# Stokes' Law: Definition & Application

Instructor: Matthew Bergstresser
Stokes' law relates the terminal velocity of a sphere to its radius, and the viscosity of the fluid it is moving through. This lesson will explain Stokes' law and provide applications of it to other science fields, as well as industrial applications.

## Stokes' Law

Have you ever been caught in the rain? If so, did Stokes' law come to mind when the raindrops struck you? What about driving or walking through fog? Stokes' law applies there too. Let's look at the basics of the law named after its founder, Sir George G. Stokes.

Imagine gently dropping a 2 mm diameter metallic ball bearing into a jar of honey. Did you visualize the ball bearing slowly falling through the honey? If so, you imagined that scenario correctly. Let's think about what might be at play in our thought experiment.

The ball bearing reached terminal velocity almost immediately when it began its journey to the bottom of the honey jar. In order for something to reach terminal velocity when falling down under gravity's pull, there must be an equal force acting up on it. That force is provided by the honey, and it might be different if our honey jar was in the refrigerator versus being stored a cabinet because the viscosity of the honey would be different in each of those situations. Viscosity is the resistance of a fluid to flow due to internal friction. Finally, the only other factor in our scenario is the ball bearing itself.

Now, let's look at how Stokes' law ties all of this together in equation form.

• FD is the drag force on the sphere falling through the fluid in newtons (N)
• η is the viscosity of the fluid in kilograms-per-meter-per-second (kg/m/s)
• r is the radius of the sphere in meters (m)
• vT is the terminal velocity of the sphere in meters-per-second (m/s)

This equation tells us that the larger the ball bearing, the larger the force on it when it is moving through a fluid at terminal velocity. This is where an application of Stokes' law comes in dealing with something we have all experienced - fog and rain! Fog is a collection of extremely small drops of water that are seemingly suspended in the air. A water droplet in a mass of fog with a 1µm radius is actually falling at a 0.1 mm/s. When a water droplet reaches a radius of around 0.25 mm it can be considered a raindrop. Stokes' law applies to raindrops too.

## Industrial Applications

There are industrial applications of Stokes' law. In the early 1930s, research was done on determining the viscosity of glass by measuring the velocity of a platinum sphere falling through glass in its molten state.

In modern times, Stokes' law has been applied to the process of drilling through rock. Lubricating oil is used when drilling in order to prevent excess heat buildup due to friction between the drill bit and the rocks. The oil mixes with particles of the rock creating drilling mud which is pumped out of the drill hole to be filtered and recycled. The viscosity of the lubricating oil affects the rate at which the particulates can be separated from the oil in the recycling process.

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