In order for the particle tracing approach to be valid, the system should be a dilute or dispersed flow. This means that the particles should occupy a very small fraction of the volume of the surrounding fluid, generally less than 1%. When the volume fraction of the particles is not small, the fluid system is categorized as a dense flow and a different modeling approach is required.

Motion of microscopic and macroscopic sized particles is typically dominated by the drag force acting on particles immersed in a fluid. There are two phases in the system: a discrete phase consisting of bubbles, particles, or droplets, and a continuous phase in which the particles are immersed. The velocity of the continuous phase is sometimes entered as an expression but is most often computed using a fluid flow physics interface such as Laminar Flow (), Turbulent Flow, k-ε (), or Creeping Flow ().

In a sparse flow, the continuous phase affects the motion of the particles via the drag force, but the particles do not have enough inertia to significantly perturb the continuous phase. This is usually true when the particles in the discrete phase are very small or have relatively low number density. This is often referred to as unidirectional coupling. When modeling such a system, it is usually most efficient to solve for the continuous phase first, then compute the trajectories of the discrete particles in a separate study. For example, in the following plot of a static mixer, the stationary velocity field and pressure were first solved using a stationary study, and then the particle trajectories were computed in a time-dependent study.

In a dilute flow the continuous phase affects the motion of the particles, and the particle motion in turn disrupts the continuous phase. This is often referred to as a bidirectional coupling. Compared to a sparse flow, the particles in a dilute flow typically have more inertia, due to the particles being bigger, denser, or more numerous.

In a dispersed flow, the density of particles is greater than in the dilute flow described in the previous section, but their volume fraction is still lower than in a dense flow. The dispersed flow is the upper limit of the applicability of a Lagrangian particle tracking approach. In addition to the bidirectional fluid-particle interaction mentioned in the previous section, particle-particle interactions may also need to be taken into account. This is sometimes referred to as four-way coupling. Particle-particle interactions can be included in models but the following limitations apply:

Molecular diffusion is most significant for extremely small particles, on the order of 0.1 μm or smaller. When such small particles are in a fluid of nonzero absolute temperature, the thermal motion of individual atoms or molecules in that fluid causes them to collide randomly with the particles in directions other than that of the mean flow velocity. These random collisions can cause particles to spread out over time, a phenomenon known as Brownian motion.

However, if the flow is turbulent, then the fluid velocity u at the particle’s position consists of a mean flow term and an instantaneous fluid velocity perturbation,

where ζ is a random unit vector and k is the turbulent kinetic energy. Some turbulence models, such as the k-ε model, solve for k as an additional degree of freedom; it represents the amount of energy associated with eddies in the flow.

If the time step taken by the solver is made extremely small, then eventually the turbulent perturbation term between successive iterations no longer consists of uncorrelated random vectors. The built-in discrete random walk and continuous random walk models compensate for this effect by only seeding the random perturbation term at discrete times based on the turbulent dissipation rate ε, if it is available. Therefore, it is important to take a sufficiently small time step so that the changes in this random number seed can be resolved.