A Fully Decoupled Linear Second-Order Energy Stable Numerical Scheme for the Moving Contact Line Problem

Abstract

A decoupled linear second-order accurate unconditionally energy stable scheme for a phase-field model of the moving contact line (MCL) problem is proposed. The model consists of coupled Cahn-Hilliard and Navier-Stokes equations with the generalized Navier boundary condition. The scheme proposed introduces three scalar auxiliary variables to decouple nonlinear coupling terms while retaining stability and accuracy. Two of these variables represent the bulk and boundary energy, and one captures the “zero-energy-contribution” property between convection and surface tension terms. All related terms are decoupled by applying semi-explicit treatments while preserving stability and accuracy. Further coupling between velocity and pressure is removed by adopting a projection method. The overall scheme is second-order accurate, and the unconditional stability of energy is rigorously proved. Numerical results demonstrate the accuracy, efficiency, and stability of the method for the MCL problem. Besides, a consistent implementation of the contact angle hysteresis (CAH) is conducted to model important CAH phenomena.