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92 4 Properties of Aerosol Particles

j
q d p u vj
g
Re p ¼ ð4:1Þ
l
In this equation, the density is the gas density and subscript g is added in order to
differentiate it with the density of solid particles. With a coordinate fixed on the
moving particle, u vj should always be a positive quantity.
j
The particle Reynolds number can be used to estimate the flow condition around
the particles. There are four regime for particle dynamics as follows.
8
Stokes regime : Re p \1
>
>
Transient regime : 1\Re p \5
<
Turbulent regime : 5\Re p \1000
>
>
Newton's regime : Re p [ 1000
:
For Stokes regime, the flow around the particle is laminar and the frictional force
is dominant over the inertia force. In the transient regime, the flow around the
particle starts to develop turbulence, both inertial and frictional forces are important.
In the turbulent regime, the flow around the particle is turbulent and drag decreases
with Re p . The flow around the particle becomes highly turbulent when Re p [ 1000
and the drag is considered constant, and the inertial force is dominant.

4.1.2 Stokes’ Law

Stokes’ law is derived from the Navier-Stokes equations, which are nearly
insolvable due to the nonlinear partial differential equations. Stokes solved these
equations with the following assumptions:

(1) The inertial forces are negligible compared with the viscous forces.
(2) There are no walls or other particles nearby. Because of the small size of
airborne particles, the fraction of the particles near a wall is negligible.
(3) The particle is a rigid sphere. This means that there is a limitation of Stokes
law for liquid droplets or soft particles.
(4) The fluid is incompressible. When the fluid is a gas, say air, the assumption
does not imply that air is incompressible, but that it does not compress near the
surface of the particle. This is equivalent to assuming that the relative velocity
between the gas and the particle is much less than the speed of sound, and it is
valid for airborne particles.
(5) There is no slipping between the fluid and particles i.e., the fluid velocity at the
particle’s surface is zero. When the relative velocity is not zero, a correction is
needed.
(6) The motion of the particle is constant.