Calculation
v = 2·r²·(ρk−ρf)·g / (9·η)
At equilibrium, weight, buoyancy and Stokes drag exactly balance; settling velocity then stays constant.
Dubbel Mechanik B 6.6.2, Gl. (32) Stokessche Widerstandsformel FW = 3π·η·d·v (Re ≤ 1); Sinkgeschwindigkeit aus Kräftegleichgewicht
Settling velocity of a sphere via Stokes' law
At equilibrium, weight, buoyancy and Stokes drag exactly balance; settling velocity then stays constant.
MINTSI
02
Result
Select a target and calculate.
Understand the inputs
- Settling velocity v — Steady velocity of the sphere relative to the still fluid.
- Sphere radius r — Radius of the particle, assumed a rigid sphere.
- Sphere density ρk — Density of the sphere material, e.g. steel about 7,800 kg/m³.
- Fluid density ρf — Density of the surrounding fluid, e.g. water 1,000 kg/m³.
- Dynamic viscosity η — Fluid viscosity; water at 20 °C about 0.001 Pa·s.
- Gravitational acceleration g — Local gravitational acceleration, about 9.81 m/s² near sea level.
Example
r=0.05 mm, ρk=7,800 kg/m³, ρf=1,000 kg/m³ and η=0.001 Pa·s give v≈0.037 m/s.
Assumptions and limits
Valid only for creeping, laminar flow at very low particle Reynolds number (Re≪1); larger particles or velocities cause turbulent deviations from Stokes' law that are not checked here.
Understand Settling velocity of a sphere via Stokes' law
This calculator determines the steady settling velocity of a small sphere in a viscous fluid, such as a grain of sand in water or a droplet in air.
What does this quantity describe?
Under creeping, laminar flow, Stokes drag FR=6πrηv balances the resultant weight force (net of buoyancy). This gives the steady settling velocity v=2r²(ρk−ρf)g/(9η).
Formula and variables
v = 2·r²·(ρk−ρf)·g / (9·η)
v = 2r²(ρk−ρf)g/(9η)
| Symbol / input | Meaning |
|---|---|
| Settling velocity v | Steady velocity of the sphere relative to the still fluid. |
| Sphere radius r | Radius of the particle, assumed a rigid sphere. |
| Sphere density ρk | Density of the sphere material, e.g. steel about 7,800 kg/m³. |
| Fluid density ρf | Density of the surrounding fluid, e.g. water 1,000 kg/m³. |
| Dynamic viscosity η | Fluid viscosity; water at 20 °C about 0.001 Pa·s. |
| Gravitational acceleration g | Local gravitational acceleration, about 9.81 m/s² near sea level. |
Choose the inputs correctly
r is sphere radius, ρk sphere density, ρf fluid density, η the fluid's dynamic viscosity and g gravitational acceleration.
How to use the calculator
Take r from particle size, ρk and ρf from material tables, η from viscosity tables for the fluid and temperature involved.
Worked example
r=0.05 mm, ρk=7,800 kg/m³, ρf=1,000 kg/m³ and η=0.001 Pa·s give v≈0.037 m/s.
Understand the result and units
Settling velocity grows with the square of radius; doubling particle size quadruples settling velocity as long as flow stays laminar.
r is a length, ρk and ρf are densities, η a dynamic viscosity, g an acceleration. v is output as a speed.
Typical applications
Estimating sedimentation and settling velocities in process engineering, soil science and settling-tank design.
Assumptions, limits and common mistakes
Valid only at very low particle Reynolds number (Re≪1, creeping flow); larger particles, higher velocities or less viscous fluids introduce inertial effects that Stokes' law no longer captures correctly, and this is not checked here.
Common mistake: Do not use the result unchecked for large or fast-settling particles; there the formula overestimates velocity because the Re≪1 assumption is violated.
Frequently asked questions
What is “Settling velocity of a sphere via Stokes' law” used for?
Estimating sedimentation and settling velocities in process engineering, soil science and settling-tank design.
Where do the input values come from?
r is sphere radius, ρk sphere density, ρf fluid density, η the fluid's dynamic viscosity and g gravitational acceleration.
What does the result not cover?
Valid only at very low particle Reynolds number (Re≪1, creeping flow); larger particles, higher velocities or less viscous fluids introduce inertial effects that Stokes' law no longer captures correctly, and this is not checked here.