In 1851, George Gabriel Stokes derived an expression for the frictional force exerted on spherical objects with very small Reynolds numbers (e.g., very small particles) in a viscous fluid by solving the generally unsolvable Navier-Stokes equations:
F = 6πrηv
where:
F is the frictional force
r is the particle radius
η is the fluid viscosity, and
v is the particles speed
If the particles are falling in the viscous fluid by their own weight, then we can derive their settling velocity by equating this frictional force with the gravitational force:


where:
Vs is the particles settling velocity,
g is the acceleration of gravity,
ρp is the density of the particles, and
ρf is the density of the fluid