thin fluid, like water or alcohol. Viscosity can change with temperature—heating a fluid generally reduces its viscosity, allowing it to flow more freely. This relationship is important in various applications, from lubrication to chemical processing.

Think about a pipe. When you pour thick syrup into a pipe, it flows much slower compared to pouring water. The syrup's high viscosity causes it to resist flow more than water does. In practical terms, using a thicker lubricant in machinery can help reduce wear, but it may also require more energy to pump through the system.

This relationship established by Newton allows us to derive other fluid properties, such as capillary height in terms of the diameter of the capillary tube.

The relationship established by Newton's law of viscosity helps derive important phenomena in fluid mechanics, particularly for systems involving small diameters, like capillary tubes. Capillarity describes how liquid rises or falls in a narrow passage due to adhesive forces between liquid molecules and the walls of the tube. The formula for capillary rise is derived from balancing the weight of the liquid column with the surface tension acting along the liquid's perimeter. The height (h) to which a liquid rises in a capillary tube can be represented by:

$h = \frac{2\sigma \cos(\theta)}{\rho g r}$

Where:

• $\sigma$ is the surface tension,
• $\theta$ is the contact angle,
• $\rho$ is the fluid's density,
• $g$ is acceleration due to gravity,
• $r$ is the radius of the capillary tube.

Visualize how a paper towel absorbs water. The water moves up through the towel's fibers, demonstrating capillary action. This occurs because the adhesive forces between the water and the fibers (wall) are stronger than the cohesive forces between the water molecules, allowing it to rise in the towel, similar to how it would rise in a capillary tube.

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