On time-dependent settling of a dilute suspension in a rotating conical channel

Abstract

The time-dependent settling of a dilute monodisperse suspension in a centrifugal force field is considered. The settling takes place between two axisymmetric narrowly spaced conical disks that are rotating rapidly. Clear fluid and suspension are assumed to behave as Newtonian fluids of different densities. The viscosities are, for simplicity, assumed to be the same. All effects of the sediment are neglected. The fluid motion is assumed to be almost parallel with the disks and is computed by using lubrication theory. This leads to a nonlinear hyperbolic equation of first order for the location of the interface between clear fluid and suspension. Local multivaluedness of the solution is removed by inserting shocks. Such a shock is a model for a small region where the slope of the interface, as scaled in the lubrication approximation, is large. Two problems are solved: batch settling and a case where the suspension is pumped into a conical channel that is initially filled with clear fluid. In the batch-settling case, the solution is quite similar to that computed by Herbolzheimer & Acrivos for settling due to a constant gravity field in a narrow tilted channel. For large values of the Taylor number, it is found that the blocking of radial flow outside the Ekman layers leads to a somewhat slower separation process than expected. In the filling problem, the character of the solution is distinctly different for Taylor numbers of order unity and large values of this parameter.