Stokes' Law Calculator - Terminal Velocity and Settling
Enter a particle diameter, densities, and fluid viscosity to get the terminal settling velocity, drag force, Reynolds number, and settling time. You can also reverse-solve: pick what you want to find and supply the other values. Results update instantly, and the regime validity indicator tells you whether Stokes' law applies to your specific case.
Formula
Worked example
A quartz sand grain (d = 100 micrometers, rho_p = 2650 kg/m3) settling in water at 20 C (rho_f = 998 kg/m3, mu = 0.001 Pa*s): v_t = 9.807 * (1e-4)^2 * (2650 - 998) / (18 * 0.001) = 9.807 * 1e-8 * 1652 / 0.018 = 8.99 mm/s. Re = 998 * 0.00899 * 1e-4 / 0.001 = 0.897, which is just below 1 - Stokes' law is approximate here but gives a good first estimate.
What is Stokes' law?
Stokes' law describes the force exerted on a small sphere moving through a viscous fluid at low velocity. George Gabriel Stokes derived it in 1851 by solving the Navier-Stokes equations for the limiting case where inertial forces are negligible compared with viscous forces - the so-called creeping-flow or Stokes-flow regime. The settling (terminal) velocity formula is v_t = g d^2 (rho_p - rho_f) / (18 mu), where g is gravitational acceleration, d is the particle diameter, rho_p and rho_f are the particle and fluid densities, and mu is the dynamic viscosity of the fluid. The drag force formula is Fd = 3 pi mu d v, which is linear in velocity unlike the quadratic drag that applies at higher Reynolds numbers.
Reynolds number and validity limits
Stokes' law is only accurate when the particle Reynolds number Re = rho_f v d / mu is well below 1. Most references consider the formula reliable for Re < 0.1, where the error is under about 1 percent. For 0.1 < Re < 1, the Oseen correction (multiplying the Stokes velocity by 1/(1 + 3Re/16)) reduces the error from roughly 6 percent to under 1 percent. Between Re = 1 and Re = 1000, use the intermediate drag law, and above Re = 1000 use Newton's drag law with Cd ~ 0.44. This calculator always computes Re and flags which regime you are in - if the indicator shows orange or red, treat the velocity as a rough upper bound rather than an accurate figure.
Applications and real-world examples
Stokes' law underpins sedimentation analysis (pipette method, hydrometer test), the design of settling tanks and clarifiers in water treatment, cyclone and centrifuge sizing, particle-size analysis in soil science (the Stokes-based hydrometer method is still an ASTM standard), air quality modelling of fine particulate matter (PM2.5 and PM10), and viscosity measurement by timing a falling ball through a reference fluid (falling-ball viscometry). In medical laboratories it describes how red blood cells settle in plasma - the erythrocyte sedimentation rate (ESR) test. In geology, Stokes settling governs how different grain sizes sort during the deposition of sedimentary layers.
How to use this calculator
Choose what you want to solve for in the top dropdown. By default the calculator finds the terminal velocity: enter particle diameter, particle density, fluid density, and dynamic viscosity, then read off the velocity, drag force, Reynolds number, and (if you supply a settling distance) the settling time. Switch to any other output to reverse-solve: for example, set 'Solve for: Fluid viscosity' and enter a measured terminal velocity to back-calculate viscosity - the principle behind falling-ball viscometry. The unit toggle switches between SI (metres, kilograms, Pascal-seconds) and CGS (centimetres, grams, Poise). The chart shows how settling velocity scales with particle size across a range around your current diameter.
Typical particle and fluid properties
| Material | Density (kg/m³) | Notes |
|---|---|---|
| Quartz / sand | 2650 | Most common sediment |
| Calcite | 2710 | Limestone, chalk particles |
| Kaolinite clay | 2600 | Fine clay mineral |
| Coal | 1300-1800 | Varies with rank and ash |
| Iron / steel | 7800 | Dense industrial particles |
| Air bubble | 1.2 | Rising, not settling |
| Water (20 °C) | 998 | Common carrier fluid |
| Seawater (20 °C) | 1025 | Slightly denser than freshwater |
| Glycerol | 1261 | High-viscosity reference fluid |
| Motor oil (SAE 30) | 875 | Used in viscometry experiments |
Common values for quick reference. Use these as starting points.
Frequently asked questions
When is Stokes' law not valid?
Stokes' law fails when the particle Reynolds number Re = rho_f v d / mu exceeds about 0.1. This happens for large or dense particles, low-viscosity fluids, or high velocities. At Re between 0.1 and 1, the Oseen correction gives better results. Above Re = 1 you should use the Schiller-Naumann or Allen drag correlations; above Re = 1000 use Newton's law with Cd ~ 0.44. The calculator always shows Re so you can judge validity.
What is the difference between terminal velocity and settling velocity?
The terms are interchangeable in this context. Terminal velocity is the constant speed a particle reaches when the gravitational force (minus buoyancy) exactly balances the drag force. In sedimentation, this is called the settling velocity. For particles rising through a fluid (e.g. air bubbles in water), the same formula applies but the sign of the density difference is reversed.
Why does Stokes' law use diameter instead of radius?
The standard form of Stokes' settling law uses diameter, not radius, because the drag formula is Fd = 3 pi mu d v (the 3pi comes from the integration over a sphere of circumference pi d). Some textbooks write it with radius r; in that form the 3 becomes 6 and d^2 becomes 4r^2, which gives the same result. This calculator uses diameter throughout for consistency with analytical instruments and engineering practice.
How do I convert between SI and CGS units for viscosity?
1 Poise (P) = 0.1 Pa*s. Water at 20 C has a dynamic viscosity of about 0.001 Pa*s (1 mPa*s) in SI, or 0.01 P (1 cP) in CGS. The calculator handles the conversion automatically when you toggle the unit system.
Can I use Stokes' law for non-spherical particles?
Not directly. Stokes' law assumes a perfect sphere. For non-spherical particles, multiply the diameter by a dynamic shape factor (also called the Stokes correction factor or drag correction factor), which accounts for the increased drag. Common shape factors: cubes ~ 0.8-0.9, cylinders ~ 0.7-0.9 depending on aspect ratio, flat discs ~ 0.6-0.8. The equivalent Stokes diameter from a laser diffraction or sedimentation experiment is the diameter of the sphere that would settle at the same velocity as the real particle.
How do I find particle size from a settling experiment?
Use the 'Solve for: Particle diameter' mode. Measure the time for a known mass of particles to settle through a known distance, compute the velocity (distance / time), then enter that velocity along with the known densities and viscosity. The calculator rearranges Stokes' law to d = sqrt(18 mu vt / (g deltaRho)). This is the basis of the Stokes-law hydrometer method standardized in ASTM D422.