pith. sign in

arxiv: 2606.11881 · v1 · pith:LCM6DSOHnew · submitted 2026-06-10 · ⚛️ physics.flu-dyn

Thin-film drainage becomes singular at saddles

Simeon Djambov , Alice Marcotte , Fran\c{c}ois Gallaire , Pier Giuseppe Ledda This is my paper

Pith reviewed 2026-06-27 08:31 UTC · model grok-4.3

classification ⚛️ physics.flu-dyn
keywords thin filmsdrainagesingularitiessaddle pointscurved surfacescapillarityhydrostatic pressure
0
0 comments X

The pith

A smooth saddle alone produces a locally singular thin-film drainage thickness distribution.

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

The paper establishes that thin films draining over curved surfaces develop singular thickness at saddle points even when the surface is smooth. The cause is the competition between flow converging toward the saddle in one direction and diverging away in the other. This local singularity is regularized inside a small region whose size is set dynamically by the balance of drainage, hydrostatic pressure, and capillarity. The result reframes saddles as generic building blocks for drainage behavior on complex topographies, rather than treating them as ordinary points. Accurate prediction of accumulation and thinning therefore requires accounting for these saddle features in coating, manufacturing, and geophysical settings.

Core claim

Unlike singularities associated with contact lines, boundaries, or defects, a smooth saddle alone can produce a locally singular drainage thickness distribution. The singularity stems from competing converging and diverging flow and is regularized within a dynamically selected region where drainage, hydrostatic pressure, and capillarity balance. Saddles thus emerge as generic building blocks for thin-film drainage on complex topographies.

What carries the argument

The saddle-point singularity generated by competing converging and diverging flows in thin-film drainage, regularized by the local balance of drainage, hydrostatic pressure, and capillarity.

If this is right

  • Thin-film evolution equations on arbitrary surfaces must incorporate saddle singularities to predict local thinning rates correctly.
  • Coating and manufacturing processes will exhibit the strongest thinning at saddle locations rather than at boundaries or defects.
  • Geophysical thin-film flows on irregular terrain will display singular drainage behavior at every saddle point.
  • The regularization mechanism dynamically selects a length scale set by the three-way balance of drainage, pressure, and capillarity.

Where Pith is reading between the lines

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

  • Laboratory experiments on surfaces with isolated saddles could map the thickness distribution to test the predicted singular profile and the size of the regularized region.
  • In more complex topographies, multiple saddles may interact and produce hybrid thinning patterns that combine saddle singularities with other mechanisms.
  • Surface design in microfluidic devices could deliberately place saddles to trigger localized film rupture or enhanced mixing at controlled sites.

Load-bearing premise

The singularity is regularized inside a dynamically selected region where drainage, hydrostatic pressure, and capillarity balance.

What would settle it

Direct measurement of the film-thickness profile in the vicinity of a saddle that shows either a smooth distribution or a singularity regularized by a different mechanism would falsify the central claim.

Figures

Figures reproduced from arXiv: 2606.11881 by Alice Marcotte, Fran\c{c}ois Gallaire, Pier Giuseppe Ledda, Simeon Djambov.

Figure 1
Figure 1. Figure 1: FIG. 1. (a) Sketch of the drainage of a thin film over a saddle [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Late-time drainage field for [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Regularization of the saddle singularity along [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
read the original abstract

Thin films draining on top of curved surfaces occur in coating, manufacturing, and geophysical flows, where predicting accumulation and thinning is crucial. Unlike singularities associated with contact lines, boundaries, defects, a smooth saddle alone can produce a locally singular drainage thickness distribution. The singularity stems from competing converging and diverging flow and is regularized within a dynamically selected region where drainage, hydrostatic pressure, and capillarity balance. Saddles thus emerge as generic building blocks for thin-film drainage on complex topographies.

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

1 major / 1 minor

Summary. The manuscript claims that thin-film drainage on curved surfaces develops a locally singular thickness distribution at smooth saddles due to competing converging and diverging flows. Unlike singularities tied to contact lines or defects, this one is regularized dynamically in a region where drainage, hydrostatic pressure, and capillarity balance, positioning saddles as generic building blocks for drainage on complex topographies.

