Find the drag force on a small sphere moving through a viscous fluid, its settling (terminal) velocity, or solve Stokes' law for a missing variable — F = 6πµrv.
Sphere and fluid properties
100 µm = 0.0001 m. Stokes' law is only accurate for small, slow spheres.
Drag force
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Enter viscosity, radius, and velocity to compute.
Settling velocity inputs
20 µm = 0.00002 m — a fine silt-sized grain.
Quartz sand ≈ 2650 kg/m³.
Settling velocity
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Enter radius, particle density, fluid density, and viscosity to compute.
Reynolds number—
Stokes' law valid?—
Solve for a missing variable
Solved value
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Choose a variable, set the target drag force, and enter the other two values.
4 min read3 steps7 terms3 examples6 FAQsF = 6πµrv
In 1851, George Gabriel Stokes worked out the drag force on a small sphere creeping through a viscous fluid — a result that still underpins sediment analysis, viscometry, and particle-size measurement today.
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Walk-through
How to Use This Calculator
3 steps▸
1
Enter the sphere and fluid properties
On the Drag Force tab, enter the sphere's radius (m), the fluid's dynamic viscosity (Pa·s), and the relative velocity between the sphere and fluid (m/s). The result card updates instantly with the Stokes drag force F = 6πµrv.
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Find the settling velocity
Switch to the Settling Velocity tab and enter the sphere's radius, particle density, fluid density, and the fluid's viscosity. The calculator returns the terminal (settling) velocity, whether the particle sinks or rises, and a Reynolds-number check on whether Stokes' law is still valid at that speed.
3
Solve for a missing variable
On the Solve tab, pick the variable you don't know (radius, viscosity, or velocity), set a target drag force, and fill in the other two — the calculator rearranges F = 6πµrv to find the missing value.
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Reference
Formula & Methodology
3 formulas▸
Stokes' drag force
F = 6πµrv
Dynamic viscosity µ (Pa·s) times 6π times the sphere's radius r (m) times its relative velocity v (m/s) through the fluid, giving drag force F in newtons. Valid only for small spheres moving slowly enough that the flow around them stays laminar.
Settling (terminal) velocity
v = 2r²(ρp − ρf)g / (9µ)
Balances gravity minus buoyancy against Stokes drag. r is the sphere radius (m), ρp and ρf are particle and fluid density (kg/m³), g = 9.81 m/s², and µ is dynamic viscosity (Pa·s). A negative result means the particle is less dense than the fluid and rises instead of sinking.
Low-Reynolds-number validity condition
Re = ρf·v·(2r)/µ ≪ 1
Stokes' law assumes creeping (laminar, low-Re) flow around the sphere. This calculator computes Re using the sphere's diameter as the characteristic length and flags results where Re climbs above roughly 1, where the formula starts to underestimate real drag.
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Glossary
Key Terms Explained
7 terms▸
Stokes' law ↗A formula describing the drag force on a small sphere moving slowly through a viscous fluid, F = 6πµrv, derived by George Gabriel Stokes in 1851 for the creeping-flow (low Reynolds number) regime.
Drag force ↗The resistive force a fluid exerts on an object moving through it, opposing the direction of relative motion. In the Stokes regime it grows linearly with velocity, radius, and viscosity.
Settling velocity ↗Also called terminal velocity — the constant speed a particle reaches when the downward pull of gravity (minus buoyancy) exactly balances the upward drag force, so acceleration stops.
Sedimentation ↗The process by which particles suspended in a fluid settle out under gravity. Stokes' law is the classical model used to predict how fast particles of a given size settle, e.g. in water treatment or soil analysis.
Viscosity ↗A fluid's resistance to flow or shear, measured here as dynamic viscosity µ in pascal-seconds (Pa·s). Water is about 0.001 Pa·s at room temperature; honey and glycerin are far higher.
Sphere ↗Stokes' law is derived specifically for a smooth, rigid sphere. Non-spherical particles (flakes, fibers, irregular grains) experience different drag and require a shape-correction factor not modeled here.
Low Reynolds number ↗The flow regime (Re ≪ 1, sometimes called creeping or Stokes flow) where viscous forces dominate over inertial ones. Stokes' law is only accurate in this regime — above roughly Re = 1, the flow begins to separate from the sphere and actual drag exceeds the Stokes prediction.
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Scenarios
Real-World Examples
3 worked examples▸
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Drag on a falling sphere
A tiny bead moving through glycerin
Radius 0.0005 mViscosity 1.49 Pa·sVelocity 0.02 m/s
F = 6π × 1.49 × 0.0005 × 0.02 ≈ 0.000281 N. Even a slow-moving, tiny sphere experiences measurable drag in a highly viscous fluid like glycerin — this is the same physics used in falling-ball viscometers to measure a fluid's viscosity.
