Stokes' Law & Terminal Velocity Calculator
Calculate terminal velocity for spheres falling through viscous fluids using Stokes' law. Find drag force and settling time.
Stokes' Law Guide
Stokes' Law Equation
Drag force on a sphere in viscous laminar flow: F = 6πηrv. Where η = dynamic viscosity (Pa·s), r = sphere radius (m), v = velocity (m/s). At terminal velocity: drag force = weight − buoyancy. v_t = 2gr²(ρ_s − ρ_f) / (9η). Where ρ_s = sphere density, ρ_f = fluid density. Example: 2mm steel ball bearing (ρ=7800) in glycerine (ρ=1260, η=1.412). v_t = 2 × 9.81 × (0.002)² × (7800−1260) / (9 × 1.412) = 2 × 9.81 × 4e-6 × 6540 / 12.7 = 0.0404 m/s = 4 cm/s. Slow enough to time with a stopwatch — making t
Reynolds Number and Validity
Stokes' law only applies for laminar flow at low Reynolds number. Re = ρ_f × v × d / η. Where d = sphere diameter. Stokes' law valid: Re < 1. Above Re ~ 1: significant errors. Above Re ~ 1000: completely different drag regime (use C_d coefficient). At terminal velocity in glycerine (very viscous): Re typically < 0.1 — Stokes' law applies. In air or water with mm-sized particles: Re often > 1 — Stokes' law gives wrong answers. Example: 5mm diameter raindrop in air. Stokes prediction: v_t = 75 m/s
Applications
Measuring viscosity: drop a calibrated ball through unknown fluid. Time the fall. Rearrange Stokes equation: η = 2gr²(ρ_s − ρ_f) / (9v_t). Falling ball viscometer: standard laboratory equipment. Sedimentation in geology: predicting how quickly silt and sand particles settle in still water. Air pollution: settling rate of dust particles determines residence time in atmosphere. PM2.5 (particles < 2.5µm): terminal velocity microns per second — stay airborne for days. PM10: settle in minutes to hour
Beyond Stokes' Law
For larger particles or higher Re: turbulent drag dominates. F_drag = ½ × C_d × ρ_f × A × v². Where C_d = drag coefficient (~0.5 for spheres at high Re), A = cross-sectional area. Terminal velocity becomes: v_t = √(4gr(ρ_s − ρ_f) / (3 × C_d × ρ_f)). Skydivers: terminal velocity ~60 m/s (belly-down) or 90+ m/s (head-down — smaller A). Cannot be calculated by Stokes' law — drag is fully turbulent. Raindrops, hailstones, falling bodies all in turbulent regime. Stokes' law applies almost exclusively
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