Stokes Settling Velocity Calculator
Introduction to Stokes Settling Velocity
Stokes settling velocity is the terminal speed a small sphere reaches as gravity pulls it through a viscous fluid and drag builds until the two balance. This calculator turns that balance into a quick estimate for particles such as mineral grains, droplets, cells, or pigment spheres, making it useful anywhere you need to know whether material will stay suspended or separate on its own. In treatment tanks, lab cylinders, product formulations, and environmental samples, the question is often not whether a particle can settle, but how long that settling will take.
When the particle Reynolds number built from the Stokes speed is below 0.1, that Stokes speed is the reported terminal velocity. Above 0.1 the page solves a Schiller–Naumann drag balance and reports that speed, still showing the Stokes comparison. The two answers are a screening tool for sizes, fluids, and density contrasts before a bench test. The Reynolds number is part of the result because the choice of law depends on it, and because a rising particle must not be scored with a negative Reynolds number.
How to Use This Calculator for Particle Settling
Enter the particle radius in meters, the particle and fluid densities in kilograms per cubic meter, the fluid viscosity in pascal-seconds, and local gravity in meters per second squared. The calculator uses SI units throughout, so micron-sized particles must be converted before you type them in; 10 micrometers, for example, is 1e-5 m. If your values come from a data sheet, density is usually straightforward, while viscosity often depends on temperature and composition.
Once you compute the result, read the reported terminal velocity as the speed after the particle has stopped accelerating. A positive value means the particle sinks; a negative value means buoyancy wins and the particle rises. The result always includes the Stokes speed and, unless the particle is neutrally buoyant, the drag-corrected speed, both Reynolds numbers, the drag coefficient, and the residual of the drag balance.
Stokes’ Law Formula for Settling Velocity
For a sphere moving slowly through a Newtonian fluid, the viscous drag is proportional to speed, and balancing that drag against the particle’s effective weight gives the standard Stokes expression:
In the equation, is particle density, is fluid density, is gravitational acceleration, is particle radius, and is dynamic viscosity.
Several practical lessons fall directly out of that formula while the Reynolds number stays small. The speed depends on the density difference between particle and fluid, not on particle density alone, so a heavy particle can still settle slowly if the surrounding fluid is nearly as dense. The radius appears squared, which means size has a very strong influence: doubling radius makes the Stokes velocity four times larger if everything else stays fixed. Viscosity sits in the denominator, so thicker fluids slow the Stokes velocity in direct proportion. If viscosity rises by a factor of ten, that Stokes speed falls by a factor of ten. Those relationships explain why fine clay can remain suspended in water, why larger sand grains drop out much faster, and why particles crawl through syrups, oils, or polymer solutions. Once inertia matters, the radius-squared rule is no longer the terminal speed.
Worked Example: a 10 µm Mineral Particle in Water
Consider a mineral sphere with radius 10 µm, density 2500 kg/m³, settling through water with density 1000 kg/m³ and viscosity 0.001 Pa·s at g = 9.81 m/s². The radius is 1e-5 m and the density contrast is 1500 kg/m³. Stokes’ law gives 3.270 × 10-4 m/s. The particle Reynolds number built from that speed is 0.00654, well below 0.1, and the Schiller–Naumann terminal speed is only 0.472% slower. The form opens on this case.
The same formulas at a radius of 100 µm are no longer a small correction. Stokes’ law gives 0.03270 m/s and a Reynolds number of 6.54, and it is 42.69% faster than the Schiller–Naumann speed. A rigid sphere of density 800 kg/m³ and radius 1 mm rises. Stokes’ law gives −0.436 m/s. The signed Reynolds number is −872. A test that asks whether that signed number is below 0.1 says the Stokes regime is valid. It is not. The speed-based Reynolds number of the drag-corrected rise is about 152.
If you keep the same particle but double the radius to 20 µm, the r² term makes the predicted speed four times larger, not just twice as large. If instead you keep the 10 µm radius and raise the viscosity from 0.001 to 0.01 Pa·s, the settling speed drops by a factor of ten. Those two comparisons are often more helpful than a single output because they show whether size, viscosity, or density contrast is the main lever in your system.
