Stokes’ Law Derivation

Stokes’ law predicts the velocity of a spherical particle moving through a fluid medium with a finite velocity. The radius of such a particle is assumed to be very small. The force acting on the particle was found to be proportional to the radius of the particle, the viscosity of the medium and the velocity of the particle when entering the medium. When the particle accelerates in the medium due to gravitational pull, the drag force on it also increases proportionally, until the particle has zero net force acting on it. Then the particle settles into uniform motion with a terminal velocity.

This law was derived by George Gabriel Stokes around 1851.

The fluids where Stokes’ law can be applied are said to be in Stokes flow. Stokes flow is a type of fluid flow where it is assumed that the viscous forces that act in a fluid are greater than the flow inertia of the fluid. Flow inertia can be considered the same as the inertia of a solid object-- it defines the reluctance of any substance in changing from a state of motion or rest to vice versa. In everyday life, Stokes flow can be observed in fluids that have a very large velocity or are moving very slow. It can also be observed over short distances in more free-flowing fluids. For example, stokes flow can be easily observed in honey. Honey has a very large viscosity and when it flows, it moves with a very small velocity. As a result, honey has very low turbulence when it is flowing.

Derivation

The statement of Stokes’ law can be made as: The drag forces, which involve the viscous and frictional forces, acting on a spherical particle moving through a fluid is given by the equation:

F = 6πηrv

Where,
F is the drag force acting on the particle
π is a mathematical constant
η is the viscosity of the fluid
r is the radius of the particle
v is the velocity of the particle when entering the fluid

Stokes assumed the following conditions before deriving his equation:

• The fluid is flowing in a laminar flow. Laminar flow happens when the flow of the particle has very low disturbance, which results in a clear, smooth flow. Practically, this is seen when a fluid retains the shape in which it was flowing before being let out by the dispenser.
• The particle moving through the fluid is spherical.
• The fluid is homogeneous in nature and is not made up of a mixture of two different fluids.
• All the surfaces involved are smooth.
• All the particles keep out of each other’s way and do not interfere in the movement of the other.

The above equation can be derived in the following way:

We know that the drag force acting on a particle moving through a fluid depends upon the following factors:

• The radius of the sphere
• Coefficient of viscosity
• The velocity of the sphere when entering the fluid

Therefore, the general statement of proportionality can be written as:

F ∝ η^ar^bv^c

To remove the sign of proportionality, we have to introduce a constant K.

F = Kη^ar^bv^c ………..{1}

Now, replacing the physical dimensions of each of the quantities mentioned in the Eq. {1}, we get:

[MLT^-2] = [ML^-1T^-1]^a [L]^b [LT^-1]^c ……….{2}

K is a dimensionless constant and therefore we do not have to consider it while equating the dimensions of other quantities. Simplifying Eq. {2}, we get:

[MLT^-2] = [M^aL^{-a +b +c}T^{a -c}]

Mass, length and time are considered independent quantities in classical mechanics, so we can freely compare their corresponding powers. Doing so, we get the following equations:

a = 1 ………{3}
-a +b +c = 1 ……….{4}
-a -c = -2 ……….{5}

Replacing Eq. {3} in {5}

c = 1 ……….{6}

Further replacing Eq. {3} and {6} in Eq. {4}

b = 1 ……….{7}

Replacing the values of the found powers back in Eq. {1}, we get:

F = Kηrv ……….{8}

The value of the constant K was found to be 6π experimentally. Therefore, the final Stokes’ equation takes the form:

F = 6πηrv ……….{9}

Eq. {9} is called Stokes’ law.

Stokes’ law can be used to design a falling-sphere viscometer, which is a device that is used to measure the viscosity of a fluid. It consists of a fluid stored in a vertical glass tube. A sphere is allowed to travel through the liquid in controlled conditions. The density and radius of the sphere are known beforehand. The sphere attains terminal velocity. This data combined with the density and radius of the sphere and also the density of the fluid is used to determine the viscosity of the fluid.