Stokes' Law Calculator

Stokes' law gives the drag force on a rigid sphere moving slowly through a viscous Newtonian fluid: F_d = 6πμrv, where μ is dynamic viscosity (Pa·s = N·s/m²), r is sphere radius (m), and v is speed relative to the fluid. The drag is linear in v and opposite to motion — characteristic of creeping flow (Reynolds number Re < 1).

Reynolds number Re = ρvD/μ = ρv(2r)/μ compares inertial to viscous forces. Stokes law is valid for Re ≲ 1. For Re > 1, inertia dominates, drag scales roughly as v² (see drag coefficient C_d), and Stokes underestimates resistance.

At terminal velocity in a gravitational field, forces balance: (ρ_s − ρ_f) × (4/3)πr³g = 6πμrv_t. Solving gives v_t = 2r²(ρ_s − ρ_f)g/(9μ). Larger, denser spheres fall faster; higher viscosity slows settling. Steel (ρ ≈ 7800 kg/m³) in water (ρ = 1000) falls much faster than in glycerin (μ ≈ 1.5 Pa·s).

Buoyancy F_b = ρ_f V g acts upward on submerged volume V = (4/3)πr³. The net driving force is weight minus buoyancy, proportional to (ρ_s − ρ_f). In air, ρ_f is small so buoyancy is often neglected for dense particles.

Applications in viscometry: drop a steel ball in an unknown liquid, measure v_t, and solve for μ. Conversely, known μ gives particle size from settling rate — used in sedimentation analysis and air quality monitoring.

George Gabriel Stokes (1851) derived this from the Navier–Stokes equations by neglecting inertia. It underpins aerosol physics, blood sedimentation (ESR tests), fog and cloud microphysics, and microfluidics at low Re.