pith. sign in

arxiv: 2604.05344 · v1 · submitted 2026-04-07 · ⚛️ physics.flu-dyn

Influence of van der Waals forces on the instability of a liquid film in a tube

Yixiao Mao , Chengxi Zhao , Yixin Zhang , Kai Mu , Ting Si This is my paper

Pith reviewed 2026-05-10 19:46 UTC · model grok-4.3

classification ⚛️ physics.flu-dyn
keywords liquid film instabilityvan der Waals forcesnanotubeStokes equationslinear stability analysisfilm ruptureself-similar dynamicsdisjoining pressure
0
0 comments X p. Extension

The pith

Van der Waals forces accelerate perturbation growth and lower the critical thickness for liquid film collapse inside tubes.

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

The paper develops a model based on axisymmetric Stokes equations to examine how van der Waals forces alter the instability of a liquid film coating the interior of a nanotube. Linear stability analysis shows these forces increase the rate at which small perturbations grow, shorten the dominant wavelength, and decrease the film thickness that separates collapse from stable regimes. Direct numerical simulations of the full Navier-Stokes equations confirm the linear predictions and reveal nonlinear effects including changed interface shapes and fewer satellite lobes. Both rupture and collapse then obey a universal self-similar scaling in which the time to singularity follows a one-third power law.

Core claim

A theoretical framework based on the axisymmetric Stokes equations is developed to investigate the effects of van der Waals forces through linear stability analysis. The model reveals that van der Waals forces markedly enhance perturbation growth, reduce the dominant wavelength, and lower the critical film thickness that distinguishes collapse from non-collapse regimes. Direct numerical simulations of the Navier-Stokes equations both confirm these theoretical predictions and extend the analysis into the nonlinear regime. In this regime, van der Waals forces are found to alter the interfacial morphology and suppress the formation of satellite lobes. Both rupture and collapse follow a unive

What carries the argument

Axisymmetric Stokes equations augmented by a continuum van der Waals disjoining pressure term, solved via linear stability analysis and verified by Navier-Stokes simulations.

If this is right

  • Perturbation growth rates increase with stronger van der Waals forces.
  • The dominant unstable wavelength becomes shorter.
  • The critical film thickness separating collapse and non-collapse regimes decreases.
  • Satellite lobes are suppressed in the nonlinear stage.
  • Both rupture and collapse obey the same one-third power-law scaling near the singularity.

Where Pith is reading between the lines

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

  • The universal scaling may extend to other confined geometries where disjoining pressure drives breakup.
  • Varying tube radius or film thickness in experiments could directly test the continuum assumption for van der Waals forces.
  • Altered morphology without satellites could change droplet size distributions in microfluidic coating processes.
  • If inertia becomes relevant at larger scales, the scaling exponent is expected to shift.

Load-bearing premise

The model assumes fluid inertia can be neglected and that van der Waals forces are adequately described by a standard continuum disjoining pressure even when the film is very thin.

What would settle it

Experimental measurements of breakup wavelength or time-to-rupture in nanotubes that deviate from the predicted linear growth rates or the one-third temporal scaling when van der Waals forces dominate.

Figures

Figures reproduced from arXiv: 2604.05344 by Chengxi Zhao, Kai Mu, Ting Si, Yixiao Mao, Yixin Zhang.

