TERMINAL VELOCITY STOKES SPHERE

Stokes showed in 1851 that a sphere of radius r moving slowly through a fluid of viscosity η experiences a drag force F = 6πηrv. The drag coefficient b = 6πηr captures all the geometry and fluid properties in a single constant; multiplied by velocity it gives the force. With r = 0.003 m and η = 0.001 Pa·s: b = 6π × 0.001 × 0.003 ≈ 5.655 × 10⁻⁵ kg/s. This is a very small number, consistent with the fact that water is not particularly viscous and the sphere is tiny. Terminal velocity is reached when gravity and drag balance: m·g = b·v_t, giving v_t = m·g/b = 1.5 × 10⁻⁴ × 9.807 / 5.655 × 10⁻⁵ ≈ 26.0 m/s. This is surprisingly large for a 3 mm sphere — a reminder that the Stokes regime (Re ≪ 1) may not actually hold at this speed in water, where the Reynolds number would be Re ≈ ρvr/η ≈ 1000 × 26 × 0.003 / 0.001 ≈ 78 000. In practice the sphere would enter the quadratic regime long before reaching this speed. The Stokes formula is exact for very slow motions; here it sets the scale. The time constant τ = m/b = 1.5 × 10⁻⁴ / 5.655 × 10⁻⁵ ≈ 2.65 s. The approach is exponential: v(t) = v_t·(1 − e^(−t/τ)). After 5τ ≈ 13 s the sphere is within 1 % of terminal velocity.