For low values of the Reynolds number (<1), true for most aerosol motion, Stokes law describes the force of resistance on a solid spherical particle in a fluid. However, Stokes law is only valid when the velocity of the gas at the surface of the particle is zero. For small particles (< 1 μm) that characterize aerosols, however, this assumption fails. To account for this failure, one can introduce the Cunningham correction factor, always greater than 1. Including this factor, one finds the relation between the resisting force on a particle and its velocity:
where
: is the resisting force on a spherical particle
: is the dynamic viscosity of the gas
: is the particle velocity
: is the Cunningham correction factor.
This allows us to calculate the terminal velocity of a particle undergoing gravitational settling in still air. Neglecting buoyancy effects, we find:
where
: is the terminal settling velocity of the particle.
The terminal velocity can also be derived for other kinds of forces. If Stokes law holds, then the resistance to motion is directly proportional to speed. The constant of proportionality is the mechanical mobility (B') of a particle:
A particle traveling at any reasonable initial velocity approaches its terminal velocity exponentially with an e-folding time equal to the relaxation time:
where:
: is the particle speed at time t
: is the final particle speed
: is the initial particle speed
To account for the effect of the shape of non-spherical particles, a correction factor known as the dynamic shape factor is applied to Stokes' law. It is defined as the ratio of the resistive force of the irregular particle to that of a spherical particle with the same volume and velocity:
where:
: is the dynamic shape factor