Stokes Settling Velocity Calculator

Introduction

Stokes settling velocity is the ideal terminal speed of a very small spherical particle moving through a viscous fluid under gravity. In plain language, it tells you how quickly a grain, droplet, cell, or pigment particle will sink once gravity pulling downward is balanced by drag pushing upward. That balance matters in water treatment, sediment transport, aerosol science, mineral processing, pharmacy, food formulation, and many laboratory separation steps. A particle that settles too slowly may remain suspended for a long time; a particle that settles quickly may separate out in seconds, minutes, or hours depending on the depth of the fluid and the size of the vessel.

This calculator gives a fast estimate for that speed using Stokes’ law, which applies when the particle is small enough and the flow around it remains smooth rather than turbulent. The result is most useful as a first-pass engineering or scientific estimate. It can help you compare materials, test whether a small change in radius will matter, or judge whether a fluid is viscous enough to keep particles suspended. It also reports the particle Reynolds number, because Stokes’ law is only reliable in the creeping-flow regime where Reynolds number stays very low.

How to Use This Calculator

Enter the particle radius in meters, the particle density and fluid density in kilograms per cubic meter, the fluid viscosity in pascal-seconds, and the local gravitational acceleration in meters per second squared. The inputs use SI units throughout, so a micron-scale particle must be converted before you type it. For example, 10 micrometers is 10 × 10^-6 meters, or 1e-5 m. If you are working from a materials data sheet, density is often already listed in kg/m³, while viscosity may be tabulated at a specific temperature. Temperature matters because many fluids become much less viscous when warmed.

After you click the compute button, the page calculates the terminal settling velocity and then estimates Reynolds number using that velocity. Read the settling velocity as the speed reached after the initial acceleration has died away. If the result is very small, the particle will settle slowly and may appear to remain suspended for a long time. If the value is larger, the particle will clear a column of fluid more quickly. If the particle density is lower than the fluid density, the calculated velocity becomes negative, which you can interpret as upward motion rather than settling. That is the same physics seen when light droplets or bubbles rise instead of sink.

Stokes’ Law Formula

For a sphere moving through a viscous fluid at very low Reynolds number, the drag force is proportional to velocity. Balancing drag against the particle’s effective weight leads to the standard Stokes expression:

In that equation, ${\rho }_p$ is particle density, ${\rho }_f$ is fluid density, $g$ is gravitational acceleration, $r$ is particle radius, and $\eta$ is dynamic viscosity.

Several practical lessons fall directly out of that formula. First, the speed depends on the density difference between particle and fluid, not just on particle density alone. A heavy particle in a dense liquid may settle only modestly faster than a lighter particle in air or water if the density contrast is small. Second, the radius appears squared, which means size matters strongly. Doubling radius makes the settling velocity four times larger if everything else stays fixed. Third, viscosity sits in the denominator, so thicker fluids slow motion in direct proportion. If viscosity rises by a factor of ten, settling speed falls by a factor of ten. These relationships are why fine clay can stay suspended in water for a long time, while larger sand grains drop out quickly, and why particles move far more slowly in oils, syrups, or polymer solutions than in water.

Worked Example

Suppose a mineral particle has radius 10 µm, density 2500 kg/m³, and is settling in water with density 1000 kg/m³ and viscosity 0.001 Pa·s at ordinary Earth gravity. Converting the radius gives 1e-5 m. The density contrast is 1500 kg/m³. Substituting those values into Stokes’ law gives a settling velocity of about 3.27 × 10^-4 m/s, which is 0.327 mm/s. That is small on a human scale, but it is not negligible in a settling test. In a still 10 cm column of water, such a particle would need a few minutes to travel the full depth once terminal conditions were reached.

Now change only the radius from 10 µm to 20 µm. Because radius is squared, the predicted speed becomes four times larger rather than merely twice as large. By contrast, if you keep the 10 µm radius but increase viscosity from 0.001 to 0.01 Pa·s, the speed becomes ten times smaller. That kind of comparison is often more informative than a single calculation, because it shows which design variable truly controls the process. In many real systems, particle size distribution dominates performance more strongly than modest density changes.

