The authors present a numerical scheme for solving a system of conservation laws that model the one-dimensional flow of two disperse phases through a continuous one. This system is of particular interest for applications such as the movement of aggregate bubbles and solid particles in flotation columns, which are crucial in various industrial processes. The paper addresses a triangular system of conservation laws with discontinuous flux, which is a significant challenge in the field due to the inherent discontinuities and the triangular structure of the system. The problem formulation includes primary and secondary disperse phases, with the primary phase’s movement independent of the secondary one’s local volume fraction.
The authors present a numerical scheme that approximates solutions to this model, focusing on its monotonicity, an invariant-region property ensuring that approximate volume fractions of the phases remain within physical bounds, and partial convergence analyses under certain assumptions. The proposed numerical scheme, notable for its simplicity and ease of implementation, is validated through theoretical arguments and numerical experiments. The paper’s theoretical contributions include proofs that the scheme satisfies the invariant-region property, ensuring that approximate volume fractions remain between zero and one, and convergence to a suitably defined entropy solution for the primary disperse phase under certain conditions. Furthermore, under additional assumptions of constant cross-sectional area and absence of flux discontinuities, the scheme for the secondary disperse phase is shown to converge to a weak solution of the corresponding conservation law.
The numerical results demonstrate the scheme’s effectiveness in modeling counter-current and co-current flows of the two disperse phases, with numerical errors tending to zero as the mesh is refined. These results underscore the practical applicability and accuracy of the proposed scheme in simulating real-world phenomena. The significance of this research lies in its advancement of numerical methods for systems of conservation laws with discontinuous flux, particularly in applications involving multiphase flows. The manuscript contributes to the theoretical understanding of such systems and provides a robust numerical tool for their simulation. This work is expected to have broad implications for the modeling and analysis of multiphase flows in various industrial and environmental processes. In summary, the paper makes a significant contribution to the field of numerical analysis of conservation laws with discontinuous flux, offering valuable insights into the modeling of three-phase flows. The proposed numerical scheme enhances the ability to simulate complex multiphase flow phenomena accurately, with implications for both theoretical research and practical applications in various engineering and scientific domains.