Stokes’ Law Calculator

A small sphere sinking through a viscous fluid soon stops accelerating and settles at a constant speed — the speed where Stokes’ drag exactly balances its net weight. Enter the sphere and the fluid, and this calculator returns that terminal velocity and the drag it lives under.

Drag Proportional to Speed: Stokes' Bargain

In the slow, syrupy regime where a fluid flows smoothly around a sphere, George Stokes showed in 1851 that the drag is F = 6πηrv — linear in the speed, the viscosity and the radius, with no dependence on density at all. Set that drag against the sphere's buoyancy-corrected weight, (4/3)πr³(ρ_p − ρ_f)g, and the terminal velocity falls out: v_t = 2r²g(ρ_p − ρ_f) ÷ (9η). The r² is the formula's character: halve the particle and it settles four times slower, which is why clay suspensions defy gravity for weeks and why centrifuges (whose g this hub's circular-motion page can raise a hundred-thousandfold) exist to hurry them. The calculator evaluates both sides of the balance — the settling speed, and the drag force that defines it — because Stokes' law is really two statements wearing one name.

Worked Example: A Steel Ball in Glycerin

The defaults recreate the classic falling-ball viscometer: a steel ball of radius 1 mm in glycerin (η = 1.41 Pa·s, density 1,260 kg/m³). Terminal velocity = 2 × (0.001)² × 9.81 × (7,800 − 1,260) ÷ (9 × 1.41) = 10.11 mm/s — a stately centimetre per second, slow enough to time with a stopwatch between two marks, which is the entire point of the instrument: rearrange the formula and the measured v_t yields the fluid's viscosity, η = 2r²g(ρ_p − ρ_f) ÷ (9v_t). The drag at that speed, 6πηrv = 0.269 mN, exactly equals the ball's net weight — the calculator's two outputs auditing each other, as a balance sheet should.

The Millikan Connection

Stokes' law has an Nobel laureate hiding in it: Robert Millikan's oil-drop experiment balanced single electrons' worth of charge on microscopic oil droplets, and the droplets' sizes — needed to compute their charge — were measured by timing their fall and inverting this page's terminal-velocity formula (with a correction for the droplets being comparable to the air molecules' mean free path, the Cunningham slip factor, which Millikan's student Harvey Fletcher helped nail down). The elementary charge, e = 1.602 × 10⁻¹⁹ C, thus rests partly on a viscous-drag formula from 1851 — and on the r² scaling this page emphasises, since the droplets' radii were the experiment's softest number. Few schoolroom formulas can claim a fundamental constant among their grandchildren.

Blood, Paint and the Settling Shelf

The erythrocyte sedimentation rate — the humble ESR blood test, a century old and still ordered daily — is this formula with red cells as the spheres: inflamed blood's cells clump into rouleaux of larger effective radius, the r² quadruples their settling, and the millimetres-per-hour the laboratory reports are Stokes' law read off a graduated tube. Paint chemists fight the same equation from the other side, engineering thixotropic paints whose effective viscosity at rest is enormous (the pigment must not settle in the can) and tiny under the brush. Settling is never abolished in a fluid product — only priced, slowed and scheduled — and this page is where the pricing starts, whether the sphere is steel, silt or a red cell.

The Volcano's Ledger

Volcanologists read eruption columns with this formula in hand: ash particles' settling speeds (Stokesian for the fine fraction, which dominates the hazard's reach) decide how long a plume's dust stays aloft and how far the wind carries it before it lands on runways, lungs and solar farms. The 2010 Icelandic eruption's airspace closure was, at bottom, a settling calculation — how fast the fine ash would clear the flight levels — run by weather offices with the same r² this page prints. Grain size by grain size, the formula sorts the sky's new sediment exactly as it sorts a laboratory cylinder: the coarse ash falls on the volcano, the fine fraction visits another continent, and the boundary between them is a radius squared.

Clarifiers and Cream Separators

Industry runs this equation in steel: the clarifier tank in a water plant is a Stokes settler sized so that particles above a design radius reach the bottom before the flow crosses — the tank's surface area, not its depth, is the figure of merit, a counterintuitive result that falls straight out of v_t being a velocity. The dairy centrifuge's disc stack shortens the settling distance to millimetres, the same acceleration trick the laboratory centrifuge plays with g itself. Everywhere a mixture must separate by density, an engineer has priced the separation with 2r²gΔρ ÷ 9η before cutting metal — this page is that pricing, unpatented since 1851. The formula's endurance needs no mystery: it is the exact solution of an exact problem, and industries are built on the few of those that fluid mechanics has deigned to provide.

