The Stokes' law calculator above finds the terminal settling velocity of a small sphere moving slowly through a viscous fluid, along with the viscous drag force acting on it and the Reynolds number that decides whether the answer can be trusted. It covers the creeping-flow regime: particles small enough, or fluids thick enough, that inertia in the surrounding fluid is negligible and viscosity alone resists the motion. That is the regime of settling tanks, sediment analysis, aerosols and viscometry.
Arb Digital builds free calculators that tell you when they stop applying, which matters more here than in almost any other physics tool. Stokes' law is exact in its limit and badly wrong outside it, and the difference is not obvious from the inputs. Every result on this page carries a Reynolds number and a plain-English verdict on validity, so you never take a number from a regime the equation does not describe.
What This Stokes' Law Calculator Does
It balances two forces. Gravity pulls the sphere down with a force set by its own weight less the buoyancy of the fluid it displaces, and viscous drag resists it with a force proportional to velocity. At terminal velocity those are equal, and rearranging gives a closed-form answer with no iteration needed. The tool reports that velocity, the drag force at it, and the time the particle would take to fall through the depth you enter.
Buoyancy is handled properly rather than ignored, which is why the density difference rather than the particle density drives the answer. If you enter a particle less dense than the fluid — an oil droplet in water, a gas bubble — the density difference goes negative and the tool reports a rising velocity instead of a settling one, with the sign preserved. That case is physically real and is exactly how flotation separation works.
How to Use It
- Enter the sphere diameter. Millimetres, micrometres and metres are all accepted. This is the single most sensitive input, because velocity scales with its square.
- Set the two densities. Only their difference matters for the velocity, so a small error in the fluid density becomes a large error when the two are close together.
- Enter the dynamic viscosity. Centipoise, pascal-seconds and poise are all available. Water at 20 °C is 1.002 mPa·s and roughly halves by 55 °C, so temperature matters here more than most people expect.
- Add a settling depth if you want a time. The depth divides into the velocity to give a settling time and never affects the velocity itself.
- Read the Reynolds number before the velocity. If it is above about one, the note under the results will tell you that Stokes' law is overstating the speed, and by roughly how much.
The Formula: How Stokes' Law Is Calculated
The viscous drag on a sphere in creeping flow is Fd = 6πμrv, where μ is the dynamic viscosity, r the sphere radius and v its velocity relative to the fluid. OpenStax University Physics Volume 1, section 6.4 on drag force and terminal speed, gives that expression directly and notes it applies to small objects moving slowly through a viscous medium.
Setting that equal to the buoyant weight, (4/3)πr3(ρp − ρf)g, and solving for velocity gives v = 2r2(ρp − ρf)g ÷ 9μ. Note the r2: the weight grows with the cube of radius while the drag grows only linearly with it, so the net effect is quadratic.
Work the default values. A 0.1 mm quartz grain has a radius of 5 × 10−5 m, so r2 = 2.5 × 10−9 m2. The density difference is 2,650 − 998 = 1,652 kg/m3. Then v = 2 × 2.5 × 10−9 × 1,652 × 9.81 ÷ (9 × 1.002 × 10−3) = 8.99 × 10−3 m/s, or about 9 mm per second. The drag force at that speed is 6π × 1.002 × 10−3 × 5 × 10−5 × 8.99 × 10−3 = 8.49 × 10−9 N, which matches the buoyant weight to three figures, as it must at terminal velocity.
The Reynolds Number Check: When the Answer Stops Being True
Stokes derived his result by dropping the inertial terms from the equations of fluid motion entirely. That is legitimate only when inertia really is negligible, and the particle Reynolds number — Re = ρfvd ÷ μ — measures exactly how good that assumption is. Below about 0.1 the result is essentially exact. Between 0.1 and 1 it is good to a few per cent. Above 1 it starts overpredicting the velocity, and by Re of 10 the error is on the order of tens of per cent.
The default case here lands at Re ≈ 0.9, which is right at the edge and is a good illustration of how small a particle has to be for this to hold in water. Push the diameter to half a millimetre and the Reynolds number climbs past 100, well into the transitional regime where the drag coefficient no longer follows 24/Re and no closed-form settling velocity exists. Our Reynolds number calculator handles that quantity on its own for pipe and external flows.
NASA Glenn Research Center's page on the drag of a sphere lays out how sharply the sphere drag coefficient varies with Reynolds number across the whole range, including the drag crisis near Re of 300,000 where the coefficient suddenly drops. Nothing in that picture is captured by Stokes' law, which is why the validity check on this page is not a formality.
How This Differs From the Drag Force and Free Fall Calculators
The boundary in one sentence: Stokes' law is the low-Reynolds, viscosity-dominated limit where drag is proportional to velocity, while the quadratic drag model is the high-Reynolds, inertia-dominated regime where drag is proportional to velocity squared. They are different physics, not different approximations of the same thing.
