Stokes Law Interactive Calculator

If you drop a particle into a viscous fluid—like sand in water, dust in air, or cells in a centrifuge—its motion is set by the balance of a few straightforward forces. George Stokes put numbers to this problem back in 1851. This calculator lets you figure out terminal velocity, drag force, particle size, viscosity, Reynolds number, and settling time based on what you know. Getting practical values here is important for things like wastewater plants, centrifuge sizing, understanding aerosols, and sorting minerals—anywhere you need honest numbers for how particles behave in fluids. Below, you’ll find keys to the math, an example settled tank calculation, discussion of when to use (and not use) Stokes, and answers to the sorts of problems that crop up in day-to-day engineering.

What is Stokes Law?

Stokes Law gives the drag force on a small sphere moving slowly through a viscous fluid. It lets you estimate how fast something sinks, or how much the fluid resists, from the size of the particle, the thickness of the fluid (viscosity), and the relative speed.

Simple Explanation

Drop a marble into honey and it barely moves. Drop the same marble into water and it drops right to the bottom. Stokes Law ties these observations together: larger particles settle faster, thicker fluids slow them down. Importantly, Stokes Law only holds while things move slow enough for the fluid to flow around the particle smoothly—before you get to swirling or turbulent wakes.

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Force Diagram: Particle Settling Under Stokes Regime

Stokes Law Interactive Calculator

How to Use This Calculator

1. Select your calculation mode from the dropdown — terminal velocity, particle diameter, viscosity, drag force, Reynolds number, or settling time.
2. Enter the required input values for your chosen mode: particle diameter (μm), particle density (kg/m³), fluid density (kg/m³), fluid viscosity (Pa·s), velocity (m/s), or settling distance (m) as applicable.
3. Check that your units match the labels — the calculator expects SI units throughout.
4. Click Calculate to see your result.

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Governing Equations

Here's the formula for Stokes drag force.

Here's the formula for terminal settling velocity.

Here's the formula for Reynolds number when you want to check if the Stokes assumption is justified.

This is how you get settling time once you know the terminal velocity.

Theory & Practical Applications

Simple Example

Inputs: Particle diameter = 50 μm, particle density = 2650 kg/m³, fluid density = 1000 kg/m³, fluid viscosity = 0.001 Pa·s (water at 20°C).

Terminal velocity: v_t = (50×10⁻⁶)² × 9.81 × (2650 − 1000) / (18 × 0.001) = 2.243 × 10⁻³ m/s ≈ 2.24 mm/s

Reynolds number: Re = 1000 × 2.243×10⁻³ × 50×10⁻⁶ / 0.001 = 0.112 — marginal Stokes regime.

Fundamental Physics of Stokes Flow

Stokes Law is what you get when viscous forces completely outweigh inertia—something you can check using Reynolds number. For a sphere moving at low velocity through a viscous fluid, flow stays symmetric and reversible; there’s no wake, no separation, just drag from viscous shear across the surface. That’s why drag here is linear with velocity, unlike the squared relationship you run into with bigger or faster objects where forms of turbulent drag dominate.

The terminal velocity goes up fast with the square of particle diameter. If one particle is ten times bigger, it’ll settle a hundred times faster under otherwise identical conditions. This is why fines linger in water while coarse particles drop out quickly, and why gravity settling is useless for dust control: for tiny particles, other mechanics are needed.

What matters in the formula is not the particle’s absolute density, but the difference to the fluid density. If the two are close, the particle takes a very long time to settle regardless of its true mass.

Reynolds Number Limitations and Transitional Behavior

You only get true Stokes behavior at Re much less than 1—ideally below 0.1. Once Reynolds number comes up toward one, things change: you see asymmetry, a wake behind the particle, and extra resistance. Drag starts shifting from proportional to v, to proportional to v². For in-between cases (Re ≈ 1–800), engineers sometimes apply empirical tweaks like the Schiller-Naumann correction, or use C_d = 24/Re which stays reasonable up to Re ≈ 0.5. By about Re = 1000, drag coefficient settles out to something closer to a flat plate in turbulent flow.

The actual process: run Stokes Law, check what Reynolds number comes out, then if it’s too high, re-calculate with an appropriate drag model. In clean water (η = 0.001 Pa·s, ρ = 1000 kg/m³), Stokes regime covers silica up to about 100 μm. If you go bigger or increase the density contrast, you’re outside that limit and corrections are necessary.

Industrial Sedimentation and Clarification Processes

Stokes Law is where clarifier and sedimentation basin design starts. The main sizing variable is overflow rate: flow per unit surface area. If you want to reliably pull out all particles larger than a set size, you need the overflow rate to be less than their terminal velocity. For example, to pull all 50 μm particles, overflow should be limited to the Stokes velocity for 50 μm in your fluid—usually 0.2 to 0.4 m/hr for water. Overloading the clarifier (flow too high for the area) allows fines to slip through even though there’s enough time—they just don’t have speed to settle down in the available space.

Lamella and tube settlers improve throughput for the same footprint by stacking inclined baffles or tubes, shrinking the vertical distance a particle must travel. Instead of settling through the full depth, the key horizon is the gap between plates—e.g., 25–50 mm. Since all those surfaces combine for more effective area, you can get more capacity from the same tank.

