# An energy stable $C^0$ finite element scheme for a quasi-incompressible phase-field model of moving contact line with variable density

## TL;DR

This paper introduces an energy stable C0 finite element scheme for simulating two-phase flows with moving contact lines and variable density, ensuring mass conservation and energy stability.

## Contribution

It develops a thermodynamically consistent phase-field model with a novel finite element scheme that is mass conserving and energy stable for quasi-incompressible flows.

## Key findings

- Scheme is mass conservative and energy stable
- Achieves 2nd order convergence with P1 elements
- Achieves 3rd order convergence with P2 elements

## Abstract

In this paper, we focus on modeling and simulation of two-phase flow with moving contact lines and variable density. A thermodynamically consistent phase-field model with General Navier Boundary Condition is developed based on the concept of quasi-incompressibility and the energy variational method. Then a mass conserving and energy stable C0 finite element scheme is developed to solve the PDE system. Various numerical simulation results show that the proposed schemes are mass conservative, energy stable and the 2nd order for P1 element and 3rd order for P2 element convergence rate in the sense of L2 norm.