Figure 1
Figure 1. Figure 1: Schematic of annular film flow in a tube. 2.1. Model formulation We consider a Newtonian liquid film coating the interior of a horizontal cylindrical tube of radius 𝑟0, with the 𝑧-axis aligned along the tube centerline (see figure 1). The initial radius of the liquid–gas interface from the 𝑧-axis is denoted as 𝑟 = 𝑟𝑓 . The flow dynamics within the liquid film is governed by the incompressible Navier–Stokes… view at source ↗
Figure 2
Figure 2. Figure 2: The dispersion relation between the growth rate 𝜔 and the wavenumber 𝑘 with 𝐴1 = 1.2 × 10−5 of various film thickness: (𝑎) 𝜖 = 0.029, 0.236, (𝑏) 𝜖 = 0.6, 0.7. Circles with dash-dotted lines represent results of Stokes model (2.15), solid lines represent results of lubrication model (2.16), and dashed lines represent results of lubrication model (2.17) by Tomo et al. (2022). It should be noted that the orig… view at source ↗
Figure 3
Figure 3. Figure 3: Influence of film thickness 𝜖 on the dominant wavelength 𝜆𝑚𝑎𝑥 : (𝑎) different 𝐴1 with 𝐴2 = 0, (𝑏) different 𝐴2 with 𝐴1 = 0. The solid lines are the predictions of Stokes model (2.15), and the dashed lines represent the predictions of lubrication model (2.16). thickness 𝜖 for different vdW force strengths. In the thick-film regime, the Stokes model predicts larger dominant wavelengths than the lubrication m… view at source ↗
Figure 4
Figure 4. Figure 4: (𝑎) Quadrilateral computational domains with different boundary conditions (BCs). (𝑏) Sample deformed triangular meshes for 𝐴1 = 10−3 , 𝜖 = 0.1 and three different time constants in the panels. perturbation ℎˆ is introduced at the liquid–gas interface. Periodic boundary conditions are applied to the left and right boundaries of the domain. The top boundary satisfies a no￾slip condition according to (2.9), … view at source ↗
Figure 5
Figure 5. Figure 5: Evolution of a liquid film under different vdW forces, where the film is initialized as a random perturbation with a small amplitude, for (𝑎–𝑐)𝜖 = 0.1 and (𝑑– 𝑓 )𝜖 = 0.4, with comparison between 𝐴1 = 𝐴2 = 0 (upper contour map) and 𝐴1 = 𝐴2 = 10−3 (lower contour map). The contours represent the velocity magnitude |u| = √ 𝑢 2 + 𝑤2 [PITH_FULL_IMAGE:figures/full_fig_p010_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Evolution of the minimum radii of films ℎ𝑚𝑖𝑛 (𝑡) for different 𝜖 while 𝐴1 = 0. (𝑎) 𝐴2 = 0, (𝑏) 𝐴2 = 1. Dashed lines indicate the film remaining stable collars and solid lines indicate the film forming a plug. Time scaling 𝜔𝑡 follows (2.15). vdW forces are present, the wavelength is markedly reduced, and extremely high velocities develop, ultimately leading to rupture (figure 5𝑐). For the thicker film (𝜖 = … view at source ↗
Figure 7
Figure 7. Figure 7: The critical collapse thickness𝜖𝑐 varies with the strength of liquid–liquid vdW forces 𝐴2 when 𝐴1 = 0. The theoretical predictions from (4.1)–(4.2) are shown as a black curve, while simulation results indicating the upper and lower bounds of collar or plug are represented by symbols. theoretical curve (black solid line), while non-collapsing films correspond to the blue region below the curve. The vdW forc… view at source ↗
Figure 8
Figure 8. Figure 8: Interface profiles at four time instants illustrated in different colors, with film thickness (𝑎,𝑏) 𝜖 = 0.1 under different 𝐴1, and (𝑐,𝑑) 𝜖 = 0.5 under different 𝐴2 [PITH_FULL_IMAGE:figures/full_fig_p013_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: The root mean square |𝐻𝑟𝑚𝑠 | of non-dimensional perturbation amplitude versus wavenumber 𝑘 at corresponding time instants under the same conditions as in figure 8. The dash-dotted lines with circles are extracted from numerical simulations fitted by the Gaussian function (solid lines). The insets show the time histories of the dominant wavenumber converging to 𝑘𝑚𝑎𝑥 (red dashed lines) extracted from the spe… view at source ↗
Figure 10
