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This paper provides numerical evidence that floating drops in a capillary tube can have two distinct equilibrium shapes with exactly equal potential energy, making the energy minimizer non-unique, and that in two dimensions an asymmetric wa

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2026-08-03 08:21 UTC pith:4DBX3JS7

load-bearing objection A solid, reproducible numerical study with genuinely new equal-energy observations, but the global non-uniqueness-of-minimizer claim outruns the evidence because only a restricted class of configurations is compared. the 2 major comments →

arxiv 2602.10120 v2 pith:4DBX3JS7 submitted 2026-01-23 physics.flu-dyn cs.NAmath-phmath.MPmath.NA

Comparison of the potential energy for different equilibrium configurations of symmetric and asymmetric floating drops

Mason Mault , Ray Treinen This is my paper
classification physics.flu-dyn cs.NAmath-phmath.MPmath.NA MSC 35Q3576M22
keywords floating dropscapillarityfree boundary problemYoung-Laplace equationnon-uniquenessenergy minimizersymmetry breakingspectral collocation
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

This paper tries to establish that the floating-drop problem—three immiscible fluids in a closed tube, with one lighter drop fluid—has non-unique equilibrium shapes and, more surprisingly, non-unique energy minimizers. For every parameter set the authors test, there are at least two distinct solutions to the governing Young-Laplace free-boundary equations. Computing the potential energy of centrally located and wall-bound drops, they find many parameter choices where the two configurations have exactly equal energies, which they report as the first observed non-uniqueness of presumed minimizers in this model. In the two-dimensional model they further show that an asymmetric drop attached to one wall can be the minimizer, breaking left-right symmetry. A sympathetic reader would care because the result challenges the expectation that a well-posed variational problem should pick out a unique ground state.

Core claim

For the model of a floating drop in a laterally bounded container, the paper's central numerical finding is that the Euler-Lagrange equations never have unique solutions: for every choice of the nine physical parameters considered, both a centrally located drop and a wall-bound drop exist as equilibria, and in the two-dimensional model also wall-bound and split-wall configurations. Comparing the full potential energy (surface, gravitational, and wetting terms) across these classes, the authors find many parameter values where the centrally located and wall-bound drops have the same energy, so the presumed global minimizer is not unique. They also find a two-dimensional case where the asymmet

What carries the argument

The argument is carried by a high-accuracy numerical solver for the free-boundary Young-Laplace equations, which uses Newton's method with Chebyshev spectral collocation to compute interfaces that meet at the triple junction with the prescribed contact angles, plus nested root-finding loops that match the free-boundary parameter and the drop volume. The energy functional, consisting of surface-tension, gravitational, and wetting terms, is then evaluated on each computed configuration via Clenshaw-Curtis quadrature. This allows a systematic sweep of the nine-dimensional parameter space and direct comparison of energies across configuration classes, which is the mechanism behind every non-uniq

Load-bearing premise

The conclusion that the energy minimizer is sometimes non-unique assumes that the only relevant candidate configurations are the centered, wall-bound, and split-wall drops computed here; if an off-center or fully three-dimensional asymmetric drop with lower energy exists at those parameters, the non-uniqueness claim fails.

What would settle it

At a parameter set where the computed central and wall-bound drops have equal energy, solve for an off-center drop in two dimensions (or a fully three-dimensional asymmetric drop in three dimensions) and evaluate its potential energy: a strictly lower value would disprove the claimed non-unique minimizer. Alternatively, rerun the equal-energy cases with a finer Chebyshev grid or tighter Newton tolerance and check whether the energy difference shrinks to zero or resolves to a nonzero gap.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • For open regions of parameter space, the floating-drop variational problem has at least two global minimizers with equal energy, so the ground state is not unique.
  • In the two-dimensional model, an asymmetric shape attached to a single wall can be the global minimizer, so symmetry breaking occurs even though the data and domain are symmetric.
  • The widely used heuristic that the concavity of the drop-free interface decides between centered and wall-bound minimizers is not reliable; explicit counterexamples exist.
  • Along a volume-increase path, the energies of the two symmetric configurations can cross twice, giving two distinct volumes where the minimizer switches or ties.
  • The parameter space contains curves along which equal-energy degeneracy persists, so degenerate minimizers are not isolated accidents.

