The least-squares spectral element method for phase-field models for isothermal fluid mixture

The phase-field approach has been regarded as a powerful method in numerically handling the interface dynamics in multiphase flow in several scientific and engineering applications. For an isothermal fluid mixture, the Navier–Stokes–Korteweg equation and the Navier–Stokes–Cahn–Hilliard equation have represented two major branches of the phase-field methods. We present a general discretization formulation for these two equations and conduct a comparison study of them. The formulation using a least-squares spectral element method is implemented by adopting a time-stepping procedure, a high-order continuity approximation and an element-by-element solver technique. To describe the same fluid mixtures by the isothermal Navier–Stokes–Korteweg and the Navier–Stokes–Cahn–Hilliard equations, we suggest a non-dimensionalization with the same dimensionless quantities. Numerical experiments are conducted to verify the spectral/hp least-squares formulation for the isothermal Navier–Stokes–Korteweg model. Besides, the equilibrium state of the van der Waals fluid model is calculated both analytically and numerically. Through spontaneous decomposition example, the isothermal Navier–Stokes–Korteweg system and the Navier–Stokes–Cahn–Hilliard system are compared in terms of the equilibrium pressure and the energy minimizing process. As a general example, the coalescence of two liquid droplets is studied with our solver for the isothermal Navier–Stokes–Korteweg system. The minimum discretization levels for space and time are investigated and a parametric study on Weber number is carried out.