The Riemann problem for a simplified two-phase flow model with the Chaplygin pressure law under the external force

Abstract

The exact solutions are admitted in perfectly explicit forms about the Riemann problem for a simplified two-phase flow model of the drift-flux type with the pressure law of Chaplygin gas under the effect of external force. More accurately, the Riemann solution is either the combination of three contact discontinuities or only one delta shock wave which depends on the choice of different Riemann initial data. It is of interest to find the singularity of delta shock wave Riemann solution that the Dirac delta functions are involved in both the densities of liquid and gas simultaneously. The generalized Rankine–Hugoniot conditions and over-compressive delta entropy inequality are proposed to solve such delta shock wave Riemann solution, which satisfies this model in the sense of distributions. In addition, it is also verified by some representative numerical simulations that the delta shock wave is indeed contained in the Riemann solution under some appropriate Riemann-type initial conditions, which confirms our theoretical analysis.