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Dynamics of contact points in 2D Boussinesq flow

Yunrui Zheng

TL;DR

This work analyzes the 2D Boussinesq flow in a bounded open-top vessel with a moving free surface and dynamic contact points, incorporating gravity, capillarity, and thermal diffusion. The authors develop a global well-posedness theory near equilibrium using nonlinear energy methods, transforming the moving domain to a fixed reference via a geometric map and an A-operator framework, and proving exponential decay of perturbations. A two-tier Galerkin strategy and an ε-regularization are used to build linear theory for the convected heat equation and to manage corner singularities, followed by a nonlinear fixed-point argument and a limit ε → 0 to recover the original system. The results provide rigorous control of the contact-line dynamics and energy dissipation, offering a robust mathematical foundation for thermally driven fluid interfaces in constrained geometries with moving contact lines.

Abstract

We consider the evolution of contact lines for thermal convection of viscous fluids in a 2D open-top vessel. The domain is bounded above by a free moving boundary and otherwise by the solid wall of a vessel. The dynamics of the fluid are governed by the incompressible Boussinesq approximation under the influence of gravity, and the interface between fluid and air is under the effect of capillary forces. Motivated by energy-dissipation structure in [Guo-Tice, J. Eur. Math. Soc, 2024], we develop global well posedness theory in the framework of nonlinear energy methods for the initial data sufficiently close to equilibrium. Moreover, the solutions decay to equilibrium at an exponential rate. Our methods are mainly based on the construction of solutions to convected heat equation and a priori estimates of a geometric formulation of the Boussinesq equations.

Dynamics of contact points in 2D Boussinesq flow

TL;DR

This work analyzes the 2D Boussinesq flow in a bounded open-top vessel with a moving free surface and dynamic contact points, incorporating gravity, capillarity, and thermal diffusion. The authors develop a global well-posedness theory near equilibrium using nonlinear energy methods, transforming the moving domain to a fixed reference via a geometric map and an A-operator framework, and proving exponential decay of perturbations. A two-tier Galerkin strategy and an ε-regularization are used to build linear theory for the convected heat equation and to manage corner singularities, followed by a nonlinear fixed-point argument and a limit ε → 0 to recover the original system. The results provide rigorous control of the contact-line dynamics and energy dissipation, offering a robust mathematical foundation for thermally driven fluid interfaces in constrained geometries with moving contact lines.

Abstract

We consider the evolution of contact lines for thermal convection of viscous fluids in a 2D open-top vessel. The domain is bounded above by a free moving boundary and otherwise by the solid wall of a vessel. The dynamics of the fluid are governed by the incompressible Boussinesq approximation under the influence of gravity, and the interface between fluid and air is under the effect of capillary forces. Motivated by energy-dissipation structure in [Guo-Tice, J. Eur. Math. Soc, 2024], we develop global well posedness theory in the framework of nonlinear energy methods for the initial data sufficiently close to equilibrium. Moreover, the solutions decay to equilibrium at an exponential rate. Our methods are mainly based on the construction of solutions to convected heat equation and a priori estimates of a geometric formulation of the Boussinesq equations.

Paper Structure

This paper contains 39 sections, 31 theorems, 281 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Formulation of the Problem
  3. Main Theorem
  4. Strategy of the proof
  5. Constants and Sobolev Norms
  6. Elliptic Theory for The Corner Domains
  7. Analysis for Poisson equations near the corner points
  8. $\mathcal{A}$-Poisson equation
  9. Solution to $\mathcal{A}$-linear Heat Equations
  10. Weak Solutions
  11. Construction of Strong Solutions
  12. Galerkin Approximation for Initial Data
  13. Construction of Strong Solutions to \ref{['eq:linear_heat']}
  14. Higher Regularity
  15. Step 1 -- Construction of iteration
  16. ...and 24 more sections

Key Result

Theorem 1.1

Assume that the contact angle $\omega_{eq} \in (0, \pi)$ and $\varepsilon_{max} = \min\{1, -1+ \pi/\omega_{eq}\} \in (0, 1)$. Suppose that the initial data $(u_0, p_0, \theta_0, \eta_0, \partial_tu(0), \partial_tp(0), \partial_t\theta(0), \partial_t\eta(0), \partial_t^2u(0), \partial_t^2\theta(0), \ and zero-average condition for $k=0, 1, 2$ Then there exists sufficiently small $\delta_0>0$, such

Figures (2)

  • Figure 1: A vessel $\mathcal{V}$.
  • Figure 2: The domain $\Omega(t)$.

Theorems & Definitions (61)

  • Theorem 1.1
  • Remark 1.1
  • Remark 1.2
  • Theorem 1.2
  • Lemma 1.3
  • Theorem 2.1
  • proof
  • Definition 2.1
  • Theorem 2.2
  • proof
  • ...and 51 more