Interpreting the Result

The velocity output is the ideal terminal value under the assumptions of Stokes flow. It is not the distance traveled, but you can use it to estimate time by dividing depth by speed. A compact way to make that estimate is:

Here, h is the travel distance through the fluid and v is the settling velocity. For instance, if a particle settles at 2 × 10^-5 m/s through a 5 cm layer of fluid, the travel time is roughly 0.05 / 0.00002 = 2500 seconds, or about 42 minutes. That kind of quick back-of-the-envelope estimate is often enough to determine whether sedimentation is practical in a lab jar test, clarifier, or storage tank.

Also pay attention to the sign. A positive value means net downward settling. A negative value means the particle is less dense than the fluid and will rise upward instead. If the reported velocity is extremely close to zero, buoyancy and weight nearly cancel, so the particle will remain suspended easily and even minor convection currents or Brownian motion may matter more than simple gravitational settling.

Assumptions, Validity, and Reynolds Number

Stokes’ law is elegant because it turns a complicated fluid-flow problem into a compact equation, but that convenience comes with assumptions. The particle should be spherical or close to spherical, the suspension should be dilute enough that particles do not strongly interfere with one another, the fluid should behave as a Newtonian liquid, and the surrounding flow must remain laminar at the particle scale. Wall effects should also be small, so the particle ideally settles in a container much wider than its own diameter. If any of those assumptions fail, the result becomes a rough estimate rather than a dependable prediction.

The usual first check is the Reynolds number:

When $\mathrm{Re}$ is below about 0.1, the Stokes regime is generally a good approximation. As Reynolds number grows, inertial effects become more important, the drag law changes, and the true settling velocity will deviate from the calculator’s value. This page therefore reports Reynolds number alongside the main answer so you can judge whether the low-Reynolds assumption is likely to hold. A very small Reynolds number is reassuring; a value near or above 0.1 is a signal to verify the problem with a more general drag correlation.

Why This Matters in Practice

In environmental engineering, settling calculations support the design of sedimentation basins, grit chambers, and clarifiers. The goal is often to give particles enough residence time to separate from water before the clarified liquid leaves the tank. In mineral processing, the same physics helps determine whether a slurry will classify by size under gravity. In pharmaceutical suspensions, a formulator may seek the opposite outcome: instead of rapid separation, the product should remain visually uniform on the shelf, which means slow settling is desirable. A viscosity modifier, a smaller particle size, or a reduced 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 waters are rarely perfectly still, so field conditions often depart from the idealized tank assumed by the equation. Even so, Stokes’ law remains a foundational starting point for thinking about fine sediment in lakes, quiet reaches of rivers, atmospheric particles settling from air, and the behavior of planktonic material in water columns. 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

Real particles are often flakes, rods, fibers, or irregular fragments rather than perfect spheres. Those shapes usually create more drag than a sphere of the same volume, which reduces settling speed. 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, treat the output as a benchmark for how an ideal sphere would behave under the same conditions.

Hindered Settling and Concentration Effects

Stokes’ law works best for isolated particles. In concentrated suspensions, 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 slurries 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

Good input data matters just as much as good math. Particle radius should represent the radius of the settling particle itself, not the radius of an agglomerate unless agglomeration is actually 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

If you are using the calculator to plan or interpret a bench experiment, 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 Estimate Stops Being Enough

There are many situations where this calculator still teaches something useful even if the exact numerical answer should not be used by itself. 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 gives a direction for 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

The table below shows order-of-magnitude values 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.

Conclusion

This Stokes Settling Velocity Calculator is best used as a clear, fast first estimate for small spherical particles in viscous fluids. It tells you how particle size, density contrast, viscosity, and gravity work together, and it adds Reynolds number so you can judge whether the assumptions behind the formula are likely to be valid. If the result matches a low-Reynolds, dilute, spherical-particle situation, you can use it to estimate separation time, compare formulations, or build intuition. If not, the answer still gives a useful baseline from which to move toward more detailed drag or multiphase-flow models.