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arxiv: 2605.12393 · v1 · submitted 2026-05-12 · ❄️ cond-mat.soft · cond-mat.mes-hall· cond-mat.stat-mech· math.DS· physics.flu-dyn

Recognition: 2 theorem links

· Lean Theorem

Variational approach to droplet motion on uneven solid surfaces, including contact line dynamics and evaporation

Gyula I T\'oth , David N Sibley , Agnes J Bok\'anyi-T\'oth , Dmitri Tseluiko , Andrew J Archer
Authors on Pith no claims yet

Pith reviewed 2026-05-13 03:26 UTC · model grok-4.3

classification ❄️ cond-mat.soft cond-mat.mes-hallcond-mat.stat-mechmath.DSphysics.flu-dyn
keywords variational dynamicsOnsager principleliquid dropscontact lineevaporationuneven surfacespinning and slidingfree energy
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0 comments X

The pith

Dynamical equations for liquid films and drops on uneven surfaces are generated variationally from the free energy using Onsager's principle.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes that the time evolution of liquid films and drops, including contact-line motion and evaporation or condensation, can be obtained as a variational dynamics in the isothermal overdamped limit. This is done by applying Onsager's variational principle directly to a free-energy functional that encodes substrate interactions and phase change. A reader would care because the resulting equations automatically satisfy the correct equilibrium states and can be used to recover known contact-line laws while studying pinning and sliding on inclined corrugated surfaces.

Core claim

The dynamical equations for liquid films and drops on uneven surfaces, including contact line dynamics and evaporation/condensation effects, may be formulated as a variational dynamics generated via Onsager's variational principle in the isothermal overdamped-dynamics limit. This approach constructs the equations starting from the free energy of the system and therefore naturally incorporates the correct equilibrium properties.

What carries the argument

Onsager's variational principle applied to a free-energy functional for the liquid-substrate system, which produces the governing dynamical equations including dissipation at the contact line and evaporation terms.

If this is right

  • Standard results for contact-line dynamics are recovered as special cases of the variational equations.
  • The motion of pinned and sliding drops on inclined corrugated surfaces can be computed directly from the free energy.
  • Evaporation and condensation are incorporated without additional ad-hoc terms because they enter through the free-energy variation.
  • Equilibrium configurations are automatically satisfied by the derived dynamics.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same variational construction could be applied to other soft-matter systems with dissipation at moving interfaces, such as thin films with surfactants.
  • Numerical implementations might allow efficient exploration of parameter regimes where drops transition from pinned to sliding states under evaporation.
  • The approach suggests a route to derive reduced models for larger-scale wetting problems on textured surfaces without solving the full Navier-Stokes equations.

Load-bearing premise

The system remains in the isothermal overdamped-dynamics limit and the chosen free-energy functional correctly encodes all relevant substrate interactions and evaporation physics.

What would settle it

A direct numerical comparison between the variational equations and a full hydrodynamic simulation or experiment that shows persistent mismatch in the predicted contact-line velocity or equilibrium drop shape on a corrugated incline under controlled evaporation.

Figures

Figures reproduced from arXiv: 2605.12393 by Agnes J Bok\'anyi-T\'oth, Andrew J Archer, David N Sibley, Dmitri Tseluiko, Gyula I T\'oth.

