What is Stokes' law and when does it apply?

Stokes' law gives the drag force on a sphere moving slowly through a viscous fluid: Fd = 6πμrv. It assumes laminar, creeping flow (Re ≪ 1). At the terminal (settling) velocity, drag balances gravity minus buoyancy, giving vs = (2/9)(ρp − ρf)g r² / μ. The law is reliable when Re < 0.1 and increasingly approximate up to Re ≈ 1.

What happens when Reynolds number exceeds 1?

Above Re ≈ 1, inertial effects become significant and the actual drag is higher than Stokes predicts, so the true settling velocity is lower than the Stokes result. For 1 1000, Newton's drag law is used. This calculator flags Re > 1 so you know to apply a correction factor or a full drag-coefficient model.

Why does the calculator use radius instead of diameter?

Stokes' law is written in terms of radius r. The calculator also accepts millimetres and converts internally. The Reynolds number uses the diameter (2r) as the characteristic length, which is the standard convention.

What units should I use for dynamic viscosity?

Dynamic viscosity must be in Pa·s (Pascal-seconds). Water at 20 °C is 0.001002 Pa·s. If you have a value in centipoise (cP), divide by 1000: 1 cP = 0.001 Pa·s.

Can I use this for particles denser than the fluid?

Yes. If ρp > ρf the particle sinks (positive vs). If ρp < ρf the result is negative, meaning the particle rises — this is the case for oil droplets in water or air bubbles.

Settling Velocity (Stokes Law)

Name: Settling Velocity (Stokes Law)
Availability: InStock
Author: Nham Vu

Calculate the terminal settling velocity of a spherical particle in a fluid using Stokes' law, with Reynolds number validation.

Particle & Fluid Properties

Particle Radius

Particle Density (kg/m³)

Quartz/sand ≈ 2650 · Clay ≈ 1800 · Steel ≈ 7800

Fluid Preset

Fluid Density (kg/m³)

Dynamic Viscosity (Pa·s)

cP × 0.001 = Pa·s | e.g. 1 cP = 0.001 Pa·s

Results

Settling Velocity

positive = sinks, negative = rises

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Reynolds Number

Re = ρ_f · v_s · 2r / μ

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Formula Reference

v_s = (2/9) × (ρ_p − ρ_f) × g × r² / μ

Re = ρ_f × v_s × 2r / μ

g = 9.81 m/s²

ρ_p — particle density (kg/m³)

ρ_f — fluid density (kg/m³)

r — particle radius (m)

μ — dynamic viscosity (Pa·s)

Regime Guide

Re < 0.1 Stokes regime — law is accurate.

0.1–1 Stokes law is approximate; error up to ~10%.

1–1000 Intermediate (Allen) regime — use Schiller-Naumann.

Re > 1000 Newton regime — Cd ≈ 0.44.

Summary

Calculate the terminal settling velocity of a spherical particle in a fluid using Stokes' law, with Reynolds number validation.

How it works

Enter the particle radius (m or mm) and particle density (kg/m³).
Select a fluid preset or enter fluid density (kg/m³) and dynamic viscosity (Pa·s) manually.
The calculator applies vs = (2/9) × (ρp − ρf) × g × r² / μ to compute settling velocity.
Reynolds number Re = ρf × vs × 2r / μ is computed and shown alongside the velocity.
A warning appears when Re > 1, indicating Stokes' law overestimates velocity and a drag-corrected model is needed.
Results update instantly as you type.

Use cases

Estimate how fast silt or clay particles settle in a water treatment clarifier.
Check whether a particle diameter falls within the Stokes regime before applying the law.
Design sedimentation tanks by computing settling rates for different particle sizes.
Validate lab centrifuge separation times for fine particles in viscous media.
Screen particle-size ranges where Stokes' law holds (Re < 0.1) vs. where corrections are required.
Quick sanity-check for homework problems or engineering calculations.

Frequently Asked Questions

Last updated: 2026-07-24 · Reviewed by Nham Vu