Stokes' Law (Settling Velocity) Equation, Derivation & Rearrangements

Core idea

Overview

Stokes' Law defines the terminal velocity reached by a spherical particle as it settles through a stationary, viscous liquid under the influence of gravity. In geology, it is primarily used to calculate the settling rates of fine-grained particles, providing a mathematical link between grain size and depositional energy.

When to use: This equation is valid for fine-grained particles like silt and clay where the Reynolds number is very low (less than 0.1). It assumes the particle is a perfect sphere, the fluid is homogeneous and still, and there is no interference from other nearby particles.

Why it matters: It allows geoscientists to determine how long it takes for specific sediment types to settle in bodies of water, which is crucial for interpreting paleoenvironments. It also aids in the design of settling basins for industrial water treatment and understanding the transport of atmospheric dust.

Symbols

Variables

v_s = Settling Velocity, ho_p = Particle Density, ho_f = Fluid Density, g = Gravity, R = Radius

v_{s}

Settling Velocity

m / s

h o_{p}

Particle Density

k g / m^{3}

h o_{f}

Fluid Density

k g / m^{3}

g

Gravity

m / s^{2}

R

Radius

m

m u

Dynamic Viscosity

P a \cdot s

Walkthrough

Derivation

Derivation: Stokes' Law (Settling Velocity)

Calculates the terminal velocity of a particle falling through a viscous fluid.

Spherical particle.
Laminar flow (Re < 0.1).
Infinite fluid extent.

1

Balance drag and buoyancy:

Derived by equating the gravitational force (minus buoyancy) with the viscous drag force (6πμRv).

v_{s} = \frac{2}{9} \frac{( ρ _{p} - ρ _{f} ) g R ^{2}}{μ}

Result

v_{s} = \frac{2}{9} \frac{( ρ _{p} - ρ _{f} ) g R ^{2}}{μ}

Source: University Sedimentology — Particulate Transport

Free formulas

Rearrangements

Solve for v_s

Make vs the subject

v_{s} = \frac{2}{9} \frac{( ρ _{p} - ρ _{f} ) g R ^{2}}{μ}

The formula is already solved for vs.

Difficulty: 1/5

Solve for \rho_p

Make rhop the subject

ρ_{p} = \frac{9 μ v _{s}}{2 g R ^{2}} + ρ_{f}

Rearrange Stokes' Law to solve for particle density.

Difficulty: 3/5

Solve for \rho_f

Make rhof the subject

ρ_{f} = ρ_{p} - \frac{9 μ v _{s}}{2 g R ^{2}}

Rearrange Stokes' Law to solve for fluid density.

Difficulty: 3/5

Solve for g

Make g the subject

g = \frac{9 v _{s} μ}{( ρ _{p} - ρ _{f} ) R ^{2}}

Rearrange Stokes' Law to solve for gravitational acceleration.

Difficulty: 3/5

Solve for R

Make R the subject

R = \frac{9 μ v _{s}}{2 g ( ρ _{p} - ρ _{f} )}

Rearrange Stokes' Law to solve for particle radius.

Difficulty: 3/5

Solve for \mu

Make mu the subject

μ = \frac{2 ( ρ _{p} - ρ _{f} ) g R ^{2}}{9 v _{s}}

Rearrange Stokes' Law to solve for dynamic viscosity.

Difficulty: 2/5

The static page shows the finished rearrangements. The app keeps the full worked algebra walkthrough.

Visual intuition

Graph

When plotting the square of the particle radius (R²) on the x-axis against settling velocity (vs) on the y-axis, the graph forms a straight line passing through the origin. This linear relationship occurs because the velocity is directly proportional to the square of the radius, resulting in a constant slope defined by the fluid and particle properties.

Graph type: linear

Why it behaves this way

Intuition

A small, dense sphere descends through a still, viscous fluid, creating a flow field around itself as it pushes the fluid aside, eventually reaching a constant speed when the downward pull of gravity (adjusted for

v_{s}

Terminal settling velocity of the particle

The constant speed a particle reaches when the downward gravitational force is balanced by the upward buoyant and drag forces.

ρ_{p}

Density of the spherical particle

A measure of how much mass is contained within the particle's volume; higher particle density increases the net downward force, leading to faster settling.

ρ_{f}

Density of the fluid

A measure of how much mass is contained within the fluid's volume; higher fluid density increases the buoyant force, reducing the net downward force and slowing settling.

g

Acceleration due to gravity

The constant acceleration pulling the particle downwards, contributing to the gravitational force.

R

Radius of the spherical particle

The size of the particle; larger particles experience greater gravitational force and thus settle faster, but also increased drag.

μ

Dynamic viscosity of the fluid

A measure of the fluid's internal resistance to flow; higher viscosity means the fluid is 'thicker' and creates more drag, slowing the particle's settling.

Signs and relationships

(ρ_p - ρ_f): This difference represents the effective density driving the settling. If the particle is denser than the fluid (ρ_p > ρ_f), the term is positive, and the particle sinks.
R^2: The settling velocity increases quadratically with the particle's radius. This strong dependence arises because the gravitational force (proportional to volume, R^3)
1/μ: Viscosity appears in the denominator because it represents the fluid's resistance to motion. A higher fluid viscosity leads to greater drag, which in turn reduces the terminal settling velocity of the particle.