Where Pith is reading between the lines

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

  • Beyond the paper: if the equal-energy sets are codimension-one surfaces in parameter space, then generic small perturbations of parameters will select one configuration, which could make the non-uniqueness hard to observe experimentally unless parameters are tuned.
  • Beyond the paper: the two-dimensional asymmetry result suggests that in full three dimensions, off-center or tilted drops might also compete with the symmetric ones; the paper leaves this open, and a numerical method for those PDEs could test it.
  • Beyond the paper: the equal-energy crossings resemble exchange-of-stability or pitchfork bifurcations; treating volume or density as a continuation parameter could reveal a connecting bifurcation structure not visible in the present energy profiles.
  • Beyond the paper: the claim that off-center drops are unlikely minimizers rests on reflection arguments, but a direct numerical search for such configurations at the equal-energy parameters would be a cheap and decisive check.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

2 major / 5 minor

Summary. The paper develops a numerical method, based on Chebyshev spectral collocation and Newton iteration, for computing equilibrium configurations of floating drops in laterally bounded containers, under the assumption that configurations are described by generating curves. The method is applied to centrally located and wall-bound drops in R^3, and to central, single-wall, and split-wall drops in R^2. The paper explores a nine-dimensional physical parameter space, computes potential energies of the different configurations, and reports non-uniqueness of solutions to the Euler–Lagrange equations. Its headline claim is that there are many parameter values where a centrally located drop and a wall-bound drop have equal potential energy, giving 'strong evidence of non-uniqueness of energy minimizers' and, in the authors' words, 'no uniqueness at all' for solutions of the floating drop problem.

Significance. If the equal-energy observations are correct, they constitute a striking numerical discovery in a classical capillary free-boundary problem, and the symmetry-breaking examples in R^2 are also valuable. The paper provides a reproducible computational framework (code is on GitHub), explicit quadrature error bounds, and a systematic parameter-space exploration. The main weakness is that the global interpretation of the results rests on an unproven assumption that the computed configuration classes exhaust all relevant minimizers; the authors themselves use hedged language ('presumed', 'likely') but the abstract and conclusions state the stronger claim as established. The numerical evidence for equal-energy crossings also lacks a convergence study.

major comments (2)
  1. [Abstract; §1; §4.2; §5] The central claim of non-uniqueness of energy minimizers is stated more strongly than the evidence supports. The paper only compares generating-curve families: central and wall-bound drops in R^3, and central, single-wall, and split wall-bound drops in R^2. Non-computed configurations are dismissed heuristically: 'Reflection arguments make the off-center drops unlikely to be energy minimizing' (§4.2) and R^3 asymmetric solutions are 'outside the scope' (§5). Equal energies between the computed families do not establish that the energy minimizer is non-unique in the full problem. Even within the computed families, the paper compares energies of stationary solutions but does not prove they are local or global minimizers under arbitrary perturbations. The hedged phrase '(presumed) energy minimizers' in §5 is appropriate, but the Abstract's 'strong evidence of non-uniqueness of energy minimi
  2. [§3.1, §4.3, Eqs. (31)–(35)] The equal-energy crossings in Figures 9, 23, and 24 are the quantitative basis for the paper's most striking claim, yet no convergence study is reported for the computed energy values. The paper states 14-digit Newton tolerance and 10-digit BVP tolerance and cites the Clenshaw–Curtis quadrature error bound in Eq. (32), but it does not report actual errors, mesh-refinement checks, or residual norms for the energy computations. Near a crossing, a small systematic error in either energy curve could create or remove an intersection. The authors should provide a convergence test (e.g., n+1 = 14 versus 28 or 56 collocation points) and error estimates for the energy differences at the reported crossings, or at least for one representative case from each of Figures 9, 23, and 24.
minor comments (5)
  1. [Figure 18 caption] The caption reads 'On the left is a drop adjacent to the left wall, and this is the energy minimizer. On the left is a drop evenly split...' The second 'On the left' should be 'On the right'.
  2. [§2.2] The text says Bagley and Treinen [2] present 'a united approach'; this should likely be 'a unified approach'.
  3. [§2.3–2.4] The two fzero loops (for F(ψ̄) and for volume matching) are described with initial guesses and observed sign changes, but no proof of continuity or uniqueness of the zero is given. This is acceptable for a computational paper, but a sentence noting that convergence is empirical and pointing to the code would improve reproducibility.
  4. [§2.6] The paper says the three surface tensions are 'arbitrarily normalized to add up to a value of 16'; it would help to state explicitly that this is a scaling convention with no loss of generality for the parameter studies reported.
  5. [§3.2] The heuristic for when a centrally located drop or wall-bound drop is the energy minimizer is stated concisely. The counterexample in §4.1 is a nice addition, but the caption of Figure 13 should say 'tube radius X=2' rather than 'R=2' for consistency with the R^2 setup.

