Stokes Settling Velocity Calculator
Enter the particle diameter d, the particle and fluid densities and the dynamic viscosity μ to compute the terminal settling velocity from v_s = g·d²·(ρp−ρf)/(18μ).
Input Data
Results
At a glance:The Stokes settling velocity is the constant terminal speed a small spherical particle reaches while sinking through a viscous fluid. At low Reynolds number the net gravitational force (weight minus buoyancy) is balanced by the Stokes drag 6πηrv, yielding v_s = g·d²·(ρp − ρf)/(18μ), where d is the particle diameter, ρp and ρf the particle and fluid densities, μ the dynamic viscosity and g=9.81 m/s². The formula assumes a rigid sphere, Re<1, dilute suspension and Newtonian fluid. It is the basis of sedimentation analysis: larger or denser particles settle faster (v_s∝d²Δρ), so measuring settling speed reveals particle size.
Formula
Settling velocity: v_s = g·d²·(ρp − ρf) / (18·μ)
Equivalent (radius form): v_s = 2r²(ρp−ρf)g / (9η)
$$v_s = \frac{g\,d^{2}(\rho_p-\rho_f)}{18\,\mu}$$How to Use
- Enter the particle diameter d (m), particle density ρp and fluid density ρf (kg/m³).
- Enter the fluid dynamic viscosity μ (Pa·s).
- The calculator returns the terminal settling velocity v_s.
Case Studies
Sand settling in water
A 100 μm quartz grain (ρp=2650) in water (ρf=1000, μ=0.001) settles at v_s = 9.81×1e-8×1650/(18×0.001) ≈ 8.98e-3 m/s.
Larger grains settle much faster because v_s∝d²; a 200 μm grain settles ~4× faster.
This is how sedimentation tanks and soil hydrometer tests work.
Clarifier design
A clarifier removes suspended solids by letting particles settle below the overflow weir.
Particles with v_s above the tank's surface-loading rate are captured; slower ones escape.
Raising temperature lowers μ, increasing v_s and improving removal efficiency.
FAQ
When is the Stokes settling formula valid?
For a rigid sphere at Re<1 in a dilute, Newtonian fluid. Above Re≈1 the drag deviates from 6πηrv and a correction (e.g. the Schiller-Naumann drag) is needed; very fine clays may also be affected by Brownian motion and flocculation.
Why does v_s scale with d²?
Weight grows with volume (∝d³) while Stokes drag grows with the projected area and velocity (∝d²·v_s); balancing them gives v_s∝d². Doubling the diameter makes a particle settle about 4× faster.
How does temperature affect settling?
Warmer fluid has lower viscosity μ, so v_s rises. Water at 40°C is roughly half as viscous as at 0°C, nearly doubling the settling velocity for the same particle.
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References
Content review: Calculatorism Science Team. Results are for reference only; please refer to the relevant authorities for the official figures.