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 parti
culate volume fraction (or concentration) c is assumed to be small eno
ugh for the effects of particle-particle interactions to be negligible
. The ratio delta = rho(p)/rho(f) of the particle and fluid densities
is considered large enough however, so that the momentum exchange betw
een the two phases caused by the Viscous drag forces (which is of the
order of the particulate mass loading factor c delta) is significant.
The particulate base concentration, c(0)(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 co
nsidered in the form of a horizontally propagating monochromatic wave
with wavenumber k and frequency omega(k). Analytical solutions for the
perturbations in the Limit of small particle inertia (such that omega
tau(p) much less than 1, where tau(p) is the particle response time)
are found to be similar to those for internal waves propagating in a s
tratified fluid with effective density rho(eff) = rho(f)(1 + c(0)(y) d
elta). On the other hand, it is found that in the opposite limit of la
rge particle inertia (omega tau(p) much greater than 1) the perturbati
ons are damped. As an example, we consider a suspension consisting of
two layers with uniform concentrations of particles c(1) (for y > + h/
2) and c(2) (for y < - h/2) separated by the interface layer of thickn
ess h, where the concentration gradient is substantial. The solutions
obtained in the long-wave Limit kh much less than 1 show that if the c
oncentration in the lower layer exceeds that in the upper layer (c(2)
> c(1)), the disturbance of the interface brings about wavy motions an
alogous to internal waves in a two-layer fluid. In the case of inverse
stratification (c(2) < c(1)) 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 in
ertia. (C) 1997 American Institute of Physics.