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arxiv: 2604.19132 · v1 · submitted 2026-04-21 · ⚛️ physics.flu-dyn

Marangoni modulation of coupled Rayleigh-Taylor and Faraday instabilities in vertically oscillated liquid films

Jun Gao , Senlin Zhu , Luca Brandt , Jianjun Tao , Qingfei Fu , Lijun Yang This is my paper

Pith reviewed 2026-05-10 02:27 UTC · model grok-4.3

classification ⚛️ physics.flu-dyn
keywords Marangoni effectRayleigh-Taylor instabilityFaraday instabilityliquid filmsinsoluble surfactantsparametric instabilityFloquet theorystability analysis
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0 comments X

The pith

Surfactants via the Marangoni effect selectively suppress subharmonic modes in coupled Rayleigh-Taylor and Faraday instabilities of vertically oscillated liquid films, with outcomes that depend on forcing frequency.

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

The authors examine how insoluble surfactants affect the stability of a liquid film under vertical oscillations where Rayleigh-Taylor and Faraday instabilities interact. Linear Floquet analysis shows that raising the Marangoni number eliminates subharmonic modes and shifts the system toward harmonic dominance, but this produces opposite changes to the stable region at low versus high frequencies. Long-wave asymptotics factor the critical forcing amplitude into static capillary-gravity and dynamic elasto-inertial parts, while nonlinear simulations trace the effect to phase-controlled interfacial transport by Marangoni stresses. A sympathetic reader would care because the results indicate surfactants can enlarge or shrink stable operating windows in vibrated thin-film systems used in coating, mixing, or separation processes.

Core claim

Linear stability analysis using Floquet theory reveals that an increasing Marangoni number selectively suppresses subharmonic modes, driving the system into a harmonic-dominated regime. The interfacial response is found to be highly frequency-dependent. At low forcing frequencies, increasing Ma causes adjacent harmonic tongues to merge into a novel surfactant mode that migrates towards long wavelengths, ultimately coalescing with the RTI branch and fragmenting the dynamically stable window. Conversely, at high frequencies, surfactants monotonically elevate the harmonic instability threshold, significantly widening the stable parameter space. A long-wave asymptotic analysis demonstrates that

What carries the argument

Phase-controlled Marangoni transport at the interface, which redistributes fluid along the surface according to the phase of the vertical oscillation.

If this is right

  • At low forcing frequencies, higher Marangoni numbers merge harmonic tongues and fragment the stable window by coalescence with the RTI branch.
  • At high forcing frequencies, higher Marangoni numbers raise the instability threshold and widen the stable parameter space.
  • In the RTI regime, increasing Ma reverses the direction of Marangoni transport, driving fluid into peaks and inducing destabilization.
  • In the FI regime, Marangoni transport redistributes fluid away from peaks at high frequencies (suppressing instability) but toward peaks at low frequencies (enhancing instability).

Where Pith is reading between the lines

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

  • Surfactant concentration could be adjusted together with oscillation frequency to select desired stable regimes in industrial vibrational fluid processes.
  • The factorization of critical amplitude into static and dynamic contributions may extend to other parametric instabilities that involve surface-tension gradients.
  • Experiments that image interfacial flow fields could directly verify the predicted reversal of transport direction across the RTI and FI regimes.
  • The merged surfactant mode at low frequencies may evolve into distinct finite-amplitude patterns once nonlinear saturation is considered beyond the reduced model.

Load-bearing premise

The liquid is Newtonian, the surfactants are insoluble, and linear stability plus a reduced nonlinear model suffice to capture the dominant dynamics before full nonlinear saturation occurs.

What would settle it

An experiment that measures the dominant mode (subharmonic or harmonic) and its growth rate while increasing surfactant concentration at fixed low and high oscillation frequencies would confirm or refute the selective suppression and the predicted reversal of stability trends.

Figures

Figures reproduced from arXiv: 2604.19132 by Jianjun Tao, Jun Gao, Lijun Yang, Luca Brandt, Qingfei Fu, Senlin Zhu.

