The two-phase flow in settling vessels with walls that are inclined to the vertical is investigated. By neglecting inertial effects and the viscosity of the suspension i t is shown that the particle concentration remains constant on kinematic-wave fronts. The wave fronts are horizontal and propagate in a quasi-one-dimensional manner, but are imbedded in a two-dimensional or three-dimensional basic flow which, in turn, depends on the waves via the boundary conditions. Concentration discontinuities (interfaces) are described by kinematic-shock theory. The kinematic shocks are shown to be horizontal, with the possible exception of discontinuities that separate the suspension from the sediment.At downward-facing inclined walls conservation of mass enforces the existence of a boundary-layer flow with relatively large velocity. As G/R2→∞ and G/R4→ 0, where G and R are respectively a sedimentation Grashof number and a sedimentation Reynolds number, the entrainment of suspended particles into the boundary-layer flow of clear liquid is negligibly small. This provides an appropriate boundary condi- tion for the basic flow of the suspension. Thus, in the double limit considered, a kine- matic theory suffices to determine the convective flow of the suspension due to the presence of inclined walls.As an example batch sedimentation in vessels with inclined plane or conical walls is investigated. The settling process is terminated after a time that can be considerably smaller than the time required in a vertical vessel under the same conditions.Depending on the initial particle concentration, there are centred kinematic waves that are linked to a continuous increase of the particle concentration in the suspension. In an appendix, the flow in the boundary layer at a downward facing, inclined wall is investigated. With G/R2→∞ and G/R4→ 0, the boundary layer consists of an inviscid particle-free main part, a viscous sublayer at the wall, and a free shear sublayer at the liquid/particle interface.
Compressible two-phase boundary-layer flow with finite particulate volume fraction
