Exact Riemann solutions for the drift-flux equations of two-phase flow under gravity

Abstract

The exact Riemann solutions are constructed in completely explicit forms for the drift-flux equations of the one-dimensional inviscid, compressible and isentropic liquid-gas two-phase flow model in an inclined pipeline under gravity. It is found that the curved delta shock wave and vacuum state are involved in the Riemann solutions for the pressureless situation with the adiabatic exponent being one. It is proved rigorously that this curved delta shock wave solution satisfies the pressureless drift-flux equations in the sense of distributions. Finally, the asymptotic limits of Riemann solutions are investigated in detail when the adiabatic exponent tends to one, in which the concentration phenomenon can be observed when the non-self-similar Riemann solution consisting of curved 1-shock wave, 2-contact discontinuity and 3-shock wave converges to this curved delta shock wave solution. In addition, the cavitation phenomenon can also be inspected that all the states in the two rarefaction wave fans become the vacuum states in the limiting situation.