The Definition and Derivation of Stokes’ Law

Stokes’ law was derived by an Irish English physicist and mathematician, George Gabriel Stokes’ in the year 1851. The law states about the drag force (commonly known as the friction force) that is applied to spherical shaped particles that have very minute Reynold numbers in a viscous fluid. The law considers forces exerted on objects as they go down through the fluid due to the presence of gravity.

Derivation of Stokes’ Law

Stokes’ Law is derived by getting the solution of the Stoke’s flow limit for small Reynold numbers of the Navier-Stokes equations. Consider the Stokes’ Law viscosity equation, in which the force that is acting on a spherical particle is directly proportional to:

• Coefficient of viscosity,

• Velocity of the object,

• The radius of the sphere.

These can be mathematically represented as,

F ∝ μarbvc……………(1)

1. Now, to calculate the value of the constants a, b, and c, the proportionality symbol must be removed; therefore, we introduce a new constant k, which is the constant of proportionality,

 F =kμarbvc

2. When the parameter’s dimensions are written on both the sides of the equation, the result will be,

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

3. Simplifying the equation,

[MLT^-2]=M^aL^-a+b+cT^-a-c…………………(2)

4. Comparing the power of the independent quantity in equation 2, therefore, mass(M), length(L), and time(T).

M¹=M^a

a=1

T^-2=T^–a-c

-2=-a-c

2=a+c

Substitute the value of a,

2=1+c

c=2-1

c=1

L1=L^-a+b+c

1=-a+b+c

Substitute the value of a and c,

1=-1+b+1

b=1

5. As the values of a, b, and c are known now, substitute the values in the first equation,

 F =kμarbvc

 F =kμ1r1v1

6.  The value of k is determined experimentally for a sphere body, which is equal to 6π. On substituting the value of k, the equation will become,

  F =6πμ rv

Thus, the equation of the viscous force applicable to a falling spherical shaped particle through a liquid is given by the equation:

  F =6πμ rv

Terminal Velocity

The terminal velocity can be defined as the constant speed of a freely falling object, which the object attains when the viscosity of the medium through which the object is falling prevents further acceleration of the object.

At terminal velocity, the extra force (Fg), which is the difference between the weight and the buoyancy of the sphere, is given by the formula:

Fg =(𝞀p–𝞀f)g ×Volume of sphere

Fg=(𝞀p–𝞀f)g 4/3r3

Where 𝝆p and 𝝆f are the mass density of the spherical ball and the fluid, g is the gravitational acceleration, and r is the radius of the sphere.

If the net force is calculated over a spherical particle, then the required force balance needed will be F=Fg, and solving this for the velocity v will give the terminal velocity of the object:

F=Fg

(𝞀p–𝞀f)g 4/3r3 = 6πμ rv

v = 2(𝞀p–𝞀f)/9μ gr2

If 𝞀p>𝞀f, then the flow is upwards, and if 𝞀p<𝞀f, then the flow is in the downward direction.

Application of Stokes’ law

Stokes’ law is widely used to measure the viscosity of the fluid because it is the basis of the falling sphere viscometer. The viscosity of the fluid is measured by putting a metal ball in the fluid whose viscosity needs to be measured. With the known terminal velocity (calculated using the time taken by the ball to reach the bottom of a vertical liquid container), the density of the metal ball and the fluid, and the size of the metal ball, the viscosity of the fluid can be calculated. Usually, multiple ball bearings with varying diameters are used to eliminate the error in the calculation.

A few of the important examples where Stokes’ law is used are as follows:

• To calculate the sedimentation of a minute organism and particle in water,

• To understand the swimming of microorganisms in the human body,

• To calculate the viscosity of the fluid in industries, such as oil.

The same theory can also be used to understand why raindrops, which are smaller, remain suspended in air (in the form of a cloud).

An example of Stokes’ Law

A solid metal ball is put into a long and straight liquid tube filled with oil to check the viscosity of the fluid. The ball’s terminal velocity is 10 m/s, and its radius is 7 cm. It is also known that the density of the metal ball is 7859 kg/m3 while that of the oil is 1100 kg/m3. If the required viscosity of the oil should be of 5 kg/ms for the oil to be used in the machine, then can the oil be used or not?

Solution:

The following information is known:

The radius of the sphere, r = 7 cm = 0.07 m

The density of the metal ball, 𝞀p = 7859 kg/m3 

The density of the oil, 𝞀f = 1100 kg/m3 

The terminal velocity of the metal ball, v = 10 m/s

Let the viscosity of the oil be represented by μ.

Substitute all the values in the formula of terminal velocity:

10 = 2(7859-1100)99.81 (0.07)2

10 = 649.7979

=649.797910=7.22 kg/ms

Since the viscosity of the oil is 7.22 kg/ms, the oil can not be used for the machine.

Conclusion

Stokes’ Law was derived by George Gabriel Stokes’ in 1851. It is derived by getting the solution of the Stokes’ flow limit for small Reynold numbers of the Navier-Stokes equations. It is commonly used to find the coefficient of viscosity of a fluid.