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:
- <math>F = 6 \pi r \eta </math>
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:
- <math>V_s = \frac{2}{9}\frac{r^2 g (\rho_p - \rho_f)}{\eta}</math>
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
See also:
- Navier-Stokes equations
- Stokes' theorem