Derivation of Stoke Law

Have you wondered why the raindrops falling from a great height do not harm humans? Stoke’s law can explain this. In this article, you will understand Stoke’s law and its derivation in detail.

What is Stoke’s Law?

Stoke’s Law is a mathematical equation that expresses the settling velocities of the small spherical particles in a fluid medium. The law is derived considering the forces acting on a particular particle as it sinks through the liquid column under the influence of gravity. The force that retards a sphere moving through a viscous fluid is directly proportional to the velocity and the radius of the sphere, and the fluid’s viscosity.

Stoke’s Law Derivation

From Stoke’s Law viscosity equation, we know that viscous force acting on a sphere is directly proportional to the following parameters:

• the radius of the sphere (r)
• coefficient of viscosity (η)
• the velocity of the object (v)

Equating the superscripts of mass, length and time, respectively from equation (2), we get

a = 1… (3)

–a + b + c = 1… (4)

–a –c = -2 or a + c = 2… (5)

Substituting (3) in (5), we get

1 + c = 2

c = 1 (6)

Substituting the value of (3) & (6) in (4), we get

–1 + b + 1 = 1

b = 1 (7)

Substituting the value of (3), (6) and (7) in (1),

The value of k for a spherical body was experimentally obtained

Therefore, the equation gives the viscous force on a spherical body falling through a liquid.

Terminal Velocity Formula

In the case of raindrops, initially, it is due to gravity that it accelerates. As the velocity increases, the retarding force also increases. Finally, when viscous force and the buoyant force is equal to the force due to gravity, the net force becomes zero, and so it does the acceleration. The raindrop then falls with a constant velocity, known as terminal velocity. Thus, in equilibrium, the terminal velocity v_t is given by the equation

ρ and σ are sphere and fluid mass densities, respectively.

From the equation above, we can infer that the terminal velocity depends on the square of the radius of the sphere and is inversely proportional to the viscosity of the medium.