Stokes' Law Calculator

Stokes' Settling Velocity Calculator

Particle Radius (r)

meters (m)

Particle Density (ρ_p)

kg/m³

Quartz/Sand is typically ~2650 kg/m³.

Fluid Density (ρ_f)

kg/m³

Pure water at 20°C is approx. 1000 kg/m³.

Dynamic Viscosity of Fluid (μ)

Pa·s (or kg/m·s)

Water at 20°C is approx. 0.001 Pa·s.

Gravitational Acceleration (g)

m/s²

Compute the terminal settling velocity of a spherical particle falling through a viscous fluid. Essential for applications in soil mechanics, geology, chemical engineering, and cellular biology.

Formula:

The Governing Formula

v = ²⁄₉ × ^{g × r² × (ρ_p − ρ_f)}⁄_μ

Variable Guide

v: Terminal settling velocity (m/s)
g: Acceleration due to gravity (m/s²)
r: Radius of the spherical particle (m)
ρ_p: Mass density of the particle (kg/m³)
ρ_f: Mass density of the fluid (kg/m³)
μ: Dynamic fluid viscosity (Pa·s)

Important Constraint: Stokes' Law operates under the assumption of a low Reynolds Number (Re < 0.1), signifying strict laminar flow where viscous forces dominate inertial forces.

Understanding Stokes' Law and Particle Sedimentation

Derived by the physical mathematician Sir George Gabriel Stokes, Stokes' Law mathematical expression expresses the frictional drag force exerted on spherical objects moving through a viscous fluid medium. When a particle falls under its own gravitational weight through a column of gas or liquid, it accelerates until the net downward force (gravity minus buoyant upthrust) perfectly equals the resisting upward fluid drag force. Once balanced, acceleration drops to zero and the object falls at a fixed speed termed the terminal settling velocity.

Crucial Variables and Environmental Impacts

The physical relationship demonstrates that the velocity is incredibly sensitive to particle sizing because the parameter relies heavily on the square of the radius (r²). Doubling the radius of a falling sand grain or cellular body quadruples its downward sedimentation speed. Conversely, narrowing the density differences between the settling particle and the fluid medium significantly drops the velocity baseline. If the surrounding fluid is denser than the particle, the velocity value turns negative, signaling upward buoyancy (flotation).

Real-World Scientific Applications

Industries rely heavily on Stokes' expressions across various applied operational environments:

Soil Mechanics: Determining distinct grain size distributions within complex mixtures of silt, sand, and clay using specialized hydrometer analytical tests.
Meteorology: Modeling how tiny mist cloud droplets stay suspended in air currents or combine to precipitate downwards as rainfall.
Biochemistry: Optimizing centrifuge configurations to quickly separate cellular organelles, blood fractions, or viral payloads out of liquid mixtures based on specific density differentials.