Interpreting the Result for Settling Time
The velocity output is the terminal speed, not the travel distance, so use it together with the fluid depth when you want a settling time estimate. A compact way to do that is:
For example, if a particle settles at 2 × 10-5 m/s through a 5 cm fluid layer, the travel time is about 2500 seconds, or roughly 42 minutes. That kind of estimate is often enough to judge whether a clarifier, jar test, or storage tank will separate particles on the timescale you need.
Also watch the sign. Positive values mean downward settling; negative values mean the particle is lighter than the fluid and will rise. If the velocity is very close to zero, buoyancy and weight nearly cancel, so tiny currents, agitation, or Brownian motion may influence what you see more than gravity alone.
Assumptions, Validity, and Reynolds Number for Stokes Settling
Stokes’ formula is trustworthy for a rigid sphere in an unbounded Newtonian fluid when the particle Reynolds number is small, the suspension is dilute, and the particle is close to spherical. Wall effects should be small, so the particle ideally moves in a container much wider than its own diameter. The drag-corrected result relaxes only the low-Reynolds assumption, and only through the Schiller–Naumann fit or a constant Newtonian drag coefficient. It does not repair a crowded suspension, a non-Newtonian fluid, a nearby wall, or a particle that is not rigid and round. If any of those assumptions fail, the result is a baseline rather than a prediction.
The usual first check is the Reynolds number:
The Reynolds number uses the speed, , not a signed velocity. A rising sphere has a negative Stokes velocity and a positive Reynolds number. When that Reynolds number is below about 0.1, Stokes’ law is the reported terminal speed. Above that, the page solves the Schiller–Naumann drag balance, , and reports that root together with the Stokes comparison. The correlation is the usual rigid-sphere fit for Re below about 1000. Past that range the page uses a Newtonian drag coefficient of 0.44 and says so.
Why This Matters in Practice for Sedimentation
In practice, Stokes settling velocity is a shortcut for deciding whether a suspension will separate quickly enough to matter. In environmental engineering, the same calculation helps size sedimentation basins, grit chambers, and clarifiers so particles have enough residence time to drop out before the cleaned water leaves the tank. In mineral processing, it helps estimate whether a slurry will classify by size under gravity. In pharmaceutical suspensions, the goal is often the opposite: a product should remain visually uniform on the shelf, which means slow settling is desirable. A viscosity modifier, a smaller particle size, or a lower density contrast can all help achieve that goal.
Food and consumer products provide familiar examples too. Cocoa particles settle in chocolate milk, spices separate in sauces, pigments drop in paints, and fragrance capsules can rise or sink depending on formulation density. Stokes’ law does not capture every real-world complication, but it gives immediate physical intuition. If a product suddenly separates after reformulation, the cause is often not mysterious at all: the particles may have grown larger, the fluid may have thinned, or the density contrast may have increased.
Environmental and Biological Systems
Natural and biological suspensions rarely look as tidy as a glass cylinder, but the same settling physics still helps you reason about them. Fine sediment in lakes, quiet reaches of rivers, atmospheric particles settling from air, and planktonic material in water columns all respond to the same balance between effective weight and drag. In biology and biotechnology, cells, spores, beads, and organelles may settle under similar principles during handling, washing, or low-speed centrifugation. Researchers often begin with a Stokes-style estimate before adding corrections for shape, porosity, aggregation, or flow disturbances.
Beyond Perfect Spheres
Most real particles are not perfect spheres, and that shape difference usually slows them down. Flakes, rods, fibers, and irregular fragments create more drag than a sphere of the same volume, so the calculator’s result should be treated as a benchmark rather than a promise when the particle is not round. Engineers may use a shape factor, an equivalent spherical diameter, or experimentally measured drag data when accuracy matters. If you are applying this calculator to non-spherical material, think of it as the settling speed an ideal sphere would have under the same conditions.
Hindered Settling and Concentration Effects
This calculator treats one particle at a time, so it cannot account for crowded suspensions. In concentrated slurries, neighboring particles disturb the fluid and slow one another down, a phenomenon called hindered settling. This effect becomes important in sludge blankets, thickeners, and dense process streams where the fluid displaced by one particle must weave around many others. Under those conditions, the single-particle result from this calculator is usually an upper bound rather than the true bulk settling speed.
