Particle-laden flows refers to a class of two-phase fluid flow, in which one of the phases is continuously connected (referred to as the continuous or carrier phase) and the other phase is made up of small, immiscible, and typically dilute particles (referred to as the dispersed or particle phase). Fine aerosol particles in air is an example of a particle-laden flow; the aerosols are the dispersed phase, and the air is the carrier phase. The modeling of two-phase flows has a tremendous variety of engineering and scientific applications: pollution dispersion in the atmosphere, fluidization in combustion processes, aerosol deposition in spray medication, along with many others.
Governing equations The starting point for a mathematical description of almost any type of fluid flow is the classical set of Navier–Stokes equations. To describe particle-laden flows, we must modify these equations to account for the effect of the particles on the carrier, or vice versa, or both - a suitable choice of such added complications depend on a variety of the parameters, for instance, how dense the particles are, how concentrated they are, or whether or not they are chemically reactive. In most real world cases, the particles are very small and occur in low concentrations, hence the dynamics are governed primarily by the continuous phase. A possible way to represent the dynamics of the carrier phase is by the following modified Navier-Stokes momentum equation:
∂ ρ u i d t + ∂ ρ u i u j ∂ x j = − ∂ P ∂ x i + ∂ τ i j ∂ x j + S i , {\displaystyle {\frac {\partial \rho u_{i}}{dt}}+{\frac {\partial \rho u_{i}u_{j}}{\partial x_{j}}}=-{\frac {\partial P}{\partial x_{i}}}+{\frac {\partial \tau _{ij}}{\partial x_{j}}}+S_{i},}
where S i {\displaystyle S_{i}} is a momentum source or sink term, arising from the presence of the particle phase. The above equation is an Eulerian equation, that is, the dynamics are understood from the viewpoint of a fixed point in space. The dispersed phase is typically (though not always) treated in a Lagrangian framework, that is, the dynamics are understood from the viewpoint of fixed particles as they move through space. A usual choice of momentum equation for a particle is:
d v i d t = 1 τ p ( u i − v i ) , {\displaystyle {\frac {dv_{i}}{dt}}={\frac {1}{\tau _{p}}}(u_{i}-v_{i}),}
where u i {\displaystyle u_{i}} represents the carrier phase velocity and v i {\displaystyle v_{i}} represents the particle velocity. τ p {\displaystyle \tau _{p}} is the particle relaxation time, and represents a typical timescale of the particle's reaction to changes in the carrier phase velocity - loosely speaking, this can be thought of as the particle's inertia with respect to the fluid with contains it. The interpretation of the above equation is that particle motion is hindered by a drag force. In reality, there are a variety of other forces which act on the particle motion (such as gravity, Basset history and added mass) – as described through for instance the Basset–Boussinesq–Oseen equation. However, for many physical examples, in which the density of the particle far exceeds the density of the medium, the above equation is sufficient. A typical assumption is that the particles are spherical, in which case the drag is modeled using Stokes drag assumption:
τ p = ρ p d p 2 18 μ . {\displaystyle \tau _{p}={\frac {\rho _{p}d_{p}^{2}}{18\mu }}.}
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