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.