Stokes Law Calculator

Compute the settling velocity of a spherical particle in a viscous fluid under gravity. Includes Reynolds number check for Stokes flow regime.

Sand: 2650, Oil droplet: 800
Water: 1000, Air: 1.225
0.1 mm = 1e-4 m
Water: 0.001, Air: 1.81e-5
Sand in water (0.1 mm)
Oil droplet in air (10 µm)
Red blood cell in plasma
Dust particle in air
Steel ball in water (0.5 mm)
Calculating...

What is Stokes Law?

Stokes law describes the steady‑state terminal velocity of a small sphere moving through a viscous fluid under gravity, when the flow is creeping (inertial forces negligible). It was derived by George Gabriel Stokes in 1851 and is fundamental to particle sedimentation, aerosol science, and biophysics.

Mathematical formulation:

Vt = (ρp - ρf) g d² / (18 μ)

Derived from balance between buoyancy‑corrected weight and Stokes drag (Fd = 3π μ d V).

Derivation & Underlying Assumptions

Starting from the drag force on a sphere at very low Reynolds number (Re = ρf V d / μ ≪ 1): Fdrag = 3π μ d V. Setting net force = 0: (ρp - ρf) g (π d³/6) = 3π μ d Vt. Solving for Vt gives the above formula.

Key assumptions:

  • Spherical, rigid particle.
  • Infinite, quiescent fluid (no wall effects).
  • No particle‑particle interactions (dilute suspension).
  • Creeping flow: Re < 0.1 (some sources use Re < 1).
  • Fluid is Newtonian with constant viscosity.

Reynolds Number & Regime Validity

The particle Reynolds number Re = ρf Vt d / μ indicates whether inertial effects are negligible. For Re < 0.1, Stokes law is accurate within ~2%. For 0.1 < Re < 1, the Oseen correction (Fd = 3πμdV (1 + 3Re/16)) improves accuracy. For Re > 1, empirical drag curves (e.g., Schiller‑Naumann) are needed.

Cunningham Correction (Slip Flow)

For very small particles (< 1 µm) in gases, the no‑slip condition breaks down. The Cunningham correction factor Cc multiplies the terminal velocity: Vt,Cunningham = Cc · Vt,Stokes, with Cc = 1 + Kn [A + B exp(‑C/Kn)], where Kn = λ/d is the Knudsen number. Our calculator does not apply this automatically; use for nanoparticles.

Engineering & Scientific Applications

  • Sedimentation analysis: Soil particle size distribution (hydrometer method).
  • Biomedical: Erythrocyte sedimentation rate (ESR), cell sorting.
  • Environmental engineering: Settling chambers for particulate removal, aerosol deposition.
  • Chemical engineering: Design of classifiers, fluidized beds (minimum fluidization velocity often based on Stokes).
  • Food processing: Cream separation, clarification of juices.
Case Study: Red Blood Cell Sedimentation

A typical red blood cell has ρp ≈ 1090 kg/m³, diameter ≈ 7 µm. In blood plasma (ρf ≈ 1025 kg/m³, μ ≈ 0.0015 Pa·s), Stokes law gives Vt ≈ (1090-1025)×9.81×(7e-6)²/(18×0.0015) ≈ 1.1×10⁻⁶ m/s. That's about 4 mm per hour, consistent with the Westergren method for ESR (which measures aggregation effects).

Limitations & Extensions

  • Non‑spherical particles: Use dynamic shape factor (χ). Equivalent volume diameter is used, then Vt,non‑spherical = Vt,sphere / χ.
  • Wall effects: In confined geometries (e.g., test tube), settling velocity is reduced. Correction factors (Faxén, etc.) apply.
  • Concentrated suspensions: Hindered settling (Richardson‑Zaki correlation) reduces velocity.

Typical Values of Density & Viscosity

Material Density (kg/m³) Viscosity of common fluids (Pa·s)
Quartz sand 2650 Water (20°C): 0.001
Clay mineral ~2500 Air (20°C): 1.81×10⁻⁵
Oil droplet 800–950 Glycerin: 1.49
Red blood cell 1090 Blood plasma: ~0.0015
Steel 7800 Ethanol: 0.0012

Historical Note

George Gabriel Stokes (1819–1903) derived his law as part of his work on the motion of pendulums in viscous fluids. His formula was later used by Millikan in the oil‑drop experiment to measure the electron charge, after applying the Cunningham correction.

Frequently Asked Questions

The terminal velocity becomes negative, meaning the particle rises (e.g., oil droplets in water). The formula still gives the magnitude; the sign indicates direction.

Because the driving force (gravity minus buoyancy) scales with volume (d³), while the Stokes drag scales with d. Balancing gives d². Thus larger particles settle much faster.

Temperature changes fluid viscosity (and density slightly). For liquids, viscosity decreases sharply with temperature, increasing Vt. For gases, viscosity increases with temperature, decreasing Vt.

No. In turbulent regime (Re > 1000), drag is quadratic in velocity. Use Newton's law or standard drag curve instead.

When particle size approaches the mean free path of gas molecules (Knudsen number > 0.1), the no‑slip condition fails and slip occurs, reducing drag. The Cunningham factor (Cc > 1) increases terminal velocity. It is essential for nanoparticles.

It comes from the exact solution of the creeping flow equations: drag = 3πμdV. Combined with volume factor (πd³/6) yields 18. It's exact for a sphere in an infinite fluid under Stokes assumptions.
References: Stokes, G.G. (1851). "On the effect of internal friction of fluids on the motion of pendulums". Cambridge Phil. Trans.; Clift, R., Grace, J.R., Weber, M.E. (1978) "Bubbles, Drops, and Particles".; Hinds, W.C. (1999) "Aerosol Technology". Values from CRC Handbook.