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

Raimund Bürger; Stefan Diehl; M. Carmen Martí; Yolanda Vásquez; Raimund Bürger; Stefan Diehl; M. Carmen Martí; Yolanda Vásquez

doi:10.3934/nhm.2023006

Networks and Heterogeneous Media

2023, Volume 18, Issue 1: 140-190. doi: 10.3934/nhm.2023006

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Research article

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

1.
CI²MA and Departamento de Ingeniería Matemática, Universidad de Concepción, Casilla 160-C, Concepción, Chile
2.
Centre for Mathematical Sciences, Lund University, P.O. Box 118, S-221 00 Lund, Sweden
3.
Departament de Matemàtiques, Universitat de València, Avda. Vicent Andrés Estellés s/n, Burjassot, València, Spain

Received: 29 August 2022 Revised: 05 November 2022 Accepted: 10 November 2022 Published: 28 November 2022

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.
- Conservation law,
- discontinuous flux,
- triangular hyperbolic system,
- three-phase flow,
- kinematic flow model,
- flotation,
- sedimentation
Citation: Raimund Bürger, Stefan Diehl, M. Carmen Martí, Yolanda Vásquez. A difference scheme for a triangular system of conservation laws with discontinuous flux modeling three-phase flows[J]. Networks and Heterogeneous Media, 2023, 18(1): 140-190. doi: 10.3934/nhm.2023006

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Abstract

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.

References

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