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Global dynamic stability of contact lines in fluids: 2-D droplet problem

Xiaoding Yang

TL;DR

This work develops a rigorous nonlinear stability theory for a two-dimensional Navier–Stokes droplet with a moving contact line and dynamic contact angles on a flat surface. The authors introduce a novel moving-polar-coordinate framework centered at a time-dependent point $\mathfrak{n}(t)$ to remove translational degeneracy in the energy, and they map the moving domain to a fixed reference via a geometric transformation, enabling an energy–dissipation analysis. By deriving a perturbation system around the gravity–capillary equilibrium $\rho_0$, expanding nonlinear terms, and establishing a coercive $1,\Sigma$ inner product after removing the translation mode, they prove a small-data, global-in-time result: solutions exist globally and decay exponentially to a horizontally shifted equilibrium with quantified bounds on the energy and dissipation. The results extend prior Stokes/vessel analyses to the full Navier–Stokes regime with moving contact points and dynamic contact angles, providing a mathematically rigorous understanding of contact-line dynamics in 2D droplets. The framework combines variational minimization for the equilibrium, moving-coordinate techniques to handle degeneracy, and a careful nonlinear energy-dissipation structure to yield global stability for near-equilibrium initial data.

Abstract

In this paper, we investigate the dynamics of an incompressible viscous Navier-Stokes fluid evolving above a one-dimensional flat surface. The fluid is subject to a uniform gravitational field and capillary forces acting along the free boundary. The interface between the fluid and the surrounding air is a free surface whose motion is driven by gravity, surface tension, and the fluid velocity field. The triple-phase intersections where the fluid, the air above the vessel, and the solid vessel wall meet are referred to as contact points, and the angles formed there are called contact angles. The model under consideration incorporates boundary conditions that allow for full motion of the contact points and dynamic contact angles. Under these conditions, \cite{Yang} established the existence of equilibrium configurations for the model. These equilibria consist of a quiescent fluid occupying a domain whose upper boundary can be represented as the graph of a function in polar coordinates, minimizing a gravity-capillary energy functional subject to a fixed mass constraint. The equilibrium contact angles may take any value in $(0,π)$ depending on the choice of capillary parameters. In the present work, we develop a framework of a priori estimates for this model. We prove that, for initial data sufficiently close to equilibrium, the system admits global solutions that converge exponentially fast to a (horizontally) shifted equilibrium state.

Global dynamic stability of contact lines in fluids: 2-D droplet problem

TL;DR

This work develops a rigorous nonlinear stability theory for a two-dimensional Navier–Stokes droplet with a moving contact line and dynamic contact angles on a flat surface. The authors introduce a novel moving-polar-coordinate framework centered at a time-dependent point to remove translational degeneracy in the energy, and they map the moving domain to a fixed reference via a geometric transformation, enabling an energy–dissipation analysis. By deriving a perturbation system around the gravity–capillary equilibrium , expanding nonlinear terms, and establishing a coercive inner product after removing the translation mode, they prove a small-data, global-in-time result: solutions exist globally and decay exponentially to a horizontally shifted equilibrium with quantified bounds on the energy and dissipation. The results extend prior Stokes/vessel analyses to the full Navier–Stokes regime with moving contact points and dynamic contact angles, providing a mathematically rigorous understanding of contact-line dynamics in 2D droplets. The framework combines variational minimization for the equilibrium, moving-coordinate techniques to handle degeneracy, and a careful nonlinear energy-dissipation structure to yield global stability for near-equilibrium initial data.

Abstract

In this paper, we investigate the dynamics of an incompressible viscous Navier-Stokes fluid evolving above a one-dimensional flat surface. The fluid is subject to a uniform gravitational field and capillary forces acting along the free boundary. The interface between the fluid and the surrounding air is a free surface whose motion is driven by gravity, surface tension, and the fluid velocity field. The triple-phase intersections where the fluid, the air above the vessel, and the solid vessel wall meet are referred to as contact points, and the angles formed there are called contact angles. The model under consideration incorporates boundary conditions that allow for full motion of the contact points and dynamic contact angles. Under these conditions, \cite{Yang} established the existence of equilibrium configurations for the model. These equilibria consist of a quiescent fluid occupying a domain whose upper boundary can be represented as the graph of a function in polar coordinates, minimizing a gravity-capillary energy functional subject to a fixed mass constraint. The equilibrium contact angles may take any value in depending on the choice of capillary parameters. In the present work, we develop a framework of a priori estimates for this model. We prove that, for initial data sufficiently close to equilibrium, the system admits global solutions that converge exponentially fast to a (horizontally) shifted equilibrium state.
Paper Structure (24 sections, 47 theorems, 809 equations, 2 figures)

This paper contains 24 sections, 47 theorems, 809 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. Equilibrium and Minimizer
  4. The Positivity Argument
  5. Geometric reformulation
  6. Moving polar coordinates
  7. Perturbation form
  8. Main results and the structure of this paper
  9. Previous work
  10. Basic computation
  11. The stability of the steady state
  12. The case when $[\![\gamma]\!]<0$
  13. The case when$[\![\gamma]\!]>0$
  14. The transformation of the domain(Without moving polar)
  15. The construction of the transformation
  16. ...and 9 more sections

Key Result

Theorem 1.1

The following statements hold $\bullet$Existence of a $\epsilon$-regularized minimizer For any $\epsilon>0$, there exists a minimizer $\rho_{0}^{\epsilon}$ of the following energy functional $E^{\epsilon}$ subject to the constraint where $V$ is the same constant as in equ:C. $\bullet$Uniform boundedness. When $M$ is fixed, $\rho_{0}^{\epsilon}(\theta)'$ is uniformly bounded for all $\epsilon>0$ a

Figures (2)

  • Figure 1: A droplet.
  • Figure 2:

Theorems & Definitions (92)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Lemma 2.1
  • proof
  • Theorem 2.2
  • proof
  • Theorem 2.3
  • proof
  • Theorem 3.1
  • ...and 82 more