Stokes' Law Calculator

Stokes' Law describes the drag force experienced by a small spherical particle moving through a viscous fluid at low Reynolds numbers (Re < 1). Formulated by Sir George Gabriel Stokes in 1851, it states that the drag force Fd = 6πμRv, where μ is the dynamic viscosity, R is the sphere's radius, and v is its velocity. It is widely used in sedimentation analysis, falling ball viscometry, and particle sizing. See also our Hydraulic Radius Calculator.

The terminal velocity formula from Stokes' Law is: v = g × d² × (ρp − ρm) / (18 × μ), where g is gravitational acceleration, d is the particle diameter, ρp is the particle density, ρm is the medium density, and μ is the dynamic viscosity of the fluid. Terminal velocity is reached when the drag force balances gravity and buoyancy, resulting in zero net acceleration.

Dynamic viscosity (μ) measures a fluid's resistance to shear stress — essentially how 'thick' or 'sticky' it is. The SI unit is the pascal-second (Pa·s). For reference, water at 20°C has a dynamic viscosity of about 0.001 Pa·s, while honey can be around 2–10 Pa·s. Higher viscosity fluids produce greater drag forces, resulting in lower settling velocities.

For a 1 cm aluminum sphere (density ≈ 2700 kg/m³) in motor oil (density ≈ 870 kg/m³, viscosity ≈ 0.1 Pa·s), the terminal velocity using Stokes' Law would be approximately v = 9.81 × (0.01)² × (2700 − 870) / (18 × 0.1) ≈ 0.1 m/s. Note that at this size and speed, the Reynolds number may exceed 1, meaning Stokes' Law assumptions might not fully hold. You might also find our Darcy Friction Factor useful.

Stokes' Law is valid only for laminar flow conditions, which correspond to a particle Reynolds number (Re = ρm × v × d / μ) less than approximately 1. At higher Reynolds numbers, inertial effects become significant and the drag coefficient deviates from the Stokes prediction. The law also assumes the particle is a perfect sphere, the fluid is incompressible and Newtonian, and the particle is far from any walls.

Terminal velocity and settling velocity are effectively the same thing in the context of Stokes' Law — both refer to the constant velocity reached by a particle falling through a fluid when the net gravitational force (gravity minus buoyancy) exactly equals the viscous drag force. 'Settling velocity' is more commonly used in environmental and geotechnical engineering, while 'terminal velocity' is the broader physics term.

Stokes' Law assumes: (1) the particle is a rigid, smooth sphere; (2) the fluid is Newtonian and incompressible; (3) flow around the particle is purely laminar (Re < 1); (4) the particle is small enough that its weight is negligible compared to viscous forces; (5) there is no interaction between particles (dilute suspension); and (6) the particle is far from any container walls. Violations of these assumptions require corrective factors.

Stokes' Law has many practical applications: determining particle size distributions in soils (hydrometer analysis), designing sedimentation tanks in water treatment, measuring fluid viscosity using falling ball viscometers, understanding dust and aerosol settling in air, analyzing blood sedimentation rates in medicine, and studying sediment transport in rivers and oceans. It is a foundational concept in fluid mechanics, chemical engineering, and environmental science.