Figure 1
Figure 1. Figure 1: Schematic of subplate thin film flow under the combined effects of surfactants [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Separate effects of surfactants and vertical vibration on the stability of a liquid [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Marangoni-induced suppression of subharmonic modes. [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Marangoni-driven evolution and reorganization of instability modes at low [PITH_FULL_IMAGE:figures/full_fig_p010_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: High-frequency (50Hz) modulation of instability regions by Marangoni effects. [PITH_FULL_IMAGE:figures/full_fig_p011_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Comparison between the long-wave analytical solution and numerical results, [PITH_FULL_IMAGE:figures/full_fig_p014_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Prediction of the critical Marangoni number Ma [PITH_FULL_IMAGE:figures/full_fig_p015_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: (𝑎) Temporal evolution of the liquid film interface, showing the development of a dry spot. (𝑏) Spatial distributions of the gravity 𝑓𝐺 and capillary forces 𝑓𝐶, at 𝑡 = 0.4. (𝑐) Spatial distributions of the Marangoni stress 𝑓𝑀 and viscous forces 𝑓𝑉 , at 𝑡 = 0.4. edges and a peak at the centre. The accumulation of fluid at the peak results in significant depletion in the trough region, ultimately giving rise… view at source ↗
Figure 9
Figure 9. Figure 9: Saturated state results and force analysis in the RTI regime at low frequencies, [PITH_FULL_IMAGE:figures/full_fig_p019_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: (𝑎) Spatial distribution of the interface profile 𝜂 over one oscillation period. Instantaneous spatial distributions of 𝑊𝑂, 𝑊𝑉 , 𝑊𝐶, and 𝑊𝐼 are shown. Panels (𝑏–𝑑) correspond to 𝑡1–𝑡3 during the growth stage of the disturbance, while (𝑒– 𝑓 ) correspond to 𝑡4–𝑡5 during the decay stage. 5.3. Role of Marangoni effects in the nonlinear dynamics In this subsection, three phenomena are examined that require fur… view at source ↗
Figure 11
Figure 11. Figure 11: Variation of the difference between the maximum and minimum time-averaged [PITH_FULL_IMAGE:figures/full_fig_p022_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Spatial distributions of the time averaged [PITH_FULL_IMAGE:figures/full_fig_p023_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Comparison of neutral stability curves for harmonic (H) and subharmonic (SH) [PITH_FULL_IMAGE:figures/full_fig_p028_13.png] view at source ↗
read the original abstract

We investigate the Marangoni modulation of coupled Rayleigh-Taylor and Faraday instabilities in a vertically oscillated Newtonian liquid film carrying insoluble surfactants. Linear stability analysis using Floquet theory reveals that an increasing Marangoni number (Ma) selectively suppresses subharmonic modes, driving the system into a harmonic-dominated regime. The interfacial response is found to be highly frequency-dependent. At low forcing frequencies, increasing Ma causes adjacent harmonic tongues to merge into a novel surfactant mode that migrates towards long wavelengths, ultimately coalescing with the RTI branch and fragmenting the dynamically stable window. Conversely, at high frequencies, surfactants monotonically elevate the harmonic instability threshold, significantly widening the stable parameter space. To uncover the underlying mechanisms, a long-wave asymptotic analysis is performed, demonstrating that the critical forcing amplitude factorizes into a static capillary-gravity margin and a dynamic elasto-inertial modulation, yielding a scaling law for the critical mode balance. Finally, nonlinear simulations based on a rigorous weighted-residual reduced model are utilized to dissect the spatial work performed by individual forces, which shows that surfactants modulate stability through phase-controlled Marangoni transport. In the RTI regime, increasing Ma reverses the transport direction and drives fluid into the peaks, inducing a transition from stabilization to destabilization. In the Faraday instability (FI) regime, the response exhibits a strong frequency dependence, governed by Marangoni transport that redistributes fluid away from interfacial peaks at high frequencies but toward them at low frequencies, thereby suppressing or enhancing the instability accordingly.

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 investigates Marangoni modulation of coupled Rayleigh-Taylor and Faraday instabilities in vertically oscillated Newtonian liquid films with insoluble surfactants. Floquet linear stability analysis shows that increasing the Marangoni number Ma selectively suppresses subharmonic modes, shifting the system toward harmonic-dominated regimes with pronounced frequency dependence: at low frequencies, adjacent harmonic tongues merge into a surfactant mode that migrates to long wavelengths and coalesces with the RTI branch, fragmenting the stable window; at high frequencies, the harmonic threshold rises and widens the stable region. Long-wave asymptotics factorize the critical forcing amplitude into a static capillary-gravity margin and dynamic elasto-inertial modulation, producing a scaling law for mode balance. Nonlinear simulations with a weighted-residual reduced model dissect spatial work integrals, revealing that surfactants modulate stability via phase-controlled Marangoni transport, which reverses direction in the RTI regime and exhibits frequency-dependent redistribution in the Faraday regime.

