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# Stokes' law

For a theorem in differential geometry, see Stokes' theorem.

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 continuous viscous fluid by solving the small fluid-mass limit of the generally unsolvable Navier-Stokes equations:

$\vec{F} = -6 \pi r \eta \vec{v} \,$

where:

• $\vec{F}$ is the frictional force,
• r is the Stokes radius of the particle,
• η is the fluid viscosity, and
• $\vec{v}$ is the particle's velocity.

If the particles are falling in the viscous fluid by their own weight, then a terminal velocity, also known as the settling velocity, is reached when this frictional force combined with the buoyant force exactly balance the gravitational force. The resulting settling velocity is given by:

$V_s = \frac{2}{9}\frac{r^2 g (\rho_p - \rho_f)}{\eta}$

where:

• Vs is the particles' settling velocity (m/s) (vertically downwards if ρp > ρf, upwards if ρp < ρf),
• r is the Stokes radius of the particle (m),
• g is the standard gravity (m/s2),
• ρp is the density of the particles (kg/m3),
• ρf is the density of the fluid (kg/m3), and
• η is the (dynamic) fluid viscosity (Pa s).