# Numerical Approximations for a Phase-Field Moving Contact Line Model with Variable Densities and Viscosities

## TL;DR

This paper develops and analyzes energy-stable numerical schemes for a complex two-phase fluid model with variable densities, viscosities, and moving contact lines, verified through spectral-Galerkin discretization and numerical tests.

## Contribution

It introduces two novel unconditionally energy stable schemes for a nonlinear coupled phase-field model with variable properties and contact line dynamics.

## Key findings

- Schemes are unconditionally energy stable and efficient.
- Spectral-Galerkin method verifies accuracy and efficiency.
- Numerical results confirm robustness of the schemes.

## Abstract

We consider the numerical approximations of a two-phase hydrodynamics coupled phase-field model that incorporates the variable densities, viscosities and moving contact line boundary conditions. The model is a nonlinear, coupled system that consists of incompressible Navier--Stokes equations with the generalized Navier boundary condition, and the Cahn--Hilliard equations with moving contact line boundary conditions. By some subtle explicit--implicit treatments to nonlinear terms, we develop two efficient, unconditionally energy stable numerical schemes, in particular, a linear decoupled energy stable scheme for the system with static contact line condition, and a nonlinear energy stable scheme for the system with dynamic contact line condition. An efficient spectral-Galerkin spatial discretization is implemented to verify the accuracy and efficiency of proposed schemes. Various numerical results show that the proposed schemes are efficient and accurate.