Sharp-interface limits of a phase-field model with a generalized Navier slip boundary condition for moving contact lines

TL;DR

This paper investigates the sharp-interface limits of a phase-field model with a generalized Navier slip boundary condition for moving contact lines, combining asymptotic analysis and numerical simulations to understand boundary effects.

Contribution

It derives the sharp-interface limits for various boundary relaxation parameters and demonstrates the convergence behavior through numerical simulations.

Findings

Sharp-interface limit corresponds to two-phase Navier-Stokes with slip boundary conditions.

Convergence rates depend on the scaling of the mobility number and relaxation parameter.

Numerical results validate the asymptotic analysis and show different convergence behaviors.

Abstract

The sharp-interface limits of a phase-field model with a generalized Navier slip boundary condition for moving contact line problem are studied by asymptotic analysis and numerical simulations. The effects of the {mobility} number as well as a phenomenological relaxation parameter in the boundary condition are considered. In asymptotic analysis, we focus on the case that the {mobility} number is the same order of the Cahn number and derive the sharp-interface limits for several setups of the boundary relaxation parameter. It is shown that the sharp interface limit of the phase field model is the standard two-phase incompressible Navier-Stokes equations coupled with several different slip boundary conditions. Numerical results are consistent with the analysis results and also illustrate the different convergence rates of the sharp-interface limits for different scalings of the two parameters.