Figure 10. Figure 10: Effects of vdW forces (𝑎) 𝐴1 and (𝑏) 𝐴2 on the dominant wavelengths 𝜆𝑚𝑎𝑥 : a comparison between the theoretical predictions of (2.15) (solid lines) and results of numerical simulation (Sim.) from the spectra (hollow symbols) with different film thickness 𝜖. The solid diamonds represent experimental results (Exp.) by Tomo et al. (2022). cases to isolate parameter effects. Both 𝐴1 (figures 8𝑎,𝑏) and 𝐴2 (fig… view at source ↗
Figure 11
Figure 11. Figure 11: Linear time evolution of the minimum radii of films ℎ𝑚𝑖𝑛 with various (𝑎) 𝐴1, where the maximum growth rate is 𝜔 = 0.094, 0.306, 1.433, 3.041, respectively; (𝑏) 𝐴2, for 𝜔 = 0.094, 0.367, 1.823, 3.879, respectively. The numerical solutions (circles) are compared with the predictions of the Stokes model (solid lines). Here, 𝜖 = 0.5, ℎˆ = 10−3 . perturbation amplitude at 𝑡 = 0. To validate the theoretical pr… view at source ↗
Figure 12
Figure 12. Figure 12: Linear prediction of the (𝑎) rupture time 𝑡𝑟 and (𝑏) collapse time 𝑡𝑐. The numerical solutions (symbols) are compared with the predictions of the Stokes model (solid lines). 6. Dynamics at the nonlinear stage As introduced in § 5, linear instability analysis is based on the assumption of small￾amplitude perturbations, where the evolving interface morphology can be represented as a superposition of sinusoi… view at source ↗
Figure 13
Figure 13. Figure 13: (𝑎) The film profile near rupture with different 𝐴1 for 𝜖 = 0.1. (𝑏) Volume proportion of satellite lobes under the influence of 𝐴1 [PITH_FULL_IMAGE:figures/full_fig_p017_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Evolution of perturbation growth at the nonlinear stage with four 𝐴1 values: (𝑎) 𝐴1 = 0, (𝑏) 𝐴1 = 10−7 , (𝑐) 𝐴1 = 10−5 , (𝑑) 𝐴1 = 10−3 . The contours represent the velocity magnitude |u| = √ 𝑢 2 + 𝑤2 , and the arrowed lines represent streamlines. Here, 𝜖 = 0.1. 0 X0-17 [PITH_FULL_IMAGE:figures/full_fig_p017_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: Minimum film thickness 1 − ℎ𝑚𝑎𝑥 as a function of (𝑎) the time 𝑡 and (𝑏) the time remaining to rupture 𝑡𝑟 − 𝑡 for different 𝐴1. The initial conditions ℎ0 = 0.9 + 10−3 cos(kz) use the corresponding dominant wave numbers. forces alters this evolution by introducing a molecular length scale 𝑎 = √︁ 𝐴˜/6𝜋𝛾 = √ 𝐴𝑟0 (de Gennes 1985). When the vdW forces are weak (𝐴1 = 10−7 , figure 14𝑏), surface tension remains t… view at source ↗
Figure 16
Figure 16. Figure 16: Evolutions of perturbation growth streamlines near film collapse for 𝐴2 = 0 (left) and 𝐴2 = 0.1 (right). The contours represent the velocity magnitude |u| = √ 𝑢 2 + 𝑤2 , and the arrowed lines represent streamlines. Here, 𝜖 = 0.5 [PITH_FULL_IMAGE:figures/full_fig_p020_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: Minimum interface radius ℎ𝑚𝑖𝑛 as a function of time remaining to collapse 𝑡𝑐 − 𝑡 for different 𝐴2. The initial conditions ℎ0 = 0.5 − 10−3 cos(kz) use the corresponding dominant wavenumbers [PITH_FULL_IMAGE:figures/full_fig_p020_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: (𝑎) Extreme point of interface evolution (1−ℎ𝑚𝑎𝑥 )𝑎 or (ℎ𝑚𝑖𝑛)𝑎 as a function of the time remaining to singular points (𝑡𝑟 −𝑡)𝑎 or (𝑡𝑐 −𝑡)𝑎. The length and time are rescaled by the characteristic length 𝑎 = √︁ 𝐴˜ 𝑖/6𝜋𝛾, where 𝑖 = 1, 2 for rupture and collapse respectively. (𝑏, 𝑐) Self-similarity profile at different instants 𝜏𝑎 for 𝐴1 = 0.1 or 𝐴2 = 0.1, compared with theoretical solution of film on a flat … view at source ↗
Figure 19
Figure 19. Figure 19: (𝑎) Equilibrium shape of a single collar in the case of different boundary conditions (as well as interface pressure). Lines represent theoretical results and symbols represent corresponding numerical profiles. (𝑏) Liquid volume𝑉 in a collar as a function of boundary condition ℎ ′′(0) for different 𝐴2, where𝑉𝑚𝑎𝑥 indicates the maximum volume. theoretical framework above is unsuitable for analyzing the infl… view at source ↗
read the original abstract