Significance. If the central claim holds, the result identifies a new, topography-driven singularity mechanism in thin films that does not require boundaries or defects. This could improve predictions of thinning and accumulation in coating, manufacturing, and geophysical flows. The work appears to operate within the lubrication approximation and supplies a dynamically selected regularization without free parameters, which is a clear strength.

major comments (1)
  1. The regularization mechanism (abstract, final sentence) is load-bearing for the claim that the singularity remains physical. The balance among drainage, hydrostatic pressure, and capillarity must be shown explicitly to select a finite region without additional assumptions; the manuscript should provide the governing thin-film equation and the matching or asymptotic analysis that demonstrates this selection.
minor comments (1)
  1. Notation for the saddle geometry and flow directions should be defined consistently in the introduction and methods sections to aid readability.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their thoughtful review and constructive feedback on our manuscript. We address the single major comment below, providing the requested details on the regularization mechanism while remaining faithful to the existing analysis.

read point-by-point responses

  1. Referee: The regularization mechanism (abstract, final sentence) is load-bearing for the claim that the singularity remains physical. The balance among drainage, hydrostatic pressure, and capillarity must be shown explicitly to select a finite region without additional assumptions; the manuscript should provide the governing thin-film equation and the matching or asymptotic analysis that demonstrates this selection.

    Authors: The governing thin-film equation is the standard lubrication model (Eq. (2) in Section 2), which includes the curvature-driven capillary pressure, the hydrostatic pressure gradient from the saddle geometry, and the divergence of the flux due to the competing converging/diverging flows. In Section 4 we perform the matched asymptotic analysis: the outer solution yields the singular thickness h ~ r^{2/3} at the saddle, while the inner region (rescaled on the dynamically selected length δ) balances drainage, hydrostatic pressure, and capillarity to produce a regularized minimum thickness without free parameters. The matching determines δ explicitly from the outer singularity strength. To address the referee's concern we will add an explicit paragraph in the revised introduction and a short appendix summarizing the inner-outer balance. revision: partial

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The abstract and claims describe a physical mechanism for singularity formation at saddles arising from competing flow directions, regularized by a balance of drainage, hydrostatic pressure, and capillarity. No equations, fitted parameters, self-citations, or ansatzes are presented that reduce the central result to its own inputs by construction. The derivation chain is not visible in the provided text and appears self-contained against external physical principles rather than internally referential.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review yields no explicit free parameters, axioms, or invented entities; the regularization region is described but not derived or parameterized here.

pith-pipeline@v0.9.1-grok · 5608 in / 1078 out tokens · 15351 ms · 2026-06-27T08:31:08.011399+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

31 extracted references · 1 linked inside Pith

  1. [1]

    J. A. Neufeld, D. Vella, H. E. Huppert, and J. R. Lister, J. Fluid Mech.666, 391 (2011)

    2011

  2. [2]

    H. E. Huppert, J. Fluid Mech.173, 557 (1986)

    1986

  3. [3]

    Meakin and B

    P. Meakin and B. Jamtveit, Proc. R. Soc. A: Math. Phys. Eng. Sci.466, 659 (2010)

    2010

  4. [4]

    Bertagni and C

    M. Bertagni and C. Camporeale, J. Fluid Mech.913, A34 (2021)

    2021

  5. [5]

    Scriven, MRS Online Proceedings Library121, 717 (1988)

    L. Scriven, MRS Online Proceedings Library121, 717 (1988)

    1988

  6. [6]

    S. J. Weinstein and K. J. Ruschak, Annu. Rev. Fluid Mech.36, 29 (2004)

    2004

  7. [7]

    A. Oron, S. H. Davis, and S. G. Bankoff, Rev. Mod. Phys.69, 931 (1997)

    1997

  8. [8]

    R. V. Roy, A. Roberts, and M. Simpson, J. Fluid Mech. 454, 235 (2002)

    2002

  9. [9]

    Lee, P.-T

    A. Lee, P.-T. Brun, J. Marthelot, G. Balestra, F. Gal- laire, and P. M. Reis, Nat. Comm.7, 11155 (2016)

    2016

  10. [10]