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Sedimentation of a silt grain
A fine silt grain settling through still water
Radius 0.00002 mParticle density 2650 kg/m³Fluid density 1000 kg/m³Viscosity 0.001 Pa·s
v = 2 × 0.00002² × (2650 − 1000) × 9.81 / (9 × 0.001) ≈ 0.00144 m/s. The Reynolds number here is only about 0.058 — comfortably inside Stokes' law's valid range — which is why the formula underpins sediment-analysis techniques like the hydrometer method for fine, silt-sized grains.
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Where Stokes' law breaks down
A larger, faster-falling particle in water
Radius 0.005 mParticle density 2650 kg/m³Fluid density 1000 kg/m³Viscosity 0.001 Pa·s
The formula predicts a fast settling velocity, but the resulting Reynolds number is far above the Re ≈ 1 creeping-flow limit — flow separates behind the particle and real drag is higher than Stokes' law predicts, so the true settling speed is lower than the calculated value. This is exactly the low-Re-only limitation the calculator's validity check exists to catch.
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Reference
Cite This Calculator
APA & MLA▸
Use either format to cite this calculator in a paper, report, or resource list.
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MLA
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Deep Dive
How Stokes' Law Predicts Drag and Settling Velocity
In 1851, George Gabriel Stokes worked out the drag force on a small sphere creeping through a viscous fluid — a result that still underpins sediment analysis, viscometry, and particle-size measurement today. This calculator applies F = 6πµrv to find drag directly, or rearranges the force balance between gravity, buoyancy, and drag to find how fast a particle settles (or rises) through a fluid.
How the Stokes' Law Calculator works
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Stokes' drag force, F = 6πµrv, comes from solving the fluid-flow equations for a sphere moving slowly enough that inertia is negligible compared to viscous friction — the so-called creeping-flow regime. The settling velocity formula extends this by setting drag equal to the net downward force of gravity minus buoyancy (which depends on the density difference between the particle and the fluid), then solving for the velocity at which those forces balance. Because that balance can favor either direction, a particle less dense than the surrounding fluid will rise rather than settle — the calculator reports this explicitly rather than treating it as an error.
Inputs and what they mean
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Radius (m) is the sphere's radius, not diameter — Stokes' law is derived specifically for spheres, so irregularly shaped particles need a shape-correction factor this calculator does not apply. Dynamic viscosity µ (Pa·s) is a property of the fluid; water is about 0.001 Pa·s while glycerin is roughly 1.49 Pa·s. Particle density and fluid density (kg/m³) only matter for the settling-velocity calculation — their difference, not their absolute values, drives the result. Velocity (m/s) on the Drag Force tab is the relative speed between the sphere and the fluid, however that motion arises.
Limits and edge cases
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Stokes' law is only accurate in the low-Reynolds-number (creeping-flow) regime, roughly Re ≪ 1, where Re = ρf·v·(2r)/µ using the sphere's diameter. Above about Re = 1 the flow starts separating from the back of the sphere, and real-world drag exceeds the Stokes prediction — the Settling Velocity tab flags this with a Reynolds-number check. The formula also assumes an infinite, still, Newtonian fluid; it does not account for nearby container walls (the 'wall effect'), particle-particle interactions in concentrated suspensions, non-spherical shapes, or compressible or non-Newtonian fluids. For fast-falling or larger particles, use a full drag-coefficient correlation instead.
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Questions
Frequently Asked Questions
6 questions▸
What is the formula for Stokes' law?+
F = 6πµrv, where µ is the fluid's dynamic viscosity (Pa·s), r is the sphere's radius (m), and v is the relative velocity between the sphere and the fluid (m/s). The result F is drag force in newtons.
How is settling velocity calculated?+
Settling (terminal) velocity balances gravity against buoyancy and drag: v = 2r²(ρp − ρf)g/(9µ), where ρp and ρf are the particle and fluid densities. Once drag exactly cancels the net gravitational/buoyant force, the particle stops accelerating and falls (or rises) at this constant speed.
When is Stokes' law valid?+
Stokes' law only holds in the low-Reynolds-number (creeping-flow) regime, roughly Re ≪ 1, which typically means small spheres (micrometers to a few millimeters) moving slowly through a viscous fluid. Above about Re = 1 the flow separates and real drag exceeds the Stokes prediction, so this calculator flags results outside that range.
Does Stokes' law apply to sedimentation?+
Yes — Stokes' law is the classical basis for predicting how fast fine particles (like silt, clay, or sand) settle out of a fluid under gravity, and it's used in techniques like the hydrometer method for soil particle-size analysis and in designing settling tanks for water treatment.
What units does this calculator use?+
Radius is in meters (m), viscosity in pascal-seconds (Pa·s), velocity in meters per second (m/s), densities in kilograms per cubic meter (kg/m³), and drag force in newtons (N). The Reynolds number itself is dimensionless.
How is Stokes drag different from other drag formulas?+
Stokes drag applies only to the low-Reynolds-number regime, where drag scales linearly with velocity (F ∝ v). At higher Reynolds numbers — larger, faster-moving objects — drag instead follows the quadratic Newtonian drag law (F ∝ v²) with a drag coefficient that depends on shape and flow conditions, a different regime this calculator does not model.
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