Stokes Law Terminal Velocity Calculator

Stokes' law describes the drag force on a small sphere moving through a viscous fluid at low Reynolds number, and by balancing this drag against gravity (minus buoyancy), it gives the constant terminal velocity the sphere eventually reaches, whether settling or rising.

v = 2r²(ρ_p−ρ_f)g/(9μ), where r is the particle's radius, ρ_p is the particle's density, ρ_f is the fluid's density, g is gravitational acceleration, and μ is the fluid's dynamic viscosity.

Stokes' law is only accurate for very low Reynolds number flow, generally Re well below 1 (some sources use Re<0.1 for high accuracy, others allow up to Re≈1). This calculator reports the resulting particle Reynolds number so you can check whether your specific case falls within this valid range.

If the particle's density is less than the surrounding fluid's density (ρ_p < ρ_f), the term (ρ_p−ρ_f) becomes negative, correctly indicating the particle is buoyant and rises rather than sinks, exactly like an air bubble in water or a light oil droplet in a denser liquid.

Gravitational (and buoyant) force on a sphere scales with its volume, proportional to radius cubed, while Stokes drag scales only with radius. Balancing these two forces at terminal velocity leaves a net radius-squared dependence, which is why particle size has such an outsized effect on settling speed.

Sedimentation analysis measures particle size distributions in a sample by timing how fast particles settle through a fluid column and applying Stokes' law in reverse (from measured settling velocity back to inferred particle radius), a classic technique in soil science, pharmaceuticals, and materials characterization.

Because Stokes' law terminal velocity scales with radius squared, very small particles (like fine dust, smoke, or fog droplets) settle extremely slowly, often slow enough that ordinary air currents keep them effectively suspended indefinitely, which is exactly why airborne dust and smog can persist for so long.

Terminal velocity is inversely proportional to viscosity (doubling viscosity halves velocity) and directly proportional to the density difference between particle and fluid, so both matter, but a highly viscous fluid can dramatically slow settling even for a fairly dense particle, which is why particles settle far more slowly in glycerin or honey than in water.

Centrifuges effectively replace gravitational acceleration g with a much larger centrifugal acceleration, dramatically increasing the effective 'g' in the Stokes' law formula and speeding up settling by orders of magnitude, which is why centrifugation can separate particles that would take impractically long to settle under normal gravity alone.

Beyond the low-Reynolds-number (Stokes) regime, drag no longer scales simply with velocity, and the terminal velocity formula requires more complex drag coefficient correlations (like those used in the general Drag Force and Drag Coefficient Calculator) instead of this simplified closed-form result.