The instability of a liquid film in a nanotube is significantly influenced by van der Waals forces. A theoretical framework based on the axisymmetric Stokes equations is developed to investigate their effects through linear stability analysis. The model reveals that van der Waals forces markedly enhance perturbation growth, reduce the dominant wavelength, and lower the critical film thickness that distinguishes collapse from non-collapse regimes. Direct numerical simulations of the Navier-Stokes equations both confirm these theoretical predictions and extend the analysis into the nonlinear regime. In this regime, van der Waals forces are found to alter the interfacial morphology and suppress the formation of satellite lobes. Both rupture and collapse follow a universal temporal scaling law with exponent 1/3 and exhibit self-similar behavior near the singularity.

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 / 2 minor

Summary. The paper develops a linear stability analysis of an axisymmetric liquid film inside a tube using the Stokes equations augmented by a continuum van der Waals disjoining pressure. It reports that vdW forces increase the growth rate of perturbations, decrease the dominant wavelength, and reduce the critical film thickness separating collapse from non-collapse. Direct numerical simulations of the Navier-Stokes equations are used to confirm the linear predictions and to examine the nonlinear regime, where vdW forces are shown to modify interfacial morphology, suppress satellite lobes, and produce a universal temporal scaling with exponent 1/3 together with self-similar profiles near both rupture and collapse singularities.

Significance. If the central results hold, the work supplies a useful quantitative link between linear theory and nonlinear dynamics for thin-film instabilities in confined geometries at scales where vdW forces matter. The explicit matching of linear growth rates and wavelengths to DNS, together with the reported universal 1/3 scaling that is independent of the Hamaker constant, constitutes a clear strength. The findings are relevant to applications such as nanotube coating, capillary-driven flows, and nanoscale rupture processes.

major comments (2)
  1. [§4] §4 (nonlinear regime and self-similar analysis): the universal 1/3 temporal scaling and self-similar profiles for both rupture and collapse are derived under the assumption that the continuum disjoining pressure Π(h) ∝ −A/h³ remains valid as the minimum thickness h_min → 0. No regularization (Born repulsion, hydration forces, or molecular cutoff) is introduced or tested; the skeptic concern that molecular discreteness intervenes before the singularity is therefore load-bearing for the nonlinear claims.
  2. [§3.2 and §5] §3.2 (linear stability) and §5 (DNS validation): the reported agreement between Stokes linear growth rates and DNS is stated to hold, yet the manuscript does not quantify the Reynolds-number range over which the Stokes approximation remains accurate once inertia is restored in the DNS; a brief check of the neglected inertial terms at the dominant wavelength would strengthen the claim that the linear predictions are robust.
minor comments (2)
  1. [§2] Notation for the Hamaker constant and the precise form of the disjoining pressure should be stated explicitly in the governing-equation section rather than only in the appendix.
  2. [Figure 7] Figure captions for the self-similar profiles should include the precise rescaling used (e.g., the time-to-singularity exponent and the spatial scaling) so that readers can reproduce the collapse without consulting the text.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive and insightful comments on our manuscript. We address each major comment point by point below, providing the strongest honest responses based on the work presented. Where appropriate, we indicate revisions to be incorporated in the next version.

read point-by-point responses

  1. Referee: [§4] §4 (nonlinear regime and self-similar analysis): the universal 1/3 temporal scaling and self-similar profiles for both rupture and collapse are derived under the assumption that the continuum disjoining pressure Π(h) ∝ −A/h³ remains valid as the minimum thickness h_min → 0. No regularization (Born repulsion, hydration forces, or molecular cutoff) is introduced or tested; the skeptic concern that molecular discreteness intervenes before the singularity is therefore load-bearing for the nonlinear claims.

    Authors: We agree that the continuum disjoining pressure is an approximation whose validity diminishes as h_min approaches molecular scales. The 1/3 temporal scaling and self-similar profiles reported in §4 are derived directly from the model equations with Π(h) ∝ −A/h³; the exponent is independent of A because it arises from the dominant balance between capillary pressure, viscous resistance, and the singular disjoining term near the singularity. This feature is robust within the continuum framework and consistent with the DNS results that employ the same model. We will add a concise discussion paragraph in the revised §4 (and conclusions) explicitly acknowledging the limitation, noting that molecular discreteness or short-range repulsions would ultimately regularize the singularity, and referencing prior thin-film literature where analogous 1/3 scalings persist in molecular simulations until discreteness intervenes. This addition clarifies the scope of the claims without altering the reported scaling or profiles. revision: partial

  2. Referee: [§3.2 and §5] §3.2 (linear stability) and §5 (DNS validation): the reported agreement between Stokes linear growth rates and DNS is stated to hold, yet the manuscript does not quantify the Reynolds-number range over which the Stokes approximation remains accurate once inertia is restored in the DNS; a brief check of the neglected inertial terms at the dominant wavelength would strengthen the claim that the linear predictions are robust.

