Stokes' Law Calculator

Enter a sphere's diameter and density, plus the surrounding fluid's density and viscosity, to find its Stokes' law terminal velocity (v = (2/9)r²(ρp − ρf)g/μ), the drag force at that speed, and the Reynolds number that checks whether Stokes' law applies.

Quick Facts

Terminal velocity
v = (2/9)r²(ρp − ρf)g/μ
Balances gravity, buoyancy, and viscous drag on a small sphere moving through a fluid.
Stokes' drag force
Fd = 6πμrv
Viscous resistance on a sphere — proportional to radius and velocity, unlike higher-speed drag.
Validity range
Re = ρfvd/μ < 1
Stokes' law holds only in slow, laminar "creeping flow"; larger Reynolds numbers need an empirical drag correction.

Your Results

Calculated
Terminal Velocity
-
v = (2/9)r²(ρp − ρf)g/μ
Motion Direction
-
Based on the sign of ρp − ρf
Drag Force at vt
-
Fd = 6πμrv, in newtons (N)
Reynolds Number
-
Re = ρfvd/μ — valid for Re < 1

Ready

Enter the sphere and fluid properties, then press Calculate.

Formula and Method for Stokes' Law

Stokes' law, published by George Gabriel Stokes in 1851, describes the viscous drag force on a small sphere moving slowly through a fluid: Fd = 6πμrv, where μ is the fluid's dynamic viscosity, r is the sphere's radius, and v is its velocity relative to the fluid. For a sphere falling (or rising) freely under gravity, that drag force grows with speed until it exactly balances the sphere's net weight — the point where it stops accelerating and falls at a constant terminal velocity. This calculator solves for that terminal velocity, the drag force at that speed, and the Reynolds number that tells you whether Stokes' law is actually valid for your inputs.

Deriving the terminal velocity from a force balance

At terminal velocity the sphere has zero acceleration, so the three forces acting on it — weight, buoyancy, and drag — sum to zero: (4/3)πr³ρpg = (4/3)πr³ρfg + 6πμrv. Solving for v cancels a factor of πr and rearranges to v = (2/9) r² (ρp − ρf) g / μ, where ρp is the sphere's density and ρf is the fluid's density. If the sphere is denser than the fluid (ρp > ρf), gravity wins and it settles downward; if it is less dense — an air bubble or an oil droplet in water, for example — buoyancy wins and it rises instead. Once v is known, plugging it back into Fd = 6πμrv gives the drag force at that terminal speed.

Checking validity with the Reynolds number

Stokes' law assumes "creeping flow," where viscous forces dominate over inertial ones and no turbulent wake forms behind the sphere. That assumption is only accurate when the particle Reynolds number, Re = ρfvd/μ (d is the diameter), stays below about 1 — ideally under 0.1 for high precision. This matters for unit choices too: keep diameter in meters, densities in kg/m³, and viscosity in Pa·s (or convert consistently) so every term cancels correctly; water at 20°C has a viscosity of about 1.002 mPa·s (0.001002 Pa·s), which is why that value is the default here. If your Reynolds number comes out above 1, Stokes' law underestimates the true drag and you should switch to an empirical drag-coefficient correlation (such as the intermediate or Newton's-law drag regimes) instead of trusting this formula.

Frequently Asked Questions

What is Stokes' law?
Stokes' law gives the viscous drag force on a small sphere moving slowly through a fluid: F = 6πμrv, where μ is the fluid's dynamic viscosity, r is the sphere's radius, and v is its velocity. Balancing that drag against gravity and buoyancy gives the terminal (settling or rising) velocity of a sphere in a fluid: v = (2/9) r² (ρp − ρf) g / μ.
When does Stokes' law apply (what is the Reynolds number limit)?
Stokes' law is only accurate in the slow, laminar "creeping flow" regime, where the particle Reynolds number Re = ρfvd/μ is less than about 1, and ideally below 0.1 for the best accuracy. Above that range, the flow separates behind the sphere, drag grows faster than linearly with velocity, and an empirical drag-coefficient correlation should be used instead.
Why does a particle rise instead of settle?
The direction depends on the density difference between the particle and the fluid. A particle denser than the fluid sinks under gravity, while a particle less dense than the fluid (such as an air bubble or an oil droplet in water) is buoyant and rises. The same Stokes' law force balance describes both cases; only the sign of the density difference changes.
What is Stokes' law used for in practice?
Common applications include sedimentation and soil particle-size analysis (the hydrometer method), measuring fluid viscosity with a falling-ball viscometer, estimating how quickly dust or aerosol droplets settle out of air, and modeling how cells or particles separate in centrifuges.

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