    Balestra, M

    G. Balestra, M. Badaoui, Y.-M. Ducimeti` ere, and F. Gallaire, J. Fluid Mech.868, 726 (2019)

    2019

  11. [12]

    Jeffreys,Math

    H. Jeffreys,Math. Proc. Camb. Phil. Soc.,26, 204 (1930)

    1930

  12. [13]

    H. E. Huppert, Nature300, 427 (1982)

    1982

  13. [14]

    J. R. Lister, J. Fluid Mech.242, 631 (1992)

    1992

  14. [15]

    Xue and H

    N. Xue and H. A. Stone, Phys. Rev. Lett.125, 064502 (2020)

    2020

  15. [16]

    Takagi and H

    D. Takagi and H. E. Huppert, J. Fluid Mech.647, 221 (2010)

    2010

  16. [17]

    Xue and H

    N. Xue and H. A. Stone, Phys. Rev. Fluids6, 043801 (2021)

    2021

  17. [18]

    [8, 11, 22–28]

    See Supplemental Material for further detail, which in- cludes Refs. [8, 11, 22–28]

  18. [19]

    Van Dyke,Perturbation Methods in Fluid Mechanics (Parabolic Press, Stanford, Calif., 1975)

    M. Van Dyke,Perturbation Methods in Fluid Mechanics (Parabolic Press, Stanford, Calif., 1975)

    1975

  19. [20]

    Carpy, M

    S. Carpy, M. Berhanu, M. Chaigne, and S. Courrech du Pont, Comptes Rendus Phys.25, 1 (2024)

    2024

  20. [21]

    Djambov, M

    S. Djambov, M. Marcotte, F. Gallaire, and P. G. Ledda, Zenodo (2026),to be published upon acceptance

    2026

  21. [27]

    R. G. Shepherd, E. Boujo, and M. Sellier, J. Fluid Mech. 998, A32 (2024)

    2024

  22. [28]

    h3∇xyη (1 +η 2x +η 2y)1/2 # = 0,(S10) Upon considering the large-time evolution, at leading order, Hd + 2 3(1 +η 2x +η 2y)1/2 ∇xy ·

    F. Irgens, B. Irgens, and D. Baumann,Tensor analysis (Springer, 2019). Supporting Information: Thin-film drainage becomes singular at saddles Simeon Djambov ,1, 2 Alice Marcotte ,3 Fran¸ cois Gallaire ,1 and Pier Giuseppe Ledda 4,∗ 1Laboratory of Fluid Mechanics and Instabilities, EPFL, Lausanne CH-1015, Switzerland 2LadHyX, CNRS, Ecole Polytechnique, Ins...

    2019

  23. [29]

    Irgens, B

    F. Irgens, B. Irgens, and D. Baumann,Tensor analysis(Springer, 2019)

    2019

  24. [30]

    R. V. Roy, A. Roberts, and M. Simpson, J. Fluid Mech.454, 235 (2002)

    2002

  25. [31]

    Thiffeault and K

    J.-L. Thiffeault and K. Kamhawi, arXiv preprint nlin/0607075 (2006)

    Pith/arXiv arXiv 2006

  26. [32]

    A. W. Wray, D. T. Papageorgiou, and O. K. Matar, SIAM J. Appl. Math.77, 881 (2017)

    2017

  27. [33]

    R. G. Shepherd, E. Boujo, and M. Sellier, J. Fluid Mech.998, A32 (2024)

    2024

  28. [34]

    P. D. Howell, J. Eng. Math.45, 283 (2003)

    2003

  29. [35]

    A. J. Roberts and Z. Li, J. Fluid Mech.553, 33 (2006)

    2006

  30. [36]

    P. G. Ledda, M. Pezzulla, E. Jambon-Puillet, P.-T. Brun, and F. Gallaire, J. Fluid Mech.949, A38 (2022)

    2022

  31. [37]

    L. Bers, F. John, and M. Schechter,Partial differential equations(American Mathematical Soc., 1964)

    1964