Our drag force calculator covers the quadratic case, taking a drag coefficient, a frontal area and a fluid density to give the resisting force on a car, a cyclist or a projectile. The free fall calculator uses the same quadratic model to give the terminal velocity of a falling object in air, which is the regime a skydiver or a hailstone lives in. Neither is applicable to a silt grain in water, and this page is not applicable to a parachute. The Reynolds number is what tells you which of the three you should be on.
Where Stokes' Law Actually Gets Used
Sedimentation analysis is the oldest application: let a soil sample settle in a column of water, sample the suspension at known depths and times, and back out a particle size distribution from the settling velocities. The method assumes spherical grains, which real soil particles are not, so the result is reported as an equivalent spherical diameter rather than a physical size.
Falling-sphere viscometry runs the equation backwards. Drop a sphere of known size and density through the fluid, time it over a marked distance, and solve for viscosity. It is simple, cheap and accurate provided the tube is wide enough that the walls do not interfere and the sphere is small enough to stay in creeping flow. Use the viscosity converter to bring your result into whatever unit your specification uses.
Aerosol and air-quality work relies on it for particle deposition rates, which is why the fine fraction of particulate matter stays airborne for hours. Water treatment uses it to size clarifiers and settling basins, and mineral processing uses the rising-particle case for froth flotation. In each of these the density difference is often small, which is why getting the fluid density right — with the water density calculator or the specific gravity calculator — matters as much as getting the particle size right.
The Assumptions Built Into the Equation
The sphere is rigid, smooth and perfectly spherical. Real particles are angular, and an angular grain settles more slowly than a sphere of the same volume because it presents more surface for a given mass. The fluid is unbounded, so a sphere settling near a wall or in a narrow tube is slowed by the boundary — a correction of several per cent applies once the sphere diameter exceeds a few per cent of the tube diameter.
The fluid is Newtonian and at rest, the flow around the sphere is steady, and the particle has already reached terminal velocity. That last one is usually harmless, because small particles accelerate to terminal velocity in a tiny fraction of a second, but it means the tool says nothing about the transient. Finally, the suspension is dilute: at high particle concentrations the neighbours interfere and hindered settling slows everything down, sometimes by an order of magnitude.
Arb Digital builds free tools like this one because useful pages earn attention. If you want tools, calculators or content built for your own audience, we can help.
Browse All Free Tools Talk to Arb DigitalCommon Mistakes to Avoid
- Ignoring the Reynolds number — the equation always returns a number, and above Re of about one that number is too high. Read the validity verdict before the velocity.
- Entering radius where diameter is asked — the field wants diameter, and using radius there makes the velocity four times too small.
- Using particle density instead of the density difference — buoyancy is not a small correction, and for a particle only slightly denser than the fluid it dominates the answer completely.
- Forgetting that viscosity is temperature dependent — water's viscosity roughly halves between 20 °C and 55 °C, which doubles the settling velocity of every particle in it.
- Applying it to a concentrated suspension — the derivation assumes one isolated sphere, and hindered settling in a dense slurry can be many times slower.
Related Free Tools From Arb Digital
For the turbulent side of the same problem use the drag force calculator or the free fall calculator, and check which regime you are in with the Reynolds number calculator. Prepare inputs with the viscosity converter, the density calculator and the water density calculator, or work from a relative figure with the specific gravity calculator. The full free online tools hub lists everything.
Frequently Asked Questions
When the particle Reynolds number is below about one, and ideally below 0.1. That means small particles, slow speeds or viscous fluids. Above that, inertia in the fluid stops being negligible, the drag coefficient no longer follows 24 divided by the Reynolds number, and the equation overstates the settling velocity.
Because the driving force is the buoyant weight, which grows with the cube of the radius, while the viscous drag grows only linearly with radius. Dividing one by the other leaves radius squared. That is why a particle ten times smaller settles a hundred times more slowly.
The density difference goes negative and the particle rises instead of settling. The tool preserves the sign and reports a rising velocity, which is the physically correct answer and the basis of froth flotation, oil-water separation and bubble rise calculations.
Not exactly. The derivation is specific to a sphere, and an angular or plate-like particle settles more slowly because it presents more drag surface per unit mass. Sedimentation results are therefore reported as an equivalent spherical diameter rather than as a physical particle size.
Stokes' law is the low-Reynolds regime where drag is proportional to velocity and viscosity dominates. The drag force calculator covers the high-Reynolds regime where drag is proportional to velocity squared and inertia dominates. They describe different physics rather than different approximations of the same thing.
Dynamic viscosity, in pascal-seconds, centipoise or poise. Kinematic viscosity is dynamic viscosity divided by density and is measured in square metres per second or centistokes, so it must be multiplied by the fluid density before it can be used in this equation.
The usual causes are a concentrated suspension, where neighbouring particles hinder each other, walls close enough to slow the sphere, non-spherical grains, or a viscosity taken at the wrong temperature. Each of these slows real settling relative to the ideal single-sphere result.
This tool is provided for educational and estimating use. It applies the idealised creeping-flow solution for a single smooth sphere in an unbounded Newtonian fluid, so treat its output as a physics result rather than a process design figure.
