Solving a Singular Limit Problem Arising With Euler–Korteweg Dispersive Waves

Abstract

Phase transition in compressible flows involves capillarity effects, described by the Euler–Korteweg (EK) equations with nonconvex equation of state. Far from phase transition, that is, in the two convex parts of the equation of state, the dispersion terms vanish and one should recover the hyperbolic Euler equations of fluid dynamics. However, the solution of EK equations does not converge toward the solution of Euler equations when dispersion tends toward zero while being nonnull: it is a singular limit problem. To avoid this issue in the case of convex equation of state, a Navier–Stokes–Korteweg (NSK) model is considered, whose viscosity is chosen to counterbalance exactly the dispersive terms. In the limit of small viscosity and small dispersion, the Euler model is recovered. Numerically, an extended Lagrangian method is used to integrate the NSK equations so obtained. Doing so allows to use classical numerical schemes of Godunov type with source term. Numerical results for a Riemann problem illustrate the convergence properties with vanishing dispersion.