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Rate independent capillary motion on a narrow Wilhelmy plate

Carson Collins, William M Feldman

TL;DR

This work develops a rate-independent energetic framework for capillary surfaces in the Wilhelmy plate setup, coupling a volume-constrained energy with a contact-line dissipation model. By employing barrier arguments for the prescribed mean-curvature problem and a viscosity-theoretic approach, the authors prove that as the container-to-plate width ratio $R$ grows, the volume constraint vanishes and a Dirichlet-driven evolution emerges, with energy solutions staying stable and well-behaved. They establish coercivity, Euler–Lagrange relations, height bounds, and almost-minimal regularity for globally stable profiles, then prove compactness and convergence of discretized minimizing movements to energy solutions. A key technical advance is the $R o ty$ limit, where they derive a precise Lagrange-multiplier decomposition, outer-approximation results, and a relativized energy that converges to a finite limiting energy $ ext{E}_ ty$, yielding a well-posed limiting evolution for the capillary system under Dirichlet forcing. The results provide a rigorous bridge between volume-constrained and Dirichlet-driven evolutions in capillarity, with implications for uniqueness, regularity, and asymptotic behavior of the contact-line dynamics in wide-container regimes.

Abstract

We study a rate independent energetic model of the Wilhelmy plate experiment in capillarity. The evolution is driven by vertical motions of the plate. We show stability of energy solutions to the evolution, in the sense used in the rate-independent systems literature, as the ratio between container width and plate width goes to infinity. In particular, we show that the volume-constraint for the finite-ratio problem disappears in the limit. This leads to a volume-unconstrained Dirichlet-forced evolution, a setting where monotonicity, uniqueness, and contact line regularity properties have been established in previous literature. Our result is based on using comparison principle techniques for the prescribed mean curvature equation with capillary contact angle condition that characterizes the liquid surface at equilibrium. Through barrier arguments, we are able to develop asymptotics for the energy which give us control independent of the container-to-plate ratio.

Rate independent capillary motion on a narrow Wilhelmy plate

TL;DR

This work develops a rate-independent energetic framework for capillary surfaces in the Wilhelmy plate setup, coupling a volume-constrained energy with a contact-line dissipation model. By employing barrier arguments for the prescribed mean-curvature problem and a viscosity-theoretic approach, the authors prove that as the container-to-plate width ratio grows, the volume constraint vanishes and a Dirichlet-driven evolution emerges, with energy solutions staying stable and well-behaved. They establish coercivity, Euler–Lagrange relations, height bounds, and almost-minimal regularity for globally stable profiles, then prove compactness and convergence of discretized minimizing movements to energy solutions. A key technical advance is the limit, where they derive a precise Lagrange-multiplier decomposition, outer-approximation results, and a relativized energy that converges to a finite limiting energy , yielding a well-posed limiting evolution for the capillary system under Dirichlet forcing. The results provide a rigorous bridge between volume-constrained and Dirichlet-driven evolutions in capillarity, with implications for uniqueness, regularity, and asymptotic behavior of the contact-line dynamics in wide-container regimes.

Abstract

We study a rate independent energetic model of the Wilhelmy plate experiment in capillarity. The evolution is driven by vertical motions of the plate. We show stability of energy solutions to the evolution, in the sense used in the rate-independent systems literature, as the ratio between container width and plate width goes to infinity. In particular, we show that the volume-constraint for the finite-ratio problem disappears in the limit. This leads to a volume-unconstrained Dirichlet-forced evolution, a setting where monotonicity, uniqueness, and contact line regularity properties have been established in previous literature. Our result is based on using comparison principle techniques for the prescribed mean curvature equation with capillary contact angle condition that characterizes the liquid surface at equilibrium. Through barrier arguments, we are able to develop asymptotics for the energy which give us control independent of the container-to-plate ratio.

Paper Structure

This paper contains 34 sections, 45 theorems, 275 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Notation for Wilhelmy plate setting
  3. Energy formulation
  4. Sign conventions
  5. Parameters and ellipticity hypotheses
  6. Outline of the paper
  7. Acknowledgments
  8. Basic properties of globally stable profiles
  9. Energy coercivity and existence of globally stable states
  10. Euler-Lagrange equation
  11. Perturbations of globally stable states
  12. Viscosity solution properties and height bound
  13. Viscosity solutions of the capillary problem
  14. Viscosity solution properties of globally stable states
  15. Vertical bound of the the free surface
  16. ...and 19 more sections

Key Result

Theorem 1.3

Let $(R_n)$ be given with $R_n\to \infty$ as $n\to\infty$, and let $L_{R_n}(t)$ be a family of energy solutions for $\mathcal{E}_{R_n}$ under the same external forcing. Then there exists a subsequence $n_k$ and a measurable path $L_\infty(t)$ such that

Figures (1)

  • Figure 1: Left: a 3D view of the liquid-vapor-plate configuration. Right: a cross-sectional view of the configuration near the liquid-vapor-plate contact line, labeling the free surface $\Sigma^{f}_L$, the liquid-plate contact surface $\Sigma^{p}_L$, the liquid-vapor-plate contact line $\gamma^{p}_L$, and the liquid-plate contact angle $\theta_{\gamma^{p}_L}$, which may differ from the optimal Young angle $\theta^{p}_{Y}$ due to hysteresis.

Theorems & Definitions (95)

  • Definition 1.1
  • Remark 1.2
  • Theorem 1.3: see Corollary \ref{['c.compactness_of_energy_solutions_in_r']}
  • Definition 1.4
  • Definition 1.5
  • Theorem 1.6
  • proof
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • ...and 85 more