Figure 1
Figure 1. Figure 1: Sketch of a liquid droplet on an uneven solid surface. The solid surface has height profile ϕ(x, y). The liquid-gas interface is determined by the function h(x, y, t), which gives the distance from the solid surface to the liquid-gas interface in the z direction. The contact area on the surface is shaded red and its projection onto the xy-plane is denoted Γ(t), while the projection of the contact line (whe… view at source ↗
Figure 2
Figure 2. Figure 2: Two-dimensional liquid ridge spreading on a homogeneous surface without gravity. Comparison between (i) asymptotic results exploiting the thin (i.e. long-wave approximation |hx| ≪ 1), slow (|X˙ | ≪ 1) and small-slip (λ ≪ 1) scenario (black dashes), and (ii) a na¨ıve evolution formed from using the equivalent static (parabolic) profile in the dynamic contact angle condition (black dots), against (iii) the f… view at source ↗
Figure 3
Figure 3. Figure 3: Ridge motion down inclined corrugated surfaces. The equilibrium contact angle θs = 25◦ , the slip length λ = 0.01, the surface mobility coefficient M0 Γ = 0.01 [see Eq. (2.52)] and the Bond number B = 1.25. The substrate surface profiles are given in Eq. (4.25), with inclination slope of 0.15 and oscillations with wavelength 2/5 and amplitudes of (a,d) A = 0; (b,e) A = 0.0025; (c,f) A = 0.005. Initial cond… view at source ↗
Figure 4
Figure 4. Figure 4: Axisymmetric droplet motion on surface with sinusoidal height profile (also axisymmetric). Panel (a) shows the t = 0 initial condition (red profile) together with the subsequent drop profile over time (blue profiles), for the times t = 1, 100, 350, 600, 850, 1000 and 10000. Panel (b) shows the drop radius over time, while (c) shows the contact angle difference θ − θs over time. The equilibrium contact angl… view at source ↗
Figure 5
Figure 5. Figure 5: Two-dimensional liquid drop (liquid ridge) experiencing evaporation or condensation on a flat surface, with Bond number B = 0. Panels (a)–(c) show the variation over time of the cross-sectional area A, ridge width 2a and the contact angle θ, for three different values of pG. In each case, the black dotted line shows the approximate evolution given by Eqs. (4.34) and (4.41) with (4.38), which assumes the dr… view at source ↗
read the original abstract

We show how dynamical equations for liquid films and drops on uneven surfaces, including contact line dynamics and evaporation/condensation effects, may be formulated as a variational dynamics, generated via Onsager's variational principle. The theory applies in the isothermal overdamped-dynamics limit. We apply this general approach to obtain several well-known results on contact line dynamics and to study drops pinning and sliding on inclined corrugated surfaces. This approach constructs the dynamical equations starting from the free energy of the system and therefore has the advantage that it naturally incorporates the correct equilibrium properties.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 3 minor

Summary. The manuscript presents a variational approach to the dynamics of liquid films and droplets on uneven solid surfaces using Onsager's variational principle. In the isothermal overdamped limit, dynamical equations—including those governing contact-line motion and evaporation/condensation—are generated from a free-energy functional. The authors recover several known results on contact-line dynamics and apply the framework to study pinning and sliding of drops on inclined corrugated surfaces. The central advantage claimed is that equilibrium properties are automatically satisfied because the dynamics derive directly from the free energy.

Significance. If the construction is robust, the work supplies a systematic route to thermodynamically consistent dynamical models for wetting on textured substrates with mass transfer. This could streamline analysis of phenomena such as evaporative droplet motion or capillary-driven transport on rough surfaces, where separate ad-hoc rules for contact-line velocity are otherwise needed. Recovery of known limits provides a consistency check, and the free-energy starting point is a genuine strength for avoiding equilibrium violations.

major comments (2)
  1. [§3] §3 (Onsager variational formulation and Rayleighian): the dissipation functional is assembled by supplementing the standard viscous integral with a regularized contact-line term and a separate evaporation flux contribution. No derivation from the same microscopic model underlying the free energy is supplied; the regularization length appears as an auxiliary parameter. Because the contact-line velocity is obtained by varying this Rayleighian, the dynamical content of the central claim rests on this choice rather than emerging uniquely from the variational principle.
  2. [Applications section] Applications to contact-line dynamics (following §3): known results such as the Cox-Voinov relation or Tanner's law are recovered, yet the recovery follows once the dissipation form is adopted. Without an independent test (e.g., comparison to full hydrodynamic simulations or molecular dynamics that do not presuppose the same dissipation), it remains unclear whether the framework predicts new behavior or reproduces limits by construction.
minor comments (3)
  1. [Abstract and §1] The abstract and introduction would benefit from an explicit statement of the regularization procedure and its physical interpretation before the applications are presented.
  2. Notation for the substrate corrugation amplitude and wavelength is introduced piecemeal; collecting all geometric parameters in a single table or appendix would improve readability.
  3. [Introduction] A brief comparison paragraph with earlier variational treatments of wetting (e.g., those employing the same Onsager principle for thin films) would help situate the novelty.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments. We address each major point below and have revised the manuscript to incorporate clarifications where appropriate.