Circularity Check

0 steps flagged

No significant circularity: the equal-energy and non-uniqueness results emerge from solving boundary-value problems and independent energy quadrature, not from fitting parameters to the target conclusion.

full rationale

The paper's central claims are (i) non-uniqueness of Euler-Lagrange solutions and (ii) common crossings where central and wall-bound configurations have equal potential energy. Both are obtained by solving the underlying boundary-value problems for fixed physical parameters, then evaluating the energy functional (5) using Clenshaw-Curtis quadrature. No parameter is fitted to make the energies equal: the equal-energy points arise as intersections of independently computed energy curves in parameter sweeps (Figures 9, 11, 23, 24). The paper explicitly states it makes no attempt to fit solutions to experimental data (Section 2.6), and the volume-matching and free-boundary matching use fzero on equations (16)-(19) derived from the model, not on energy targets. The self-citations are to a spectral solver [36], existence results [33], and symmetry results [35]; these are tools and supporting theory, not the target non-uniqueness/equal-energy claim, and the solver is accompanied by public code. The restriction to generating-curve configurations and the treatment of off-center and fully 3D asymmetric drops as outside scope is an acknowledged completeness limitation, not a circular derivation. Thus no step reduces by construction to its inputs.

Axiom & Free-Parameter Ledger

7 free parameters · 6 axioms · 0 invented entities

The central numerical observations rest on a standard three-fluid capillary model plus the solver's accuracy; no new physical entities are postulated. The nine physical parameters are inputs chosen by hand, and the solution unknowns (\bar\psi, \bar r, λ) are determined by solving the equations, not fitted to the claimed equal-energy result. The main unstated burden is the restriction to ODE-configurable shapes.

free parameters (7)
  • Drop volume Vol = 0.4, 0.8, 1, 1.5, 2 in examples
    Chosen by hand across parameter scans; the volume-matching fzero solves for \bar r, but Vol itself is an input parameter.
  • Tube radius R (R3) / half-width X (R2) = 2
    Fixed in all experiments; part of the nine-parameter space.
  • Drop density ρ1 = 0.01 to 14.9 (ρ0=0, ρ2=15)
    Varied to explore gravitational energy balance; selected to make striking examples.
  • Surface tensions σ01, σ02, σ12 = triples normalized to sum 16, e.g. (3,7,6), (7.9999,2.0001,6)
    Chosen by hand; normalization is arbitrary and affects contact angles via the Neumann triangle relation (2).
  • Plate angles γ2_0p, γ1_0p = e.g. π/2, 2, 0.01; γ2_1p derived from equation (4)
    Input wetting parameters; chosen to explore wetting regimes.
  • Chebyshev collocation points n+1 = 14
    Fixed discretization resolution; error estimate (32) is cited but no systematic convergence study is reported for the energy crossings.
  • Initial-guess factor for \bar r volume loop = 0.2 (adjustable)
    Algorithmic tuning constant in Section 2.4; changed adaptively in some automation.
axioms (6)
  • domain assumption The three-fluid energy functional (5) and Young-Laplace free boundary conditions (6)-(9) from [8] correctly describe floating-drop equilibrium.
    The entire computation is built on this variational model; it is not re-derived in this paper.
  • standard math Force balance and the Neumann triangle relation (1)-(2) hold at the triple junction.
    Assumed from capillarity theory [4,8,10,22].
  • domain assumption Wetting energies satisfy Finn's contact-angle relation (4), giving the third plate angle from two.
    Taken from [12]; central to the wall-bound drop boundary conditions.
  • domain assumption Configurations can be reduced to generating curves: radial symmetry in R3 and planar curves in R2.
    Excludes full 3D asymmetric solutions; acknowledged in the Introduction and Section 5.
  • standard math Existence and symmetry of the considered configurations follow from [33], [35], [37].
    Used to justify that the candidate configurations are legitimate solutions.
  • domain assumption The spectral solver [36] returns numerical solutions within the stated 14-digit/10-digit relative tolerances.
    All downstream observations depend on this; the paper provides no independent verification.

pith-pipeline@v1.3.0-alltime-deepseek · 15848 in / 12770 out tokens · 133157 ms · 2026-08-03T08:21:32.634053+00:00 · methodology

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read the original abstract

We provide a numerical method for computing solutions to a free boundary problem arising from the equilibrium state of a floating drop. This numerical method is based on a Newton's method for the underlying nonlinear boundary value problems, and at each iterative step a Chebyshev spectral collocation method is employed. The problems considered here are those that can be described by using generating curves, and include problems in $\mathbb{R}^2$ and $\mathbb{R}^3$. The resulting nine-dimensional space of physical parameters is explored, and examples are given that highlight the potential energy of centrally located drops, wall-bound drops, and asymmetrical configurations in $\mathbb{R}^2$. Non-uniqueness of solutions to the corresponding Euler-Lagrange equations is displayed, and also strong evidence of non-uniqueness of energy minimizers is given.

Figures

Figures reproduced from arXiv: 2602.10120 by Mason Mault, Ray Treinen.