Free study cues

Insight

Canonical usage

This equation is used to calculate a velocity, so all input units must consistently resolve to a unit of length per unit of time (e.g., meters per second or centimeters per second).

Common confusion

A common error is mixing units from different systems (e.g., using density in g/cm3 with viscosity in Pa·s). All input units must be consistent within either the SI or CGS system for accurate results.

Unit systems

v_sm/s (SI), cm/s (CGS) · The terminal settling velocity of the particle.

\rho_pkg/m^3 (SI), g/cm^3 (CGS) · The density of the spherical particle. Ensure consistency with fluid density units.

\rho_fkg/m^3 (SI), g/cm^3 (CGS) · The density of the fluid. Ensure consistency with particle density units.

gm/s^2 (SI), cm/s^2 (CGS) · Acceleration due to gravity. Standard value near Earth's surface is approximately 9.81 m/s^2 or 981 cm/s^2.

Rm (SI), cm (CGS) · The radius of the spherical particle. In geology, particle sizes are often given in millimeters (mm) or micrometers (μm), requiring conversion to meters or centimeters for calculation.

\muPa·s or kg/(m·s) (SI), Poise (P) or g/(cm·s) (CGS) · The dynamic viscosity of the fluid. 1 Poise (P) = 0.1 Pa·s.

One free problem

Practice Problem

A quartz silt grain with a radius of 0.00001 m is settling in water at 20°C. Given the density of quartz is 2650 kg/m³, the density of water is 1000 kg/m³, and the dynamic viscosity of water is 0.001 Pa·s, calculate the terminal settling velocity.

Particle Density2650 kg/m³

Fluid Density1000 kg/m³

Gravity9.81 m/s²

Radius0.00001 m

Dynamic Viscosity0.001 Pa·s

Solve for: vs

Hint: Subtract the fluid density from the particle density first, then multiply by the constant 2/9.

The full worked solution stays in the interactive walkthrough.

Where it shows up

Real-World Context

A fine sand grain (R=0.1mm) settling in water.

Study smarter

Tips

Always convert grain diameter to radius by dividing by two before calculating.
Ensure all units are in the SI system: meters, kilograms, and seconds.
The density of quartz (2650 kg/m³) is a common default for rhop in sedimentology.
Check that the settling velocity is slow enough to maintain laminar flow.

Avoid these traps

Common Mistakes

Using diameter instead of radius.
Forgetting the viscosity unit (Pa·s).

Keep going

Related Formulas

Common questions

Frequently Asked Questions

Calculates the terminal velocity of a particle falling through a viscous fluid.

This equation is valid for fine-grained particles like silt and clay where the Reynolds number is very low (less than 0.1). It assumes the particle is a perfect sphere, the fluid is homogeneous and still, and there is no interference from other nearby particles.

It allows geoscientists to determine how long it takes for specific sediment types to settle in bodies of water, which is crucial for interpreting paleoenvironments. It also aids in the design of settling basins for industrial water treatment and understanding the transport of atmospheric dust.

Using diameter instead of radius. Forgetting the viscosity unit (Pa·s).

A fine sand grain (R=0.1mm) settling in water.

Always convert grain diameter to radius by dividing by two before calculating. Ensure all units are in the SI system: meters, kilograms, and seconds. The density of quartz (2650 kg/m³) is a common default for rhop in sedimentology. Check that the settling velocity is slow enough to maintain laminar flow.

References

Sources

Bird, R. Byron, Stewart, Warren E., Lightfoot, Edwin N. Transport Phenomena. John Wiley & Sons.
Incropera, Frank P., DeWitt, David P., Bergman, Theodore L., Lavine, Adrienne S. Fundamentals of Heat and Mass Transfer. John Wiley & Sons.
Wikipedia: Stokes' Law
Britannica: Stokes' law
Bird, R. Byron; Stewart, Warren E.; Lightfoot, Edwin N. (2007). Transport Phenomena (2nd ed.). John Wiley & Sons.
Incropera, Frank P.; DeWitt, David P.; Bergman, Theodore L.; Lavine, Adrienne S. (2007). Fundamentals of Heat and Mass Transfer (6th ed.).
Bird, R. Byron, Stewart, Warren E., and Lightfoot, Edwin N. Transport Phenomena. 2nd ed. John Wiley & Sons, 2002.
Boggs, Sam. Principles of Sedimentology and Stratigraphy. 5th ed. Pearson Prentice Hall, 2012.

Stokes' Law (Settling Velocity)

Overview

Variables

Derivation

Balance drag and buoyancy:

Rearrangements

Graph

Intuition

Insight

Practice Problem

Real-World Context

Tips

Common Mistakes

Related Formulas

Reynolds Number (Porous Flow)

Kozeny-Carman Equation

Darcy's Law (Specific Discharge)

Frequently Asked Questions

Sources