A diffuse-interface approach to two-phase isothermal flow of a Van der Waals fluid near the critical point

A novel numerical method for simulations of isothermal, compressible two-phase flows \textbf{of one fluid component} near the critical point is presented on the basis of a diffuse-interface model and a Van der Waals equation of state. Because of the non-convexity of the latter, the nature of the set of governing equations is mixed hyperbolic-elliptic. This prevents the application of standard numerical methods for compressible flow. Moreover, the Korteweg capillary stress tensor, characteristic for the diffuse-interface approach, introduces third-order spatial derivatives of mass density in the Navier-Stokes equation, resulting in a dispersive behavior of the solution. Our computational method relies on a transformation of the conserved variables, which controls dispersion, stabilizes the numerical simulation and enables the use of coarser grids. A one-dimensional simulation shows that this method provides better stability and accuracy than without transformation of variables. Two- and three-dimensional simulations for isothermal liquid-vapor flows, in particular the retraction of a liquid non-spherical drop in vapor and the binary droplet collision in vapor, show the applicability of the method. The surface tension calculated from the numerical results is in good agreement with its theoretical value if the computational grid is sufficiently fine.