Related Tools

More from the Physics Calculators hub: the Circular Motion Calculator — for when settling is too slow and a centrifuge takes over — and the Velocity Calculator for the speeds themselves.

Frequently Asked Questions

When does Stokes' law stop working?

When the flow stops being laminar: the Reynolds number for the sphere, Re = 2ρ_fvr ÷ η, must stay below roughly 1. Beyond that the wake turns inertial, drag grows toward the quadratic (Newton) regime, and this page's terminal velocity overestimates the truth — a steel ball in water, for instance, is far outside the law's jurisdiction (the worked example's glycerin is chosen precisely because its viscosity keeps Re tiny: check it — 2 × 1,260 × 0.0101 × 0.001 ÷ 1.41 ≈ 0.018, comfortably lawful). The calculator prints the law's answer; the Reynolds check, which the inputs make a ten-second computation, is the reader's licence to believe it.

Why do clouds not fall?

Their droplets are micrometre-scale, and the r² executes them: a 10 µm droplet in air settles at about 1.2 cm/s by this formula (with slip correction), a 1 µm droplet at a hundredth of that — slower than the gentle updrafts and Brownian jostling of ordinary air, so the cloud is effectively permanent until the droplets grow. Rain is the formula's escape clause: droplets that coalesce to millimetre size settle at metres per second (in the quadratic regime, beyond Stokes), and the cloud's water arrives as weather. Fog, mist and the haze over a valley are all the r², admired from inside.

How does sedimentation analysis use this?

By timing the fall: a soil or pigment sample dispersed in a cylinder separates by size exactly as the formula dictates — the coarse fraction lands first, the fines last — and sampling the suspension's density at a depth and time yields the particle-size distribution (the hydrometer and pipette methods of soil science are Stokes bookkeeping). Pharmaceutical suspensions, paint pigments and ceramic slips are all specified this way, because settling is their shelf life: a medicine that must be shaken is a Stokes failure at the design stage, priced by this page before the fact.

What does the negative sign mean if the sphere rises?

Exactly what it says: if ρ_p < ρ_f, the formula returns a negative terminal velocity — the sphere's steady speed is upward. Cream globules in milk, air bubbles in resin, and hot-air balloon ballast calculations all live on that branch, and the widget says so in plain words when it happens. The physics is symmetric (buoyancy wins, drag still balances), and the sign is the formula's way of keeping the direction in the answer rather than leaving it to the reader's intuition — a small courtesy more calculators should copy.

Can I measure an unknown viscosity with this?

That is the falling-ball viscometer, a standard instrument: drop a sphere of known radius and density through the fluid in a graduated tube, time it over a measured fall once it reaches terminal speed, and invert — η = 2r²g(ρ_p − ρ_f) ÷ (9v_t). Commercial instruments (the Höppler viscometer and its descendants) refine it with inclined tubes and calibrated balls. The calculator runs the forward direction; enter your measured speed by adjusting η until the printed v_t matches, and you have performed the inversion by iteration — the same method, with the algebra worn smooth.

Does the container's size matter?

Slightly, and honestly: a sphere settling near a wall feels extra drag (the wall's presence thickens the effective flow resistance — Faxén's corrections quantify it), so precision viscometry uses wide tubes or applies the wall factor. The calculator gives the infinite-fluid figure, which for a tube ten radii across is within a few percent of the truth — good for settling estimates, with the correction available in the literature when the audit demands it.

Who was Stokes?

George Gabriel Stokes (1819–1903), Lucasian Professor at Cambridge (Newton's chair), who derived the drag law in 1851 while working on pendulums' damping — the same mind behind the Navier–Stokes equations' second half and the Stokes shift in fluorescence. The law's longevity is its regime's honesty: it never claimed the fast water of rivers, only the slow world of syrup, silt and oil drops — and in that world, 170 years of measurement have found it exact. This page inherits both the formula and the modesty: it computes the laminar answer and tells you the Reynolds number at which to stop believing it.