Stokes’ Law Definition Equation Formula with Examples

Stokes’ law explains why rainwater falling from the sky does not harm us on Earth. Stokes’ principle is an essential part of physics. Stokes’ law is a mathematical equation relating the drag force experienced by small spherical particles moving through a dense fluid medium. It deals with the resistive (frictional) force applied to a body under the action of gravity when falling through a liquid or air.

Historically, Stokes’ law was named after Anglo-Irish physicist, and mathematician George Gabriel Stokes formulated an expression for the drag force in 1851. In Stokes’ law, the upward drag force acting against the fall is equal to F 6πrηv, where r is the sphere’s radius, η is the fluid’s viscosity, and v is the velocity of the fall. Stokes’ law discusses the active force exerted on a body when it falls into a fluid. Initially, due to low viscous force, the velocity of the falling body is low.

Stokes Law

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According to Stokes law, the drag force Fd experienced by a spherical particle flowing through a viscous fluid is given by the following formula:

Stokes Law Formula

F_d = 6πηrv

Where,

η is the viscosity of the fluid.

r is the radius of the particle.

v is the velocity of the particle relative to the fluid.

Assumptions of Stokes Law:

Stokes’s law is valid under the following conditions.

Cells should be firm, smooth and spherical.

Particles should be of uniform density.

The particles must be large enough (>0.001 mm) compared to the liquid molecules so that the thermal (Brownian) motion of the liquid molecules does not affect the particles.

Particles should not interfere with each other during the collapse.

Stokes Law Derivation

Stokes’ law can be derived using dimensional analysis. Consider the forces acting on a spherical particle as it sinks through a liquid column under the action of gravity. The viscous drag force Fd acting on the particle is proportional to the following factors.

• The radius of the sphere, r
• The viscosity of the fluid, η
• The velocity of the particle, v

This proportionality can be represented as,

F_d ∝ η^a r^b v^c

Replacing the proportionality sign with the equality sign,

F_d = k η^a r^b v^c

Where k is the dimensionless proportionality constant

Writing the above equation in terms of dimensions,

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

Or, [MLT^-2] = M^a L^-a+b+c T^-a-c

Equating the superscripts of mass, length, and time on both sides.

1 = a

1 = – a + b + c

– 2 = – a – c

Solving the above set of equations.

a = 1

b = 1

c = 1

Therefore,

F_d = k η r v

Experimentally, the value of k was found to be 6π. Hence,

F_d = 6πηrv

Stokes law application

Sedimentation is the tendency for suspended particles to settle from a liquid into deposits called sediments. Stokes’ law is essential in understanding the sedimentation of small particles under the force of gravity. It can determine the sedimentation rate, which is the terminal velocity of the sediment.

Stokes’ law can explain the buoyancy of clouds. Since raindrops are tiny in size, their terminal velocities are small. Hence, they are suspended in the air in the form of clouds. Their terminal velocities increase as they increase in size and begin to fall as rain. It helps in finding sediments in freshwater. When jumping from an airplane, a parachute helps a person land slowly and softly.

Conclusion:

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