Implementing contact angle boundary conditions for second-order Phase-Field models of wall-bounded multiphase flows

TL;DR

This paper introduces a general formulation for implementing contact angle boundary conditions in second-order Phase-Field models for multiphase flows, ensuring mass conservation and physical accuracy in complex, large-density-ratio scenarios.

Contribution

A novel, consistent, and conservative formulation for contact angle boundary conditions in second-order Phase-Field models applicable to N-phase moving contact line problems.

Findings

The formulation accurately reproduces exact and asymptotic solutions.

It effectively captures complex dynamics in large-density-ratio flows.

Numerical results demonstrate the method's robustness and physical fidelity.

Abstract

In the present work, a general formulation is proposed to implement the contact angle boundary conditions for the second-order Phase-Field models, which is applicable to $N$-phase $(N \geqslant 2)$ moving contact line problems. To remedy the issue of mass change due to the contact angle boundary condition, a source term or Lagrange multiplier is added to the original second-order Phase-Field models, which is determined by the consistent and conservative volume distribution algorithm so that the summation of the order parameters and the \textit{consistency of reduction} are not influenced. To physically couple the proposed formulation to the hydrodynamics, especially for large-density-ratio problems, the consistent formulation is employed. The reduction-consistent conservative Allen-Cahn models are chosen as examples to illustrate the application of the proposed formulation. The numerical scheme that preserves the consistency and conservation of the proposed formulation is employed to demonstrate its effectiveness. Results produced by the proposed formulation are in good agreement with the exact and/or asymptotic solutions. The proposed method captures complex dynamics of moving contact line problems having large density ratios.