Particle-laden flows

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. 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. 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: 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: where u i {displaystyle u_{i}} represents the carrier phase velocity and v i {displaystyle v_{i}} represents the particle velocity. τ p {displaystyle au _{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: Here d p {displaystyle d_{p}} is the particle diameter, ρ p {displaystyle ho _{p}} , the particle density and μ {displaystyle mu } , the dynamic viscosity of the carrier phase. More sophisticated models contain the correction factor: where R e p {displaystyle Re_{p}} is the particle Reynolds number, defined as: If the mass fraction of the dispersed phase is small, then one-way coupling between the phases is a reasonable assumption; that is, the dynamics of the particle phase are affected by the carrier phase, but the reverse is not the case. However if the mass fraction of the dispersed phase is large, the interaction of the dynamics between the two phases must be considered - this is two-way coupling.