On a phase field model for binary mixtures of micropolar fluids with non-matched densities and moving contact lines
Kin Shing Chan, Baoli Hao, Kei Fong Lam, Björn Stinner
TL;DR
This work develops a thermodynamically consistent phase-field model for binary mixtures of incompressible micropolar fluids with non-matched densities and moving contact lines. It couples a volume-averaged Navier–Stokes–Cahn–Hilliard system for $(\bm{u}, p, \phi, \mu)$ with a micropolar equation for $\bm{\omega}$, and derives the model from conservation laws under an energy-dissipation framework. The paper presents both local (logarithmic) and nonlocal/free-energy formulations, and establishes global weak solutions in 3D for the singular potentials, including a deep quench limit to the obstacle potential. The analysis adopts a two-stage approximation with implicit time discretisation, yielding robust existence results and laying groundwork for numerical analysis of moving-contact-line two-phase flows in micropolar fluids.
Abstract
We introduce a new phase field model for binary mixtures of incompressible micropolar fluids, which are among the simplest categories of fluids exhibiting internal rotations. The model fulfils local and global dissipation inequalities so that thermodynamic consistency is guaranteed. Our model consists of a Navier--Stokes--Cahn--Hilliard system for the fluid velocity, pressure, phase field variable and chemical potential, coupled to an additional system of Navier--Stokes type for the micro-rotation. Our model accounts for non-matched densities as well as moving contact line dynamics, and serve as a generalisation to earlier models for binary fluid flows based on a volume averaged velocity formulation. We also establish the existence of global weak solutions in three spatial dimensions for the model equipped with singular logarithmic and double obstacle potentials.