read point-by-point responses

  1. Referee: [§3] §3 (Onsager variational formulation and Rayleighian): the dissipation functional is assembled by supplementing the standard viscous integral with a regularized contact-line term and a separate evaporation flux contribution. No derivation from the same microscopic model underlying the free energy is supplied; the regularization length appears as an auxiliary parameter. Because the contact-line velocity is obtained by varying this Rayleighian, the dynamical content of the central claim rests on this choice rather than emerging uniquely from the variational principle.

    Authors: We agree that the dissipation functional is introduced phenomenologically, as is standard when applying Onsager's principle in the overdamped limit. The free-energy functional is constructed from the equilibrium interface properties, while the Rayleighian incorporates bulk viscous dissipation together with regularized contributions for contact-line friction and evaporation/condensation; these terms are motivated by known physical mechanisms and regularized with an auxiliary length to remove the contact-line singularity. The variational procedure then yields the dynamical equations consistently from the chosen free energy and dissipation. We have added a paragraph to the revised §3 that discusses the physical motivation for each dissipation term, its relation to microscopic cut-off scales, and references to similar constructions in the literature on variational wetting models. revision: partial

  2. Referee: [Applications section] Applications to contact-line dynamics (following §3): known results such as the Cox-Voinov relation or Tanner's law are recovered, yet the recovery follows once the dissipation form is adopted. Without an independent test (e.g., comparison to full hydrodynamic simulations or molecular dynamics that do not presuppose the same dissipation), it remains unclear whether the framework predicts new behavior or reproduces limits by construction.

    Authors: The referee correctly notes that recovery of the Cox-Voinov relation and Tanner's law serves as a consistency check once the dissipation functional is specified. This check is valuable because it confirms that the variational construction reproduces established limits without violating equilibrium conditions. The framework's utility is demonstrated by its application to droplet pinning and sliding on inclined corrugated surfaces, where the interplay between the free-energy landscape and the variational dynamics produces new predictions for critical inclination angles and evaporation effects that are not presupposed by the dissipation choice. We have added a brief remark in the conclusions of the revised manuscript acknowledging that direct comparisons with independent full hydrodynamic or molecular-dynamics simulations would provide further validation and identifying this as a direction for future work. revision: partial

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained via standard Onsager principle

full rationale

The paper derives dynamical equations for films and drops from a free-energy functional via Onsager's variational principle in the isothermal overdamped limit. This is a standard construction that automatically enforces correct equilibrium properties by design of the free energy while generating dynamics from an auxiliary dissipation functional; the approach reproduces known contact-line results as applications rather than by tautological redefinition. No load-bearing step reduces to a fitted parameter, self-citation chain, or ansatz smuggled from prior work by the same authors. The central claim remains independent of the target outputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on Onsager's variational principle and the overdamped isothermal limit; no free parameters or new entities are introduced in the abstract.

axioms (2)
  • domain assumption Onsager's variational principle generates the correct dynamical equations from the free energy
    Invoked to formulate the time evolution of films and drops.
  • domain assumption The system is in the isothermal overdamped-dynamics limit
    Explicitly stated as the regime where the theory applies.

pith-pipeline@v0.9.0 · 5419 in / 1210 out tokens · 71365 ms · 2026-05-13T03:26:10.123051+00:00 · methodology

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Reference graph

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