Stokes drag & terminal velocity
Drop spheres of different sizes and densities into a column of viscous fluid. At low Reynolds number (Re < 1), drag is linear in velocity — Stokes' law. At high Reynolds number (Re > 1000), drag becomes quadratic. Watch the transition from acceleration to constant terminal velocity, and compare multiple spheres falling simultaneously.
When an object moves through a fluid, it experiences a drag force that opposes its motion. In 1851, George Gabriel Stokes derived the exact drag force on a sphere moving slowly through a viscous fluid: Fdrag = 6πμRv, where μ is the dynamic viscosity, R is the sphere radius, and v is the velocity. This linear drag law is valid when the Reynolds number Re = 2ρfvR/μ is much less than 1.
A falling sphere accelerates under gravity until the drag force plus buoyancy exactly balances its weight. At this point it reaches terminal velocity: vt = 2(ρs − ρf)gR² / (9μ) in the Stokes regime. The approach to terminal velocity is exponential: v(t) = vt(1 − e−t/τ), where the time constant τ = 2ρsR² / (9μ).
At higher Reynolds numbers, the drag becomes nonlinear. In the turbulent regime (Re > 1000), drag is approximately quadratic: Fdrag = ½CDρfπR²v², where CD ≈ 0.44 for a smooth sphere. The intermediate regime (1 < Re < 1000) is modeled by empirical correlations for the drag coefficient.
This simulation uses the Schiller-Naumann drag correlation, which smoothly interpolates between Stokes drag and turbulent drag: CD = (24/Re)(1 + 0.15 Re0.687) for Re < 1000, and CD = 0.44 for Re ≥ 1000.
Try dropping a steel ball (7800 kg/m³) versus a wooden bead (600 kg/m³) into glycerin vs water. In glycerin, Stokes drag dominates and terminal velocity is low. In water, even small spheres can reach the turbulent regime.