A $$\varvec{2\times 2}$$ hyperbolic system modelling incompressible two phase flows: theory and numerics

Abstract

We propose a \(2\times 2\) hyperbolic system of conservation laws to model the dynamics of two incompressible fluids in mechanical disequilibrium. In the theoretical part of the paper we show that this 1D system is not strictly hyperbolic, that the characteristic speed can not a priori be ordered and that the characteristic fields are neither genuinely nonlinear, nor linearly degenerate. We nevertheless prove the existence and uniqueness of an admissible solution to the Riemann problem. This solution remains bounded with positive volume fractions even when one the phases vanishes. We conclude that the multiphase/single phase transition does not imply mechanical equilibrium but displays a non classical wave structure. In the numerical part of the paper we propose some approximate Riemann solvers to simulate the model, especially the multiphase/single phase transition. The classical Riemann solvers have been considered as Godunov scheme, Roe scheme with or without entropy fix. We also propose an in-cell discontinuous reconstruction method which proves to be successful, whereas the other schemes may show some spurious oscillations in some Riemann problem. Finally, as an application we study and simulate the problem of phase separation by gravity.