Analysing The Different Types Of Stokes’ Law

The Stokes’ Law was discovered by Sir George Gabriel Stokes. He was an English scientist. Stokes’ Law is a mathematical equation that was derived to calculate the drag force.

Stokes’ Law states that the force acting on a particle sinks the particle in the fluid medium under the influence of gravity. The force is directly proportional to the velocity, fluid viscosity, and the radius of the sphere.

The force is mathematically given as:

Frv

F=6 rv 

Where is the fluid viscosity, r is the radius of the sphere, and v is the sphere’s velocity. 

Assumption Of Stokes’ Law

The following assumptions are made while using the Stokes’ equation:

• The particle must be treated as solid, smooth, and spherical in shape.
• The density of the particle must be uniform.
• The radius of the spherical particle must be greater than 0.001 mm compared to fluid molecules so that the thermal (Brownian) motion of the fluid does not affect the particle.
• Particles should not interfere with each other during the fall.
• The flow should be laminar.

Some Examples

Example 1: A solid ball is falling in a liquid medium that has a terminal velocity of 10 m/s. What is the value of force if the radius of the ball is 5 cm and the viscosity is 1/ kg/ms?

Solution: 

Given

The radius of the sphere = 5 cm = 0.05 m

The viscosity of the fluid = 1/ kg/ms

The terminal velocity = 10 m/s 

We know that the force is given by the Stokes’ Law.

F=6rv

F=610.0510

F=60.0510

F=3 N

Thus, the value of the drag force is 3 Newtons.

Example 2: The value of the drag force is 6 N. Find the value of terminal velocity if the sphere of the radius is 20 cm and the viscosity of the fluid is 0.5 kg/ms.

Solution: 

Given 

The radius of the sphere = 20 cm = 0.2 m

The viscosity of the fluid = 0.5 kg/ms

The drag force = 6 N

We know that the velocity is given by the Stokes’ Law.

F=6rv 

6=60.20.5v

  v=10.20.5

  v=10 m/s 

Applications Of The Stokes’ Law

The applications of the Stokes’ Law are given as follows:

1. Settling of the sediment in freshwater: In this application, the Stokes’ Law is widely used because the sediment falls down in a liquid under the influence of gravity.
2. Measurement of the viscosity of the fluid: Stokes’ Law is used to calculate the viscosity of the fluid.

Limitations Of The Stokes’ Law

The limitations of the Stokes’ Law are as follows:

1. In the case of the colloid, the Stokes’ Law is not valid because the colloidal particle of the same mass falls slower than the others due to the size of the particles.
2. Many particles fall faster due to which they drag finer particles down along with them.
3. If the size of the particle is more than 0.08 mm, then the particle settles quickly due to turbulence.

Stokes’ Law Of Sedimentation

Stoke’s Law of Sedimentation states that the terminal velocity of the sediment particle is directly proportional to the square of particle diameter, the difference between the density of the particle and liquid phase and inversely proportional to the viscosity of the liquid.

vd2(p–f)

v=gd2(p–f)18

Where

p is the density of the particle

f is the density of the liquid

is the viscosity of the liquid 

d is the diameter of the particle 

g is the acceleration due to gravity

Example 1: What is the viscosity of the liquid if the density of the solid ball is 8050 kg/m3, the diameter of the particle is 0.1 m, and the terminal velocity is 10 m/s? Assume the density of the fluid is 1000 kg/m3 and g = 10 m/s2.

Solution:

Given 

The terminal velocity (v) = 10 m/s

The density of the particle (p) = 8050 kg/m3

The density of the liquid (f) = 1000 kg/m3

The diameter of the particle (d) = 0.1 m

We know that the Stokes’ Law of Sedimentation will be:

v=gd2(p–f)18

Then, the viscosity of the fluid is given as:

=gd2(p–f)18v

If we put all the values in the above equation, we have the following:

=100.12(8050-1000)1810

=100.017050180

= 705180

=3.92 kg/ms

Precautions While Performing The Stokes’ Law Of Sedimentation

Some precautions are given as follows:

• The diameter of the sphere must be taken very accurately.
• The ball is to be wetted thoroughly in the experimental liquid before dropping the ball in the liquid.
• The temperature of the place where the experiment is being conducted remains unchanged because the viscosity gets affected by the temperature.

Conclusion

Using the Stokes’ Law, we can predict the settling velocity of particles of different sizes. The large grain settles faster than the small size grain. Stokes’ Law of Sedimentation is generally used to calculate the value of the terminal velocity, but we can also calculate the value of the viscosity of the fluid.