The asymptotic limits of Riemann solutions for the isentropic drift‐flux model of compressible two‐phase flows

Abstract

The formation of vacuum state and delta shock wave are observed and studied in the limits of Riemann solutions for the one‐dimensional isentropic drift‐flux model of compressible two‐phase flows by letting the pressure in the mixture momentum equation tend to zero. It is shown that the Riemann solution containing two rarefaction waves and one contact discontinuity turns out to be the solution containing two contact discontinuities with the vacuum state between them in the limiting situation. By comparison, it is also proved rigorously in the sense of distributions that the Riemann solution containing two shock waves and one contact discontinuity converges to a delta shock wave solution under this vanishing pressure limit.

The Perturbed Riemann Problem for a Macroscopic Production Model with Chaplygin Gas

“…In the current paper, we only construct possible solution of the Riemann problem (1.2) and (1.3) for this certain initial condition and the uniqueness problem is still unsolved and left for the future work. In addition, it is well known that delta shock wave should be introduced in order to solve Riemann problems for some weak hyperbolic conservation systems and the theory of delta shock wave has been well established and developed in recent years, see [22][23][24][25][26][27][28] for examples. However, as far as we know, our constructed composite wave solution with the singularity of delta function cannot be found in any Riemann solution but only appears in the study [29][30][31] of wave interaction between a delta shock wave with a rarefaction wave for some hyperbolic systems of conservation laws.…”