The Cahn–Hilliard equation involves fourth-order spatial derivatives. Finite element
solutions are not common because primal variational formulations of fourth-order operators …

In this paper, we establish a binary fluid surfactant model by coupling two mass-conserved
Allen–Cahn equations and the Navier–Stokes equations and consider numerical …

We introduce provably unconditionally stable mixed variational methods for phase-field
models. Our formulation is based on a mixed finite element method for space discretization …

A phase-field moving contact line model is presented for a two-phase system with soluble
surfactants. With the introduction of some scalar auxiliary variables, the original free energy …

In this paper, we consider the numerical solution of a binary fluid–surfactant phase field
model, in which the free energy contains a nonlinear coupling entropy, a Ginzburg–Landau …

This paper is devoted to the numerical simulation of the Navier–Stokes–Korteweg
equations, a phase-field model for water/water-vapor two-phase flows. We develop a …

Efficient and accurate fracture modeling is of great importance in applications where
catastrophic outcomes under extreme scenarios are possible. The phase-field (PF) …

The binary fluid surfactant phase-field model, coupled with two Cahn--Hilliard equations and
Navier--Stokes equations, is a very complex nonlinear system, which poses many …

We propose a linear, fully decoupled, and energy stable finite difference scheme for solving
a phase-field surfactant fluid system. Inspired by the idea of multiple scalar auxiliary …

In this paper, we consider the numerical approximations for the commonly used binary fluid-
surfactant phase field model that consists two nonlinearly coupled Cahn–Hilliard equations …