Exact solutions to the Riemann problem for compressible isothermal Euler equations for two-phase flows with and without phase transition (Q2850029)

The authors consider a Riemann problem for inviscid and compressible fluids under isothermal conditions and with or without phase transition between the liquid and vapor phases. They write the local mass conservation law \(\frac{ \partial \rho }{\partial t}+\frac{\partial (\rho v)}{\partial x}=0\) and the balance law of momentum \(\frac{\partial (\rho v)}{\partial t}+\frac{\partial (\rho v^{2}+p)}{\partial x}=0\), where \(\rho \) is the mass density, \(v\) is the velocity field and \(p\) is the pressure. Across the discontinuity between the phases, they impose the jump condition \([\rho (v-W)]=0=\rho (v-W)[v]+[p]\), where \([\cdot ]\) means the jump of the quantity through the discontinuity. The pressure is linked to the density through the equation of state \(p=p(\rho )\) with \(p^{\prime }(\rho )=a^{2}\), where \(a\) is the constant speed of sound. The authors add the Riemann initial data \( \rho (x,0)=\rho _{-}\) (resp. \(\rho _{+}\)) if \(x<0\) (resp. \(x>0\)) and \( v(x,0)=v_{-}\) (resp. \(v_{+}\)) if \(x<0\) (resp. \(x>0\)). Once the model has been described, the authors prove some properties under different hypotheses on the discontinuity: rarefaction wave fans, shocks, phase transition. They first write this system in a vectorial way, leading a semilinear problem. The main part of the paper is devoted to the computation of exact solutions of the Riemann problem for the isothermal Euler equations, under these different hypotheses. Then the authors prove that if there is no solution of the Riemann problem, a nucleation phenomenon occurs. The paper ends with a short presentation of numerical results.