Stokes Law, Fluids etc

Stokes' law

(From Wikipedia)

 

Not to be confused with Stokes' theorem.

Creeping flow past a sphere: streamlines, drag force F_d and force by gravity F_g.

In 1851, George Gabriel Stokes derived an expression, now known as Stokes' law, for the frictional force – also called drag force – exerted on spherical objects with very small Reynolds numbers (e.g., very small particles) in a continuous viscous fluid. Stokes' law is derived by solving the Stokes flow limit for small Reynolds numbers of the generally unsolvable Navier–Stokes equations:^[1]

where:

• F_d is the frictional force acting on the interface between the fluid and the particle (in N),
• μ is the dynamic viscosity (N s/m²),
• R is the radius of the spherical object (in m), and
• v_s is the particle's settling velocity (in m/s).

If the particles are falling in the viscous fluid by their own weight due to gravity, 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 (or terminal velocity) is given by:^[2]

where:

• v_s is the particles' settling velocity (m/s) (vertically downwards if ρ_p > ρ_f, upwards if ρ_p < ρ_f ),
• g is the gravitational acceleration (m/s²),
• ρ_p is the mass density of the particles (kg/m³), and
• ρ_f is the mass density of the fluid (kg/m³).

Stokes' law makes the following assumptions for the behavior of a particle in a fluid:

• Laminar Flow
• Spherical particles
• Homogeneous (uniform in composition) material
• Smooth surfaces
• Particles do not interfere with each other

Note that for molecules Stokes' law is used to define their Stokes radius.
The CGS unit of kinematic viscosity was named "stokes" after his work.

Flow visualization: