3.27
m/s
—
N
6,540
3.27
m/s
—
N
6,540
The Stokes' Law Calculator computes the terminal settling velocity and drag force on a small spherical particle falling through a viscous fluid. Stokes' law is one of the cornerstones of low-Reynolds-number fluid dynamics, governing processes from sedimentation in water treatment plants to aerosol settling in the atmosphere and blood cell separation in centrifuges.
When a sphere falls through a viscous fluid at sufficiently low speed (Reynolds number Re < 1), the drag force is linearly proportional to velocity. At terminal velocity, the net gravitational force exactly balances the viscous drag, yielding the elegant Stokes settling formula: $$v_t = \frac{2r^2 \Delta\rho \, g}{9\mu}$$ and the corresponding Stokes drag: $$F_d = 6\pi\mu r v$$
This calculator also reports the particle Reynolds number so you can verify that the creeping-flow assumption (Re < 1) holds. If Re exceeds unity, corrections such as the Oseen or Schiller-Naumann correlations should be applied. Stokes' law finds applications in geology (grain-size analysis), chemical engineering (sedimentation, centrifugation), environmental science (pollutant settling), and biomedical engineering (cell sorting).
Stokes derived the drag on a sphere in 1851 by solving the linearized Navier-Stokes equations for creeping flow. The key results are:
Terminal velocity: $$v_t = \frac{2r^2 (\rho_p - \rho_f) g}{9\mu}$$ where r is the particle radius, $$\rho_p - \rho_f = \Delta\rho$$ is the density difference, g is gravity, and μ is the fluid's dynamic viscosity.
Stokes drag force: $$F_d = 6\pi\mu r v$$ This gives the resistive force at any velocity v. At terminal velocity, it equals the net weight: $$F_d = \frac{4}{3}\pi r^3 \Delta\rho \, g$$
Reynolds number check: $$Re = \frac{\rho_f v_t (2r)}{\mu}$$ Stokes' law is valid when Re < 1. The calculator uses ρ_f = 1000 kg/m³ (water) for the Re estimate; adjust interpretation for other fluids.
A larger terminal velocity means the particle settles faster. Velocity scales with the square of the radius, so doubling particle size quadruples settling speed. Higher viscosity dramatically slows settling — honey versus water, for instance. If the reported Reynolds number exceeds 1, the Stokes approximation becomes inaccurate and you should use empirical drag correlations instead.
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A 0.1 mm radius sand grain (Δρ = 1650 kg/m³) settles at about 3.6 mm/s in water. Re ≈ 0.72, near the Stokes limit — still roughly valid.
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A 10 μm radius fog droplet settles at ~1.2 mm/s in air (μ = 1.81×10⁻⁵ Pa·s). Re ≈ 0.013, well within the Stokes regime.
Stokes' law states that the drag force on a small sphere moving through a viscous fluid at low Reynolds number is F_d = 6πμrv. It was derived by George Gabriel Stokes in 1851 from the Navier-Stokes equations under the creeping-flow approximation.
Stokes' law is only valid when the particle Reynolds number Re = ρ_f v d / μ is less than about 1. At higher Re, inertial effects become significant and the drag increases faster than linearly with velocity. Empirical corrections (Oseen, Schiller-Naumann) or the full drag curve must then be used.
The gravitational force scales as r³ (volume) while the Stokes drag scales as r (first power of radius at a given velocity). Balancing these gives v_t proportional to r². This means even small increases in particle size dramatically increase settling speed.
Yes, if Δρ is negative (particle lighter than fluid), v_t becomes negative, indicating upward motion. The magnitude still follows the same formula, though internal circulation in bubbles can modify the effective drag.
Temperature primarily affects the fluid viscosity μ. For water, viscosity drops roughly in half from 10°C to 40°C, so warm water allows significantly faster settling than cold water.
Dynamic viscosity μ (Pa·s) is the ratio of shear stress to strain rate. Kinematic viscosity ν = μ/ρ (m²/s) divides out the fluid density. Stokes' law uses dynamic viscosity μ.
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