All About Applications of Stokes’ Law

Stokes’ Law is an expression derived by George Gabriel Stokes to determine the drag force exerted on a spherical body falling in a viscous fluid. The body must have a very small Reynolds number.

The law states that the force of viscosity on a sphere that is moving through a viscous liquid is given by,

Fd = 6πηRv

where:

Fd is the force of friction that acts on the interface between the fluid and the particle

η is the dynamic viscosity of the fluid

R is the radius of the spherical object

v is the flow velocity relative to the object

The statement states that the viscous force acting on a spherical body of radius r depends directly on:

• The radius of the sphere
• The velocity of the sphere
• The coefficient of viscosity of the considered liquid

Assumptions for Stokes’ Law

Stokes’ law makes the following assumptions:

• The fluid must have a laminar flow.
• The particles under study must be spherical.
• The fluid must be homogeneous.
• The surface of the spherical object must be smooth.
• The particles must not interfere with each other by any means.

Terminal velocity

Terminal velocity is the maximum velocity attained by a spherical object as it falls through something fluid in nature. This happens when the sum of the drag force and the buoyancy is equal to the force of gravity that acts in the downward direction. The object hence has zero acceleration. 

In fluid dynamics, if the object has a constant speed, it means that the body is experiencing a restraining force exerted by the fluid, and hence it has a terminal velocity. 

The drag force acting on a body is directly dependent on the body’s speed. Hence as the speed increases, so does the drag force. Then comes a moment when the speed, the drag, or the force of resistance becomes equal to the gravitational pull on the object. At that particular moment, the body stops accelerating as no net force acts on the spherical body. The speed of the spherical body becomes constant, and this constant speed is called terminal velocity.

Practical Applications of Stokes Law

Here are some practical applications of Stokes law:

1) The fall of raindrops

When the water droplets are small in size, their terminal velocities decrease. Rain droplets usually don’t acquire high velocity during their free fall. As the raindrops are small in size, their terminal velocities are also small. Therefore they remain suspended in the upper atmosphere for a long time. These suspended water droplets are usually referred to as clouds.

As the drops combine and their size increases, their terminal velocity starts to increase as well, and after a certain time, they become heavy and start falling like rain. Therefore, Stokes’ law explains two natural phenomena very well. The first is the floatation of clouds, and the second is the reason for the larger rain droplets hurting more than the smaller ones. The larger rain droplets have a larger terminal velocity, and when it hits us, we feel more pain.

2) Jumping with a parachute

A parachute is an equipment that helps a person land safely on the ground when they jump from a higher altitude. A human falling down from a higher altitude acquires terminal velocity. But if the velocity is higher, the impact will kill the person. So, a parachute is used to decrease the terminal velocity. This happens because of the drag force of the atmosphere acting on the surface area of the parachute. 

3) Bubbles in beer

The bubbles in a glass of beer show the contrasting effect of Stokes’ law as in the case of a rain droplet. As the bubbles are lighter than the surrounding fluid, they get an upward push. By Stokes’ law, the bubbles in the beer glass rise at a constant speed. However, it can be seen sometimes that bubbles move downward. This is due to the circulation of the fluid in the beer glass. As the bubble rises up, it drags some beer along with it. Hence such bubbles are dragged downwards due to gravity.

4) Sedimentation analysis

Small particles of a certain size usually settle down at the bottom of a container as multiple forces act on the body. This process is called sedimentation. 

This property of small soil particles is usually used for sedimentation analysis. In sedimentation analysis, the particle size distribution is analysed. Particle size distribution is the separation of soil into various fractions based on their sizes. 

Soil particles smaller than 75 microns can’t be analysed by sieve analysis. Hence sedimentation analysis based on Stokes’ law is used here. Two forces act on the soil particle in an upward direction – the buoyant force and drag force. These forces act in the opposite direction of the gravitational force. Subsequently, the particle reaches equilibrium condition, and the particle’s speed becomes constant. These particles fall with constant velocity, and this velocity is known as terminal velocity. The size of the particle can be estimated if you find out the terminal velocity.

Conclusion

Stokes’ law states that if the buoyant force and the drag force oppose the gravitational force, the spherical body travels at a constant speed called terminal velocity. Stokes’ law is an important concept with plenty of real-life applications. Stokes’ law has also contributed to several Nobel award wins as well.