Stokes’ Law

Sir George G. Stokes first introduced the Stokes’ law in 1851. Stokes’ law calculates the drag force experienced by a spherical object in a fluid medium. Now, drag force is the frictional force felt by an object which lies in a fluid medium due to the movement of the liquid medium.

The law further evaluates the sinking period of a spherical object in a liquid medium as it experiences the pull of gravity towards the bottom of a liquid column or liquid container. According to Stokes’ law, the drag force resisting the downward pull is equal to 6πrηv.

Where,

r denotes the radius of the spherical object

η denotes the viscosity of the liquid

and v denotes the velocity of the fall of the object

Stokes’ law equation

The viscosity of a liquid is the thickness of a liquid that affects the movement of an object which is placed within the liquid medium and can be derived from Stokes’ law. The law takes the example of a spherical object, and it calculates the downward fall of the object in the viscous medium.

Mathematically, finding the velocity of the spherical object experiencing the drag force due to the viscosity of the liquid medium was not possible until Sir George G. Stokes found the equation for calculating the drag force in the year 1851.

The equation for Stokes’ law is as follows:

F =6πrηv

Now, it is also important to note that the drag force or the velocity of the sinking spherical object is directly proportional to the radius of the sphere. People often confuse this with the drag force being directly proportional to the density of the medium, but that is certainly not the case.

Also, the drag force or the velocity of the spherical object being dragged to the bottom of the liquid column is directly proportional to the speed with which the object is moving towards the bottom of the liquid medium.

The Stokes’ law is used in the derivation of the drag force of the sediments in the fresh water and a variety of other fluid mediums. Earlier, the derivation of the drag force was limited to only certain categories of liquid mediums due to the factor of turbulence underwater. However, the law has been modified to fit the other categories and requirements as well.

Stokes’ law derivation

Sir George G. Stokes experimented to calculate the viscosity of the liquid medium with a sinking spherical-shaped object which is being dragged to the bottom of the liquid column due to the gravitational force, also more commonly known as the drag force.

The viscous force (which is expressed by the letter F) of the liquid medium is dependent on the following measurements:

• • The radius of the sphere, which is denoted by the letter r
  • The velocity of the spherical object falling downwards into the liquid medium which is denoted by the letter v

• Coefficient of viscosity of the liquid, which is denoted by the symbol n

Derivation of Stokes’ law:

F  ∝ ηarbvc

• F = kηarbvc   ………….( i)

Where k = proportionality constant

Dimension of

[F] = [MLT^-2]

[η] = [ML^-1T^-1]

[r] = [L]

[v] = [L¹T^-1]

Putting on dimensions in equation i)

[MLT^-2] = [ML^-1T^-1]a [L1] b [LT^-1]c

[MLT^-2] = [MaL-a+b+cT-a-c]

• Equating the powers of M, L, and T

a = 1 ……………… ii)

-a+b+c = 1 …………………. iii)

-a-c = -2 …………………………. iv)

From equation iv)

Put the value of a

-1-c = -2

-1+2 = c

C = 1

Put value of a and c in equation iii)

-1+b+1 = 1

b = 1

So, from equation i)

F =kn1r1v1

Where k = 6π

Hence, F = 6πnrv

Stokes’ law applications

Following are some of the common applications of Stokes’ law:

• Floatation of the clouds can be explained by Stokes’ law
• The raindrops with larger density hurt more than the raindrops with smaller density
• A person who’s coming down with the help of a parachute gains constant velocity throughout the fall due to the drag force of gravity.

Conclusion

Stoke’s law was derived by Sir George G. Stokes for the calculation of the viscosity of a liquid medium with a spherical object sinking at a certain velocity which is initiated by the drag force of the gravity on the object.

The drag force of the spherical object is directly proportional to the radius of the spherical object and also the speed of the object sinking in the liquid medium. The equation for calculation of the Stokes’ law was found in the year 1851.