The dynamics of a concentration interface in a dilute suspension of solid heavy particles

Abstract

Gravitational settling of solid heavy particles in a dilute suspension is studied analytically and numerically. The particle Reynolds number is assumed to be less than unity, for which the viscous drag force on the particle is well approximated by the linear Stokes law. The particulate volume fraction (or concentration) c is assumed to be small enough for the effects of particle–particle interactions to be negligible. The ratio δ=ρp/ρf of the particle and fluid densities is considered large enough however, so that the momentum exchange between the two phases caused by the viscous drag forces (which is of the order of the particulate mass loading factor cδ) is significant. The particulate base concentration, c0(y), is assumed to be a smooth function of the vertical coordinate y (hence, a stratified suspension) and a perturbation of the initially stationary settling regime is considered in the form of a horizontally propagating monochromatic wave with wavenumber k and frequency ω(k). Analytical solutions for the perturbations in the limit of small particle inertia (such that ωτp≪1, where τp is the particle response time) are found to be similar to those for internal waves propagating in a stratified fluid with effective density ρeff=ρf(1+c0(y)δ). On the other hand, it is found that in the opposite limit of large particle inertia (ωτp≫1) the perturbations are damped. As an example, we consider a suspension consisting of two layers with uniform concentrations of particles c1 (for y>+h/2) and c2 (for y<−h/2) separated by the interface layer of thickness h, where the concentration gradient is substantial. The solutions obtained in the long-wave limit kh≪1 show that if the concentration in the lower layer exceeds that in the upper layer (c2>c1), the disturbance of the interface brings about wavy motions analogous to internal waves in a two-layer fluid. In the case of inverse stratification (c2<c1) the disturbance grows exponentially and generates plume-like “bubbles,’’ similar to those produced due to the Rayleigh–Taylor instability in a two-layer fluid. The results of the numerical simulations show that, as expected, the waves are damped and the instability growth rate is reduced for particles having larger inertia.