Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
In this paper, we present a high-order polynomial free energy for the phase-field model of two-phase incompressible fluids. The model consists of the Navier–Stokes (NS) equation and the Cahn–Hilliard (CH) equation with a high-order polynomial free energy potential. In practice, a quartic polynomial has been used for the bulk free energy in the CH equation. It is well known that the CH equation does not satisfy the maximum principle and the phase-field variable takes shifted values in the bulk phases instead of taking the minimum values of the double-well potential. This phenomenon substantially changes the original volume enclosed by the isosurface of the phase-field function. Furthermore, it requires fine resolution to keep small shapes. To overcome these drawbacks, we propose high-order (higher than fourth order) polynomial free energy potentials. The proposed model is tested for an equilibrium droplet shape in a spherical symmetric configuration and a droplet deformation under a simple shear flow in a fully three-dimensional fluid flow. The computational results demonstrate the superiority of the proposed model with a high-order polynomial potential to the quartic polynomial function in volume conservation property.