Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Navier-Stokes/Allen-Cahn system with moving contact line

Yinghua Li, Yuanxiang Yan, Xijun Yin

TL;DR

This work analyzes a diffuse-interface Navier–Stokes/Allen–Cahn (NSAC) system modeling moving contact lines under generalized Navier boundary conditions and phase-field boundary laws. It proves local-in-time existence and uniqueness of strong solutions in both 2D and 3D with GNBC and dynamic or relaxation boundary conditions, and shows global well-posedness in 2D channels, with exponential decay near phase separation or for small interface tension. For the relaxation boundary case in 3D, it develops a δ-approximation and δ-independent estimates to obtain local results, and then demonstrates global existence under small energy assumptions, including exponential decay. The analysis relies on contraction mapping arguments, interior and boundary regularity, mass conservation of the phase field, and a careful treatment of the energy-dissipation structure of NSAC with moving contact lines, providing rigorous insights into the dynamics of two-phase flows with diffuse interfaces near solid walls.

Abstract

In this paper, we study a diffuse interface model for two-phase immiscible flows coupled by Navier-Stokes equations and mass-conserving Allen-Cahn equations. The contact line (the intersection of the fluid-fluid interface with the solid wall) moves along the wall when one fluid replaces the other, such as in liquid spreading or oil-water displacement. The system is equipped with the generalized Navier boundary conditions (GNBC) for the fluid velocity ${\boldsymbol u}$, and dynamic boundary condition or relaxation boundary condition for the phase field variable $φ$. We first obtain the local-in-time existence of unique strong solutions to the 2D and 3D Navier-Stokes/Allen-Cahn (NSAC) system with generalized Navier boundary conditions and dynamic boundary condition. For the 2D case in channels, we further show these solutions can be extended to any large time $T$. Additionally, we prove the local-in-time strong solutions for systems with generalized Navier boundary conditions and relaxation boundary condition in 3D channels. Finally, we establish a global unique strong solution accompany with some exponential decay estimates when the fluids are near phase separation states and the contact angle closes to 90 degrees or the fluid-fluid interface tension constant is small.

Navier-Stokes/Allen-Cahn system with moving contact line

TL;DR

This work analyzes a diffuse-interface Navier–Stokes/Allen–Cahn (NSAC) system modeling moving contact lines under generalized Navier boundary conditions and phase-field boundary laws. It proves local-in-time existence and uniqueness of strong solutions in both 2D and 3D with GNBC and dynamic or relaxation boundary conditions, and shows global well-posedness in 2D channels, with exponential decay near phase separation or for small interface tension. For the relaxation boundary case in 3D, it develops a δ-approximation and δ-independent estimates to obtain local results, and then demonstrates global existence under small energy assumptions, including exponential decay. The analysis relies on contraction mapping arguments, interior and boundary regularity, mass conservation of the phase field, and a careful treatment of the energy-dissipation structure of NSAC with moving contact lines, providing rigorous insights into the dynamics of two-phase flows with diffuse interfaces near solid walls.

Abstract

In this paper, we study a diffuse interface model for two-phase immiscible flows coupled by Navier-Stokes equations and mass-conserving Allen-Cahn equations. The contact line (the intersection of the fluid-fluid interface with the solid wall) moves along the wall when one fluid replaces the other, such as in liquid spreading or oil-water displacement. The system is equipped with the generalized Navier boundary conditions (GNBC) for the fluid velocity , and dynamic boundary condition or relaxation boundary condition for the phase field variable . We first obtain the local-in-time existence of unique strong solutions to the 2D and 3D Navier-Stokes/Allen-Cahn (NSAC) system with generalized Navier boundary conditions and dynamic boundary condition. For the 2D case in channels, we further show these solutions can be extended to any large time . Additionally, we prove the local-in-time strong solutions for systems with generalized Navier boundary conditions and relaxation boundary condition in 3D channels. Finally, we establish a global unique strong solution accompany with some exponential decay estimates when the fluids are near phase separation states and the contact angle closes to 90 degrees or the fluid-fluid interface tension constant is small.

Paper Structure

This paper contains 16 sections, 43 theorems, 463 equations.

Table of Contents

  1. Introduction
  2. Notations and preliminaries
  3. NSAC system with GNBC and dynamic boundary condition
  4. Local well-posedness
  5. The linearized Allen-Cahn equations
  6. The linearized Navier-Stokes equations
  7. Proof of Theorem \ref{['local']}
  8. Global well-posedness
  9. NSAC system with GNBC and relaxation boundary condition
  10. Local well-posedness
  11. The -approximate problem
  12. -independent estimates
  13. Proof of Theorem \ref{['relaxation local']}
  14. Global well-posedness
  15. Time-independent estimates
  16. ...and 1 more sections

Key Result

Lemma 2.1

(Trace Imbedding Theorem, 9) Let $\Omega$ be a domain in ${\mathbb R}^N$ that satisfying the $C^{k-1,1}$-regularity condition, and suppose $u\in W^{k,p}(\Omega), p>1$, $k$ is an integer. If $l\le k-1$, then there is a trace operator $\rm Tr|_{\Gamma}$ such that holds. If $u|_\Gamma=\varphi$, we denote $\rm Tr|_\Gamma=\varphi$.

Theorems & Definitions (74)

  • Remark 2.1
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Lemma 2.5
  • Lemma 2.6
  • Lemma 2.7
  • Lemma 2.8
  • Lemma 2.9
  • ...and 64 more