Centrifugal Separation and Enhanced Gravity

Centrifuges speed up settling by replacing gravity with centrifugal acceleration (ω²r). Same terminal velocity equation works with this g value. Most industrial centrifuges deliver between 3000g and 15,000g, slashing settling time for small or low-density particles by 1000x or more. Take a protein molecule of 8 nm diameter: in pure water, it would take months to settle out under 1g, but only hours or less in a good centrifuge. This is the only way to make some lab or industrial separations practical.

The sigma factor (Σ) gives you a way to compare how much fluid a centrifuge can process, irrespective of its specific geometry. You can size up from pilot tests based on sigma; if a test centrifuge with Σ = 100 m² handles 500 L/hr, a full-size unit with Σ = 5000 m² will process 25,000 L/hr to the same removal specs.

Aerosol Physics and Atmospheric Applications

For aerosols (roughly 0.5–100 μm), Stokes Law tells you how long things will stay airborne versus falling out, but only above about 1 μm is gravity settling relevant. For smaller particles, turbulence and Brownian motion dominate. For air at 20°C, a 1 μm particle settles at less than 0.1 mm/s—so even in still air, these hang around. Below that, filtration must use something other than settling, like diffusion or electrostatics.

For particles below 1 μm, especially in air, the fluid “slips” around the particle more than Stokes Law expects, which cuts drag. This effect is called the Cunningham slip correction, and has to be used for anything in the nano-to-micron range, or you’ll underpredict how far small aerosols can travel or mis-calculate their aerodynamic sizes. This doesn’t matter in water, though—mean free path is so short it’s only an issue for molecules.

Worked Example: Wastewater Clarifier Design

Problem: A municipal wastewater plant needs a secondary clarifier to pull out suspended solids after biological treatment. Flow is 8500 m³/day; you must remove everything above 65 μm diameter. The suspended solids have density 1240 kg/m³, wastewater is 998 kg/m³, and dynamic viscosity is 1.15 × 10⁻³ Pa·s at 18°C. Find: (a) minimum surface area needed, (b) clarifier diameter if it’s round, (c) the cutoff size for 50% removal, and (d) if you’re actually in the Stokes regime.

Solution Part (a) - Terminal Velocity and Minimum Surface Area:

First, use Stokes Law to calculate terminal velocity for 65 μm particles:

v_t = [d²g(ρ_p - ρ_f)] / (18η)

Convert diameter: d = 65 μm = 65 × 10⁻⁶ m

v_t = [(65 × 10⁻⁶)² × 9.81 × (1240 - 998)] / (18 × 1.15 × 10⁻³)

v_t = [4.225 × 10⁻⁹ × 9.81 × 242] / (0.0207)

v_t = 1.003 × 10⁻⁵ / 0.0207 = 4.847 × 10⁻⁴ m/s

Convert to m/hr: v_t = 4.847 × 10⁻⁴ m/s × 3600 s/hr = 1.745 m/hr

Overflow rate must not exceed this speed. Minimum area is:

A_min = Q / v_t = (8500 m³/day) / (1.745 m/hr × 24 hr/day)

A_min = 8500 / 41.88 = 203.0 m²

Solution Part (b) - Circular Clarifier Diameter:

Area for a circle is πD²/4:

D = √(4A/π) = √(4 × 203.0 / π) = √(258.3) = 16.07 m

Add 25–40% safety margin for flow patterns and temperature swings. So D = 18 m gives A = 254.5 m².

Solution Part (c) - 50% Removal Particle Size:

Real plants aren’t ideal, but the typical approach: the cutoff is where terminal velocity matches surface overflow rate.

v₅₀ = Q / A_actual = 8500 / (24 × 254.5) = 1.390 m/hr = 3.861 × 10⁻⁴ m/s

Get diameter from v_t = d²g(ρ_p - ρ_f) / 18η:

d₅₀² = (18ηv₅₀) / [g(ρ_p - ρ_f)]

d₅₀² = (18 × 1.15 × 10⁻³ × 3.861 × 10⁻⁴) / (9.81 × 242)

d₅₀² = 7.986 × 10⁻⁶ / 2374.0 = 3.364 × 10⁻⁹ m²

d₅₀ = 5.800 × 10⁻⁵ m = 58.0 μm

Solution Part (d) - Reynolds Number Verification:

Check Re for the 65 μm particle:

Re = (998 × 4.847 × 10⁻⁴ × 65 × 10⁻⁶) / (1.15 × 10⁻³)

Re = 3.146 × 10⁻⁵ / 1.15 × 10⁻³ = 0.0274

With Re = 0.0274 ≪ 0.1, the Stokes assumption holds very well for this clarifier size and target. 18 m diameter with 254.5 m² area will reliably pull out particles ≥ 65 μm at 8500 m³/day.

Non-Spherical Particles and Shape Factors

In the real world, few particles are true spheres. Odd shapes run into more drag. If you know the equivalent spherical diameter, apply a shape factor ψ: F_d = 3πηd_eqvψ. Elongated or flat particles (like fibers or sheets) present more area, so drag goes up—shape factors can span from ~0.85 (rounded sand) to 10 or more (long fibers). Particle size analyzers may report different “diameters” depending on their method. When using settling-based sizes, always check the measurement basis to avoid confusion.

Frequently Asked Questions

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📹 Video Walkthrough — How to Use This Calculator