Table of Contents
What Is Stokes' Law?
Stokes' Law, derived by George Gabriel Stokes in 1851, describes the drag force experienced by a small sphere moving slowly through a viscous fluid. The law applies in the low Reynolds number regime (Re < 1), where viscous forces dominate over inertial forces. The drag force is proportional to the velocity, fluid viscosity, and sphere radius.
At terminal velocity, the drag force exactly balances the net gravitational force (gravity minus buoyancy), and the sphere settles at a constant speed. This principle governs sedimentation of particles in water treatment, rain drop formation, and blood cell separation in centrifuges.
Formulas
Where μ is dynamic viscosity (Pa·s), r is particle radius (m), v is velocity (m/s), ρ_p is particle density, ρ_f is fluid density, and g = 9.81 m/s². The Reynolds number is Re = 2rρ_f v / μ; Stokes' Law is valid for Re < 1.
Fluid Viscosities
| Fluid | Viscosity (Pa·s) | Temperature |
|---|---|---|
| Air | 1.81 × 10^-5 | 20°C |
| Water | 1.00 × 10^-3 | 20°C |
| Olive Oil | 0.081 | 20°C |
| Honey | 2-10 | 20°C |
| Glycerin | 1.49 | 20°C |
| Blood | 3-4 × 10^-3 | 37°C |
Applications
- Sedimentation: Designing settling basins in water and wastewater treatment.
- Geology: Analyzing sediment transport and grain size distribution.
- Aerosol science: Modeling airborne particle behavior and filtration.
- Biotechnology: Cell separation, protein purification by centrifugation.
- Viscometry: Measuring fluid viscosity using falling ball viscometers.
Frequently Asked Questions
When does Stokes' Law break down?
Stokes' Law is only valid for Reynolds numbers below about 1 (creeping flow). For larger or faster particles, inertial effects become important and the drag increases faster than linearly with velocity. Above Re = 1, empirical drag correlations (like the Schiller-Naumann correlation) must be used instead.
How does particle size affect settling velocity?
Terminal velocity is proportional to r², so doubling the particle radius quadruples the settling speed. This is why coarse sand settles much faster than fine clay particles, and why clay can remain suspended in water for hours or days.
What is a falling ball viscometer?
A falling ball viscometer measures fluid viscosity by timing how long a sphere of known size and density takes to fall through a column of fluid. By applying Stokes' Law, the viscosity can be calculated from the terminal velocity, particle dimensions, and density difference.