The vanishing pressure limits of Riemann solutions to the isothermal two-phase flow model under the external force

The accurate Riemann solutions in fully explicit forms are obtained for the one-dimensional inviscid, compressible, isothermal liquid–gas two-phase flow model of drift-flux type under the action of the unique external force of gravity for a pipeline placed on an inclined plane. It is also shown that the Riemann solution involves either a curved delta shock wave or the vacuum state for the pressureless case. Moreover, the asymptotic limits of Riemann solutions are investigated in detail by letting the pressure drop to zero, in which the formation of curved delta shock wave is obtained from the curved Riemann solution made up of backward shock wave, middle contact discontinuity and forward shock as well as the formation of vacuum state is also achieved from the curved Riemann solution consisting of backward rarefaction wave, middle contact discontinuity and forward rarefaction wave. In addition, some representative numerical results are offered to confirm the formation of delta shock wave and vacuum state.