The Particle Tracing for Fluid Flow Interface is designed for modeling microscopic and macroscopic particles in a background fluid. There are two phases in the system: a particle phase consisting of discrete bubbles, particles, droplets, and so forth; and a continuous phase in which the particles are immersed. In order for the particle tracing approach to be valid, the fluid system should be a dilute or dispersed flow. This means that the volume fraction of the particles is much smaller than the volume fraction of the continuous phase, 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.

It is important to realize that with the particle tracing approach, particles do not displace the fluid they occupy. In addition, the finite size of the particle is not taken into account when modeling particle-wall interactions. In other words, for purposes of detecting particle-boundary interactions, the particles are treated as point masses. The specification of particle diameter is mostly used for size-dependent forces, such as the Drag Force and Dielectrophoretic Force.

In a sparse flow, the continuous phase affects the motion of the particles but not vice versa. This is often referred to as “one-way coupling”. When modeling such a system in COMSOL Multiphysics, it is usually most efficient to solve for the continuous phase and the dispersed phase in separate studies. The fluid usually affects the particle motion through the Drag Force feature, which defines a drag force based on the fluid properties, particle properties, and the particle velocity relative to the flow. Several built-in drag laws can be used, and the optimal drag law usually depends on the size and speed of the particles. Built-in options to apply random perturbations to account for turbulence in the fluid are also available.

For example, in the following model example, the velocity field is first computed using a Stationary study, then the particle trajectories are computed using a separate 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 or “two-way coupling”. The bidirectional coupling between particles and fluids can be modeled using the Fluid-Particle Interaction multiphysics coupling node. This node can be added manually if the necessary physics interfaces are already present. Alternatively, The Fluid-Particle Interaction Interface can be used to automatically add the necessary physics interfaces and multiphysics coupling nodes.

The body force exerted on the fluid by the particle is applied in an approximate way, in that it is smeared out over a mesh element. This smearing effect makes the volume force computed by the Fluid-Particle Interaction node somewhat mesh dependent. When modeling fluid-particle interactions for which the mass flow rate of particles is not constant, the continuous phase and dispersed phase must be computed simultaneously in the same study. The computational demand is significantly higher than in the Sparse Flow case.

If the fields are stationary, as often occurs when particles are released at constant mass flow rate, it may be possible to compute the particle trajectories using a Time-Dependent solver while computing the fluid flow variables using a Stationary solver. It is also possible to create a solver loop that alternates between the Stationary and Time-Dependent solvers so that a bidirectional coupling between the trajectories and fields can be established; a dedicated Bidirectionally Coupled Particle Tracing study step is available for setting up such a solver loop. The process of combining these solvers is described in the section Study Setup.

In addition to the effects mentioned above, particle-particle interactions also need to be taken into account. This is often referred to as “four-way coupling”. Particle-particle interactions can be included in a model by adding a Particle-Particle Interaction node.

Built-in auxiliary dependent variables for the mass and temperature can be activated by selecting the Compute particle mass and Compute particle temperature check boxes, respectively, in the Additional Variables section of the Settings window for the physics interface. When the option to compute particle mass is activated, it is possible to set the initial particle mass in particle release features, such as Release, Inlet, and Release from Grid. It is also possible to select a distribution function for the initial mass. This is important when modeling separation devices where the goal is to understand the transmission probability of particles of various sizes.

Continuum methods such as FEM have one major drawback when it comes to modeling the advection and diffusion of a particulates in a fluid. The higher the Péclet number, the more numerically unstable the method becomes. The Péclet number is the ratio of the rate of advection to the rate of diffusion:

where L is the characteristic dimension, u is the advection velocity, and D is the diffusion coefficient of the particles. In general, continuum methods cannot handle systems where the Péclet number is greater than around 1000. The diffusion coefficient D (SI unit: m2/s) for spherical particles is, following Einstein’s relation,

For 100 nm diameter particles in water at room temperature, this results in a diffusion coefficient of around 4 × 10-12 m2/s. For a microfluidic device with characteristic size 1 mm and velocity of 1 m/s, this results in a Péclet number of 2.5 × 108. Handling such a large Péclet number with continuum methods is usually not practical (Ref. 1).

Particle trajectories are computed in a Lagrangian reference frame, removing the restriction on the Péclet number. The Péclet number can be anything from 0 to infinity without introducing numerical instabilities. Advection is added to the particles via the Drag Force. Molecular diffusion is added to particles by adding the Brownian Force. If the background velocity field is zero then particle motion is purely diffusive (zero Péclet number). If the Brownian Force is neglected and the background velocity is nonzero, the motion is pure advection (infinite Péclet number).

Many drag laws, such as the Stokes drag law, are based on the assumption of continuum flow, in which the particle Knudsen number Kn (dimensionless) is very small,

where λ (SI unit: m) is the mean free path of molecules in the surrounding fluid, and L (SI unit: m) is a characteristic length of the particle, which is often the particle radius or diameter. The exact definition of the characteristic length may vary depending on the source being cited and should be considered with caution.

When the particles are extremely small or they are surrounded by a rarefied gas, the assumption of continuum flow may not be valid. By selecting the Include rarefaction effects check box in the physics interface Particle Release and Propagation section, it is possible to apply correction factors to the Drag Force and Thermophoretic Force, enabling accurate modeling of particle motion in the slip flow, transitional flow, and free molecular flow regimes.

The Droplet Breakup feature and its two subnodes, Kelvin-Helmholtz Breakup Model and Rayleigh-Taylor Breakup Model, can be used to model the breakup of liquid droplets into successively smaller child droplets. The Nozzle feature can be used to release a spray of droplets at a specified location.

The Droplet Sprays in Fluid Flow Interface functions like The Particle Tracing for Fluid Flow Interface, with appropriate default features and settings for modeling the breakup of liquid droplets.