Figure 1
Figure 1. Figure 1: Schematically, a drop of oil floats on the surface of water inside a capillary tube. Shown is a [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The surface tensions are shown in a force balance form as vectors, and the contact angles [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: A typical centrally located floating drop. The physical parameters that determine this drop are [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: We adapt the above process to the left side of the configuration, where the generating curves [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: We are not restricted to symmetric interfaces here, as asymmetric configurations are possible [PITH_FULL_IMAGE:figures/full_fig_p009_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Asymmetric floating drops in R 2 . On the left is a drop adjacent to the left wall with the physical parameters that determine this drop being a volume of 1, tube radius of X = 2, densities ρ1 = 1, ρ2 = 15, surface tensions σ01 = 7.9999, σ02 = 5, and σ12 = 3.0001, and plate angles γ 1 0p = 0.01 and γ 2 1p = π−0.01. On the right is a drop split with 1/6 of the volume of liquid adjacent to the right wall, an… view at source ↗
Figure 7
Figure 7. Figure 7: A pair of axially symmetric floating drops with the same parameters and a volume of 0.8. The [PITH_FULL_IMAGE:figures/full_fig_p011_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: A similar pair of axially symmetric floating drops with the same parameters as those in [PITH_FULL_IMAGE:figures/full_fig_p012_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Comparing the energies of the centrally located drop (solid) and the wall-bound drop (dashed). [PITH_FULL_IMAGE:figures/full_fig_p012_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Comparing the energies of the centrally located drop (solid) and the wall-bound drop (dashed) [PITH_FULL_IMAGE:figures/full_fig_p014_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Comparing the energies of the centrally located drop (solid) and the wall-bound drop (dashed). [PITH_FULL_IMAGE:figures/full_fig_p014_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Here are two drops displaying a reflection symmetry of in the parameters. For both drops, [PITH_FULL_IMAGE:figures/full_fig_p015_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Here is a range of plate angles γ 2 0p where the heuristic breaks. On the left there is an atypical second crossing of the curves away from π/2. The physical parameters that determine these are volume of 0.4, tube radius of R = 2, densities ρ1 = 5 and ρ2 = 15, surface tensions σ01 = 7.9999, σ02 = 2.0001, and σ12 = 6. One plate angle is fixed as γ 1 0p = 2.1277, and the third is determined by the governing… view at source ↗
Figure 14
Figure 14. Figure 14: The wall-bound drop displayed here has lower energy than the displayed centrally located [PITH_FULL_IMAGE:figures/full_fig_p016_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: Here is an experiment where γ02 is varied over a range of values. The solid curve is the energy of the centrally located drop, the dashed curve is for the wall-bound drop, and the dot-dashed curve is for the drop split evenly on each wall. The physical parameters that determine these are a drop volume of 0.4, a tube radius X = 2, densities ρ1 = 7.5 and ρ2 = 15, plate angles γ 2 0p = π/2 and γ 1 2p = 1.252… view at source ↗
Figure 16
Figure 16. Figure 16: Here is an experiment where γ 1 0p is varied over a range of values. The solid curve is the energy of the centrally located drop, the dashed curve is for the wall-bound drop, and the dot-dashed curve is for the drop split evenly on each wall. The physical parameters that determine these are a total volume of 0.4, a tube radius of X = 2, densities of ρ1 = 1 and ρ2 = 15, surface tensions σ01 = 3, σ02 = 7, a… view at source ↗
Figure 17
Figure 17. Figure 17: Floating drop configurations with a volume of 1, tube radius of [PITH_FULL_IMAGE:figures/full_fig_p018_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: Floating drop configurations with the same physical parameters. On the left is a drop adjacent [PITH_FULL_IMAGE:figures/full_fig_p018_18.png] view at source ↗
Figure 19
Figure 19. Figure 19: The energy of the split volume drops with the portion of the volume on the left side parame [PITH_FULL_IMAGE:figures/full_fig_p019_19.png] view at source ↗
Figure 20
Figure 20. Figure 20: Floating drop configurations with a volume of 1, tube radius of [PITH_FULL_IMAGE:figures/full_fig_p019_20.png] view at source ↗
Figure 21
Figure 21. Figure 21: Floating drop configurations with the same physical parameters as Figure 20. On the left is [PITH_FULL_IMAGE:figures/full_fig_p020_21.png] view at source ↗
Figure 22
Figure 22. Figure 22: The energy of the split volume drops with the portion of the volume on the left side parame [PITH_FULL_IMAGE:figures/full_fig_p020_22.png] view at source ↗
Figure 23
Figure 23. Figure 23: Comparing the energies of the centrally located drop (solid) and the wall-bound drop (dashed). [PITH_FULL_IMAGE:figures/full_fig_p021_23.png] view at source ↗
Figure 24
Figure 24. Figure 24: An overview of parameter studies where the number of configurations with equal energy is [PITH_FULL_IMAGE:figures/full_fig_p021_24.png] view at source ↗

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