A difference scheme for a triangular system of conservation laws with discontinuous flux modeling three-phase flows

A triangular system of conservation laws with discontinuous flux that models the one-dimensional flow of two disperse phases through a continuous one is formulated. The triangularity arises from the distinction between a primary and a secondary disperse phase, where the movement of the primary disperse phase does not depend on the local volume fraction of the secondary one. A particular application is the movement of aggregate bubbles and solid particles in flotation columns under feed and discharge operations. This model is formulated under the assumption of a variable cross-sectional area. A monotone numerical scheme to approximate solutions to this model is presented. The scheme is supported by three partial theoretical arguments. Firstly, it is proved that it satisfies an invariant-region property, i.e., the approximate volume fractions of the three phases, and their sum, stay between zero and one. Secondly, under the assumption of flow in a column with constant cross-sectional area it is shown that the scheme for the primary disperse phase converges to a suitably defined entropy solution. Thirdly, under the additional assumption of absence of flux discontinuities it is further demonstrated, by invoking arguments of compensated compactness, that the scheme for the secondary disperse phase converges to a weak solution of the corresponding conservation law. Numerical examples along with estimations of numerical error and convergence rates are presented for counter-current and co-current flows of the two disperse phases.