# Fundamental physical processes and particle metrology

## THE MOTION OF PARTICLES IN A FLOWING MEDIUM

### The forces on a single particle

#### (a) Synopsis

Let us consider a single particle (Fig. 3.1) suspended in a fluid at a given point in time. Its instantaneous position is determined by its orientation and the position of its centre of gravity. We shall designate its translational velocity by *w* and its rotational velocity by cd. In the absence of the particle the fluid at the point of time in question has a definite velocity profile with a velocity *v* at the position of the centre of gravity of the particle. The relative velocity is

If the particle is small with respect to the spatial changes in the field of flow, then the flow can be considered to have a flat velocity profile relative to the particle. In this case *v _{re]}* denotes the velocity of approach, i.e. the uniform velocity with which the fluid approaches the particle (strictly speaking, from an infinite distance).

The accelerations i), *w* and i>_{rel} which must satisfy the equation

can take any direction other than those of the velocities *v*, *w* and p_{rel}.

The following types of force can act on the particle.

Section 3.1 was written with the cooperation of Dr J. Raasch.

Figure 3.1 Sketch illustrating the definitions of particle velocity w, fluid velocity *и *and relative velocity *v _{Te}.*

1. Field forces: the best known and most important example is the gravitational force

where *V* and p_{p} are respectively the volume and the density of the particle and *g* is the acceleration due to gravity. Electric and magnetic fields of force can also act on the particle.

- 2. Aerodynamic and hydrodynamic forces: as a consequence of the motion of the fluid relative to the particle, it is acted upon, in the most general case, by a torque
*M*and a force*F*. The latter can be resolved into a component in the direction of p_{rel}, which is the drag force D, and a component perpendicular to*v*which is the dynamic thrust_{re]},*T*The torque_{d}.*M*and the component forces*D*and*T*are not constant quantities. They vary according to whether or not the flow is steady state and depend,_{d}*inter alia*, on the Reynolds number and the degree of turbulence in the fluid, on its compressibility and mean free path, on the proximity of boundary walls and other particles, and on the surface roughness and the shape of the particles. These matters will be examined more thoroughly in later sections. - 3. Pressure forces: in addition to the dynamic forces, and also when there is no relative movement between fluid and particle, pressure forces
*В*can be exerted on the particle. This is always the case whenever a pressure gradient gradp exists in the field of flow. The general relation is

The pressure gradient derived from the Navier-Stokes equations is
if it can be assumed that the fluid is incompressible and that no other particles are nearby (suspension with a vanishingly small solids concentration). p_{f} and *r]* stand for the density and viscosity of the fluid respectively. In a quiescent fluid and under the effect of gravity only it follows that

The pressure force in the gravitational field is also called the static thrust (buoyancy).

4. Inertial forces: in accordance with d’Alembert’s principle an inertial force

is introduced where *V, p _{p}* and

*w*are respectively the volume, density and acceleration of the particle. If the velocity

*w*of the particle is related to a rotating system of reference, as is the case, for example, when determining the motion of a particle in a centrifuge, two additional inertial forces have to be introduced, i.e. the centrifugal force

and the Coriolis force

where *R* is the radius vector of the rotating system of reference and ft is its angular velocity; *[w* x ft] is the vector product.

- 5. Diffusion forces: arise either through the direct molecular bombardment of the particle’s surface (Brownian movement, radiation pressure) or through equalizing flows which occur in gases as a consequence of a temperature or concentration gradient.
- 6. Contact forces: forces can be transferred by bodily contact between particles and solid walls or between particles alone. In this case it is necessary to distinguish between impact forces, frictional forces and adhesional forces.