where x is the position of the particle,
m the particle mass, and
F is the sum of all forces acting on the particle. Examples of forces acting on a particle in a fluid are the drag force, the buoyancy force, and the gravity force. The drag force represents the force that a fluid exerts on a particle due to a difference in velocity between the fluid and the particle. It includes the viscous drag, the added mass, and the Basset history term. Several empirical expressions have been suggested for the drag force. One of those is the one proposed by Khan and Richardson (
Ref. 13). That expression is valid for spherical particles for a wide range of particle Reynolds numbers. The particle Reynolds number is defined as
where u is the velocity of the fluid,
up the particle velocity,
r the particle radius,
ρ the fluid density, and
μ the dynamic viscosity of the fluid. The empirical expression for the drag force according to Khan and Richardson is