On periodic traveling wave solutions with or without phase transition to the Navier–Stokes–Korteweg and the Euler–Korteweg equations

The Navier–Stokes–Korteweg and the Euler–Korteweg equations are considered in isothermal setting. These are diffuse interface models of two-phase flow. In one-dimensional setting, we show that for any period and any density average, there exists a periodic traveling wave solution with two phases having only two sharp internal layers in a periodic cell provided that the Korteweg relaxation parameter is small compared with the period. For the Euler–Korteweg equations, mass flux is allowed to be non-zero so that phase transition occurs but it is shown that for the Navier–Stokes–Korteweg equations no phase transition occurs for periodic traveling wave solution. Here, the Helmholtz (available) energy (modified by the mass flux) is assumed to be double-well type. We also show that such a periodic traveling wave solution tends to a monotone traveling wave solution as the period tends to infinity under suitable spatial translation. Our numerical experiment suggests that there is periodic traveling wave with phase transition which is stable under periodic perturbation for small viscosity but it seems that this is a transition pattern. • Non existence of a periodic traveling wave solution with phase transition for the one-dimensional Navier–Stokes–Korteweg equations. • Existence of a periodic traveling wave solution with phase transition for the one-dimensional Euler–Korteweg equations for any period and speed. • A periodic traveling wave solution for the Euler–Korteweg equations is observed as a transition pattern in a numerical test for the Navier–Stokes–Korteweg equations.