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Stokes' Law Calculator

Stokes' law governs how spheres move through viscous fluids at low Reynolds numbers, making it essential for viscometry and particle settling studies. Engineers and researchers use falling-ball viscometers to measure liquid viscosity by observing a sphere's descent rate. When a particle reaches terminal velocity—where drag force balances gravitational force—the fluid's resistance becomes quantifiable. This calculator lets you solve for either terminal velocity or viscosity depending on which parameters you have.

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Creators Davide Borchia, PhD
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Understanding Fluid Viscosity

Viscosity quantifies a fluid's resistance to flow under stress. Unlike density or temperature, viscosity describes the internal friction between fluid molecules. A viscous fluid like glycerin flows slowly; a less viscous fluid like water flows readily. This property matters in lubrication, coating processes, and any application where flow rate affects performance.

Two related concepts exist:

  • Dynamic viscosity (absolute viscosity, μ) measures force per unit area needed to shear a fluid at a given rate. Units are pascal-seconds (Pa·s).
  • Kinematic viscosity is dynamic viscosity divided by fluid density. It's useful when inertial forces matter equally to viscous ones.

Temperature dramatically affects viscosity. Honey becomes runnier when warmed; oil thickens in cold conditions. For accurate measurements with a falling-ball viscometer, maintain constant fluid temperature throughout the test.

Stokes' Law and Terminal Velocity

When a sphere falls through a viscous fluid, three forces act upon it: gravity pulling downward, buoyancy pushing upward, and drag resisting motion. At terminal velocity, these forces balance perfectly, and acceleration stops. Stokes' law describes this equilibrium state, valid for low-speed motion (Reynolds number below ~0.1).

The terminal velocity depends on particle size, density difference, and fluid viscosity:

v = g × d² × (ρₚ − ρₘ) / (18 × μ)

  • v — Terminal velocity (m/s) — the constant speed reached by the falling sphere
  • g — Gravitational acceleration (m/s²) — typically 9.81 m/s² on Earth
  • d — Particle diameter (m) — the sphere's width measured across its center
  • ρₚ — Particle density (kg/m³) — the sphere material's density
  • ρₘ — Medium density (kg/m³) — the fluid's density
  • μ — Dynamic viscosity (Pa·s) — the fluid's resistance to shear

Practical Applications of Stokes' Law

Falling-ball viscometry remains one of the simplest, most reliable methods for measuring viscosity. It requires minimal equipment and works well for opaque fluids where optical viscometers fail. Industries routinely use this approach:

  • Food and beverage — measuring honey, oils, syrups, and molasses consistency for quality control
  • Pharmaceuticals — ensuring correct viscosity for liquid medicines and suspensions
  • Polymers — monitoring melt viscosity during plastic manufacturing
  • Automotive — testing lubricant grades at various temperatures

The method also applies to particle settling in environmental contexts. When dust or sediment falls through air or water, Stokes' law predicts settling rates, essential for designing filtration and clarification systems.

Rearranging for Viscosity

If you measure terminal velocity in the lab, you can rearrange the formula to solve for fluid viscosity instead. This is the primary use case in a viscometer: drop the sphere, time its descent, and calculate the unknown viscosity.

μ = g × d² × (ρₚ − ρₘ) / (18 × v)

  • μ — Dynamic viscosity (Pa·s) — solved from the measured terminal velocity
  • g — Gravitational acceleration (m/s²)
  • d — Particle diameter (m)
  • ρₚ — Particle density (kg/m³)
  • ρₘ — Medium density (kg/m³)
  • v — Measured terminal velocity (m/s)

Common Pitfalls and Considerations

Stokes' law assumes ideal conditions that may not always hold in practice.

  1. Reynolds number validity — Stokes' law is accurate only at low Reynolds numbers (Re < 0.1). For larger spheres or faster motion, inertial effects become significant and Stokes' formula underestimates drag. Use alternative equations or empirical corrections if your Reynolds number exceeds this limit.
  2. Temperature control — Viscosity changes sharply with temperature. A 10°C variation can alter viscosity by 20–50% depending on the fluid. Always allow the viscometer tube and test sphere to reach thermal equilibrium before measuring, and note the temperature in your report.
  3. Wall effects and tube geometry — Spheres falling near a tube wall experience additional drag not captured by Stokes' law. The effect worsens as sphere diameter approaches tube diameter. Ensure the tube is wide enough—industry standards suggest the sphere diameter should not exceed 1/6 of the tube diameter.
  4. Sphere surface and deformation — The formula assumes a rigid, smooth sphere. Rough surfaces increase drag; elastic spheres may deform under pressure in viscous media, altering their effective size. Always use high-quality, certified reference spheres for accurate viscosity measurements.

Frequently Asked Questions

When is Stokes' law valid for settling calculations?

Stokes' law applies to laminar flow around a sphere at low Reynolds numbers, conventionally Re < 0.1. Beyond this threshold, inertial forces distort the flow field and Stokes' linear drag assumption breaks down. For spheres in air at room temperature, this typically limits the formula to very small particles (below ~100 μm) or very viscous fluids. Always calculate Reynolds number (Re = ρₘ × v × d / μ) to verify applicability before using Stokes' law.

How do you measure terminal velocity in a viscometer?

Fill a calibrated glass tube with the test fluid and allow it to stabilize at constant temperature. Release a sphere at the top and record the time it takes to fall between two reference marks near the tube's center. Divide the distance by the time to get velocity. Measure multiple runs (typically 3–5) to average out timing errors. Ensure the sphere has reached terminal velocity before your measurement window; this usually requires falling through at least one tube diameter first.

What is the difference between dynamic and kinematic viscosity?

Dynamic viscosity (μ) is the actual resistance to shear in units of Pa·s. Kinematic viscosity (ν) divides dynamic viscosity by fluid density: ν = μ / ρ. Kinematic viscosity describes how fast a fluid flows under gravity alone. While Stokes' law uses dynamic viscosity, kinematic viscosity appears in other fluid mechanics contexts. For water at 20°C, dynamic viscosity is about 0.001 Pa·s and kinematic viscosity is 1 × 10⁻⁶ m²/s.

Can Stokes' law predict settling of dust or sand in water?

Yes, Stokes' law works well for fine particles settling in water when Re < 0.1. Sand grains larger than ~100 μm typically exceed this limit; for them, use modified drag laws like the Oseen correction or empirical formulas. Stokes' law predicts that settling velocity increases with the square of particle diameter, so doubling grain size quadruples the settling rate. This is crucial in water treatment design, where coarser particles settle faster than fine clays, affecting clarifier tank dimensions.

How does sphere material choice affect viscometer accuracy?

Using steel or tungsten spheres (high density) allows measurement of higher-viscosity fluids because terminal velocity increases with particle density. Lighter materials like glass or plastic require very viscous media to avoid wall effects and inertial errors. Sphere density must be significantly higher than the test fluid to achieve stable, measurable terminal velocities. Laboratory standards specify exact sphere materials for each viscosity range, ensuring comparability between different instruments and operators.

What causes Stokes' law calculations to differ from actual measured viscosity?

Real fluids often deviate from ideal behavior assumed by Stokes' law. Non-Newtonian fluids (like ketchup or polymer melts) have viscosity that depends on shear rate, violating the linear assumption. Wall proximity in narrow tubes adds drag not accounted for in the formula. Temperature gradients during a long fall introduce local viscosity changes. Sphere surface roughness and imperfect sphericity contribute small errors. Finally, if terminal velocity is reached too close to the measurement zone, the sphere may still be accelerating slightly, lowering apparent viscosity. Calibration with standard reference fluids corrects most systematic errors.

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