Stokes' Law Calculator
For low-Reynolds-number flow, a sphere's drag is F=6πηrv; terminal velocity v_t=2r²g(ρ_p−ρ_f)/(9η). Water η=0.001, r=1 mm, glass ρ=2500 → v_t≈2.18 m/s.
Input Data
Results
At a glance:Stokes' law (George Gabriel Stokes, 1851, Irish mathematical physicist): at low Reynolds number (Re=ρ_f·v·d/η≪1, creeping flow), the drag on a sphere moving through a viscous fluid is F=6πηrv, where η is fluid viscosity, r is sphere radius and v is the relative velocity. Terminal velocity: when the gravity-minus-buoyancy force equals drag, (4/3)πr³(ρ_p−ρ_f)g=6πηrv_t, solved as v_t=2r²g(ρ_p−ρ_f)/(9η). Reynolds number Re=ρ_f·v·(2r)/η measures inertial/viscous force ratio; Re<1 validates the law. Derivation: in the Navier–Stokes equations at Re≪1 the inertial terms are dropped; solving the flow field around the sphere and integrating shear stress and pressure yields 6πηrv. Physical meaning: (1) terminal velocity ∝ r² (small slow, large fast); (2) v_t ∝ (ρ_p−ρ_f) (density difference drives settling); (3) v_t ∝ 1/η (viscosity resists); (4) at high Re drag becomes ∝v² (Newton drag), needing correction. History: Stokes derived it in 1851, founding low-Re fluid mechanics. Applications: (1) sedimentation to measure viscosity and particle size; (2) centrifuge separation and density gradients; (3) aerosol and dust settling; (4) blood erythrocyte sedimentation rate (ESR); (5) oil-water separation and ore dressing; (6) raindrop terminal velocity.
Formula
Stokes drag: F = 6πηrv
Terminal velocity: v_t = 2r²g(ρ_p − ρ_f)/(9η)
Reynolds number: Re = ρ_f·v·(2r)/η
Gravity-buoyancy: (4/3)πr³(ρ_p − ρ_f)g
Valid when Re ≪ 1 (creeping flow)
$$F = 6\pi\eta r v, \quad v_t = \frac{2r^2 g(\rho_p - \rho_f)}{9\eta}, \quad Re = \frac{\rho_f v \cdot 2r}{\eta}$$How to Use
- Enter fluid viscosity η (Pa·s; water 1e-3).
- Enter sphere radius r (m; raindrop 1e-3).
- Enter sphere density ρ_p (kg/m³) and fluid density ρ_f (kg/m³).
- The tool gives terminal velocity, drag and Reynolds number.
Case Studies
Raindrop terminal velocity and aerosols
Water drop r=1e-3 m in air (η=1.8e-5, ρ_f=1.225, ρ_p=1000).
v_t = 2(1e-3)²×9.81×(1000−1.225)/(9×1.8e-5) ≈ 1.2 m/s.
A 1 mm raindrop falls ~1.2 m/s; fine dust (r<10 µm) stays suspended for hours.
Content review: Calculatorism Science Team. Results are for reference only; please refer to the relevant authorities for the official figures.