Engineering
Stokes' Law Terminal Settling Velocity & Laminar Re_p <= 1 Check
Calculate creeping-flow particle settling or rising velocity v_t = (g * d^2 * (rho_p - rho_f)) / (18 * mu) (mm/s) for clarifiers, emulsions, and falling-ball viscometers—with automatic Particle Reynolds Re_p verification.
Size water-treatment sedimentation basins and verify creeping-flow validity!
Sir George Gabriel Stokes (1851) solved the Navier-Stokes creeping-flow drag F_d = 3 * PI() * mu * d * v on a small sphere of diameter d (m) and density rho_p (kg/m^3) moving through a fluid of density rho_f and dynamic viscosity mu (Pa*s)! Balancing Stokes viscous drag against net weight minus Archimedes buoyancy yields Terminal Velocity: v_t = (g * (d ^ 2) * (rho_p - rho_f)) / (18 * mu)!
Notice two critical physical insights: 1) Signed Density Difference (rho_p - rho_f) handles BOTH heavy silt settling downward (+v_t) AND light oil droplets or microbubbles rising upward (-v_t) in flotation cells! 2) Stokes' Law is ONLY accurate in laminar creeping flow where Particle Reynolds Number Re_p = (rho_f * ABS(v_t) * d) / mu <= 1.0 (above Re_p > 1, turbulent wake drag slows settling far below Stokes' d^2 prediction)!
| Settling Scenario (d = 50 um = 0.00005 m, g = 9.81 m/s^2) | Fluid Properties | Terminal Velocity v_t (mm/s) | Particle Re_p Check |
|---|---|---|---|
| Quartz Silt in Water (rho_p = 2,650) | rho_f = 998, mu = 0.001 Pa*s | +2.2508 mm/s (Sinks) | Re_p = 0.1123 (<= 1 Valid!) |
| Quartz Silt in Gear Oil | rho_f = 880, mu = 0.050 Pa*s | +0.0482 mm/s (50x slower) | Re_p = 0.00004 (Pure Stokes) |
| Coarse Gravel (d = 5 mm in Water) | d = 0.005 m | v_t -> Re_p > 1,000! | "Re_p > 1: Turbulent Wake" |
Stokes Terminal Velocity in mm/s with Creeping-Flow Re_p <= 1 Guard
Uses LET to evaluate v_t (m/s) and verify Particle Reynolds Number Re_p <= 1 before reporting mm/s.
=IF(OR(B2 <= 0, B5 <= 0), "Invalid Diameter or Viscosity", LET(vt, (9.81 * (B2 ^ 2) * (B3 - B4)) / (18 * B5), Rep, (B4 * ABS(vt) * B2) / B5, IF(Rep > 1, "Re_p > 1: Use Newton/Schiller-Naumann", ROUND(vt * 1000, 4))))Particle Reynolds Number Re_p = (rho_f * |v_t| * d) / mu
Confirms creeping laminar regime (Re_p <= 1).
=LET(vt, (9.81 * (B2 ^ 2) * (B3 - B4)) / (18 * B5), ROUND((B4 * ABS(vt) * B2) / B5, 4))Try the example
Use the original values or change them to test your understanding.