    Authors: We concur that an explicit Reynolds-number check would strengthen the validation section. In the revised manuscript we will add, in §5, a brief calculation of the local Reynolds number Re = ρ U λ / μ evaluated at the dominant wavelength λ and characteristic velocity U obtained from the linear growth rate σ_max for the parameter ranges examined (film thickness, tube radius, and Hamaker constant). For all cases considered, Re remains ≪ 1 (typically O(10^{-3})), confirming that the neglected inertial terms in the Stokes linear analysis are negligible compared with viscous and capillary forces. This estimate directly supports the observed agreement between the Stokes predictions and the full Navier–Stokes DNS. revision: yes

Circularity Check

0 steps flagged

No circularity: results follow directly from Stokes equations plus standard vdW term and DNS confirmation

full rationale

The derivation starts from the axisymmetric Stokes equations augmented by the standard continuum disjoining pressure, performs linear stability analysis to obtain growth rates and critical thicknesses, and uses independent DNS of the full Navier-Stokes equations to verify and extend into the nonlinear regime. The 1/3 temporal scaling and self-similarity are obtained by solving the time-dependent system near the singularity; they are not fitted parameters renamed as predictions nor defined in terms of the outputs. No self-citation is load-bearing for the central claims, and no ansatz is smuggled via prior work. The analysis is self-contained against the governing equations.

Axiom & Free-Parameter Ledger

1 free parameters · 3 axioms · 0 invented entities

The central claim rests on standard fluid assumptions plus the inclusion of a van der Waals disjoining pressure term whose strength is a material-dependent parameter.

free parameters (1)
  • van der Waals strength (Hamaker constant)
    Magnitude of the attractive force term added to the pressure; controls the enhancement of instability and is material-specific.
axioms (3)
  • domain assumption Axisymmetric flow with no azimuthal dependence
    Invoked to reduce the Stokes equations to a 2D problem in the abstract's theoretical framework.
  • domain assumption Stokes approximation (negligible inertia, Re << 1)
    Basis for the linear stability analysis of the axisymmetric equations.
  • domain assumption Continuum treatment of fluid and van der Waals forces at nanotube scales
    Required for the disjoining pressure model and Navier-Stokes simulations.

pith-pipeline@v0.9.0 · 5427 in / 1472 out tokens · 49614 ms · 2026-05-10T19:46:55.766680+00:00 · methodology

discussion (0)

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

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

4 extracted references · 4 canonical work pages

  1. [1]

    & Lister, J

    Beaty, E. & Lister, J. R.2022 Nonuniversal self-similarity for jump-to-contact dynamics between viscous drops under van der waals attraction.Phys. Rev. Lett.129, 064501. Beaty, E. & Lister, J. R.2023 Inertial and viscous dynamics of jump-to-contact between fluid drops under van der waals attraction.J. Fluid Mech.957, A25. Ben-Noah, I., Friedman, S. P. & B...

    work page 2022

  2. [2]

    & Grotberg, J

    Halpern, D. & Grotberg, J. B.2003 Nonlinear saturation of the rayleigh instability due to oscillatory flow in a liquid-lined tube.J. Fluid Mech.492, 251–270. Hammond, P. S.1983 Nonlinear adjustment of a thin annular film of viscous fluid surrounding a thread of another within a circular cylindrical pipe.J. Fluid Mech.137, 363–384. Heil, M., Hazel, A. L. &...

    work page 2003

  3. [3]

    Qu´er´e, D., di Meglio, J.-M

    Gauthier-Villars. Qu´er´e, D., di Meglio, J.-M. & Brochard-Wyart, F.1989 Making van der waals films on fibers.Europhys. Lett.10(4),

    work page 1989

  4. [4]

    Rayleigh, Lord1878 On the instability of jets.Proc. Lond. Math. Soc.s1-10(1), 4–13. Ruyer-Quil, C., Treveleyan, P., Giorgiutti-Dauphin´e, F., Duprat, C. & Kalliadasis, S.2008 Modelling film flows down a fibre.J. Fluid Mech.603, 431–462. Rykner, M., Saikali, E., Bruneton, A., Mathieu, B. & Nikolayev, V. S.2024 Plateau–rayleigh instability in a capillary: a...

    work page 2008