Choosing Reliable Inputs for a Settling Estimate
Getting meaningful settling estimates starts with matching the inputs to the actual particle and fluid you are studying. Particle radius should represent the settling particle itself, not the radius of an agglomerate unless aggregation is truly present in the fluid. Density should match the specific material and, when possible, the actual temperature of the experiment or process. Viscosity deserves special care because it can shift strongly with temperature and composition. A fluid that behaves like water at one condition may act much more like a syrup after cooling or after dissolved solids are added. If your answer seems surprising, check unit conversion first, then confirm whether the viscosity value truly belongs to the fluid state you are modeling.
A practical habit is to vary one input at a time and see how sensitive the result is. If a small uncertainty in radius changes the settling speed a great deal, then measuring particle size more accurately may matter more than refining density to a third decimal place. This kind of sensitivity thinking is one of the most useful outcomes of using a calculator like this. It helps you decide where to spend experimental effort and where a rough estimate is already good enough for planning.
Laboratory Tips for Stokes Settling Experiments
When you use Stokes settling velocity to plan a bench experiment, the setup details matter as much as the numbers you enter. Make sure the fluid is as quiescent as possible before timing the motion, record temperature so you can choose an appropriate viscosity, use a vessel wide enough to reduce wall effects, and measure particle size carefully because the radius-squared dependence makes size errors especially costly. A 10% uncertainty in radius becomes roughly a 20% uncertainty in settling velocity. That sensitivity is one reason particle-sizing methods are so important in suspension science.
When the Stokes Estimate Stops Being Enough
The calculator is still useful even when the full system is messier than ideal Stokes flow, because it tells you which direction each change should push the settling speed. If Reynolds number is not very small, if the particle is porous or deformable, if the fluid is non-Newtonian, or if many particles are settling together, then the system has moved beyond classical Stokes behavior. Even then, the result remains a valuable baseline. It shows the effect of changing particle size, density contrast, viscosity, or gravity, and it often tells you whether you are in the right ballpark before moving to a more advanced model or a laboratory test.
That is why Stokes’ law appears so often in teaching, screening calculations, and early-stage design. It condenses a lot of physical reasoning into one short equation without hiding what matters. Bigger particles settle faster. Stronger buoyancy contrast pushes motion harder. More viscous fluids resist motion more strongly. Stronger gravity speeds the process. Those simple statements are exactly what the calculator turns into numbers.
Example Velocities for Stokes Settling
The table below gives order-of-magnitude settling velocities for a particle density of 2500 kg/m³ and a fluid density of 1000 kg/m³ unless noted. The point is not the exact number but the pattern: increasing radius strongly speeds up settling, while increasing viscosity slows it dramatically.
| Particle Radius (µm) | Fluid | Velocity (mm/s) |
|---|---|---|
| 1 | Water | 0.0033 |
| 5 | Water | 0.0818 |
| 10 | Oil (η = 0.05 Pa·s) | 0.0065 |
Conclusion: using Stokes settling velocity as a screening tool
This Stokes settling velocity calculator is a screening tool for a rigid sphere in a Newtonian fluid. It shows how particle size, density contrast, viscosity, and gravity work together, and it reports a terminal speed only from a law that the Reynolds number still supports. Below a Stokes Reynolds number of 0.1 that law is Stokes’ formula. Above it, the reported speed is the Schiller–Naumann root, or a Newtonian drag coefficient of 0.44 once that root would sit at a Reynolds number of 1000 or more. If the particle is not spherical, the suspension is crowded, or the fluid is not Newtonian, the result is a baseline for a more detailed model or an experiment.
Use SI units for every field: radius in meters, densities in kg/m³, viscosity in Pa·s, and gravity in m/s². Example conversion: 10 µm = 1e-5 m.
Mini-Game: Clarifier Control for Settling Velocity
This optional mini-game turns Stokes settling into a timing challenge. Route each feed particle into the lane whose viscosity best matches the release timing, then try to open the collector window when the particle arrives. It mirrors the calculator's logic without changing the underlying equation.
Formula hint: settling speed rises with density contrast and the square of particle radius, and falls as viscosity increases. During special phases, temperature or gravity changes may shift the timing.