Significance. If the results hold, the work offers mechanistic insight into surfactant effects on interfacial instabilities under vertical oscillation, relevant to thin-film coating and microfluidic applications. The complementary use of Floquet theory, long-wave factorization yielding a scaling law, and work-integral analysis from the reduced model provides a coherent picture of mode selection and transport reversal. The frequency-dependent merging and stabilization behaviors, when supported by consistent numerics, constitute a clear advance over prior studies of pure RTI or Faraday problems.

major comments (2)
  1. [Long-wave asymptotic analysis] Long-wave asymptotic analysis section: The factorization of the critical forcing amplitude into static capillary-gravity and dynamic elasto-inertial contributions is central to the reported scaling law and frequency dependence, yet the manuscript does not explicitly delineate the validity range of the long-wave assumption (e.g., wavenumber cutoff) for the low-frequency merging regime where modes approach long wavelengths; this risks over-extension of the asymptotics precisely where the novel surfactant mode coalescence is claimed.
  2. [Nonlinear simulations] Nonlinear simulations and work-integral analysis: The reported reversal of Marangoni transport direction in the RTI regime and the frequency-dependent redistribution in the FI regime rely on the weighted-residual model; however, no direct comparison is shown between the nonlinear thresholds or growth rates and the Floquet linear stability boundaries for identical Ma and frequency values, leaving open whether the reduced model faithfully reproduces the linear predictions before interpreting the work integrals.
minor comments (3)
  1. [Abstract] Abstract: The term 'elasto-inertial modulation' appears without a preceding definition or reference; a short parenthetical clarification would improve accessibility.
  2. [Figures] Figure captions (stability diagrams): The boundaries separating subharmonic, harmonic, and RTI branches should be labeled more explicitly, and the color scale or line styles for different Ma values clarified to avoid ambiguity in the mode-merging regions.
  3. [Linear stability analysis] Notation: The definition of the Marangoni number Ma is introduced in the linear analysis but its precise nondimensionalization (including surfactant concentration scaling) should be restated in the asymptotic section for self-contained reading.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their positive assessment and recommendation for minor revision. The comments are constructive and help strengthen the presentation of the long-wave analysis and the validation of the reduced model. We address each point below.

read point-by-point responses

  1. Referee: [Long-wave asymptotic analysis] Long-wave asymptotic analysis section: The factorization of the critical forcing amplitude into static capillary-gravity and dynamic elasto-inertial contributions is central to the reported scaling law and frequency dependence, yet the manuscript does not explicitly delineate the validity range of the long-wave assumption (e.g., wavenumber cutoff) for the low-frequency merging regime where modes approach long wavelengths; this risks over-extension of the asymptotics precisely where the novel surfactant mode coalescence is claimed.

    Authors: We agree that an explicit statement of the long-wave validity range is needed, especially near the low-frequency coalescence. In the revised manuscript we will add a dedicated paragraph specifying that the asymptotic expansion assumes k ≪ 1 and quantifying the cutoff by comparing the magnitude of retained and neglected terms. We will also overlay the asymptotic critical amplitude with the full Floquet dispersion relation at small k to confirm that the surfactant-mode coalescence lies inside the regime where the long-wave factorization remains accurate. This addition directly addresses the concern of over-extension. revision: yes

  2. Referee: [Nonlinear simulations] Nonlinear simulations and work-integral analysis: The reported reversal of Marangoni transport direction in the RTI regime and the frequency-dependent redistribution in the FI regime rely on the weighted-residual model; however, no direct comparison is shown between the nonlinear thresholds or growth rates and the Floquet linear stability boundaries for identical Ma and frequency values, leaving open whether the reduced model faithfully reproduces the linear predictions before interpreting the work integrals.

    Authors: We acknowledge the value of an explicit validation step. In the revised manuscript we will insert a new figure (or panel) that directly compares the instability thresholds and linear growth rates obtained from the weighted-residual model with the corresponding Floquet boundaries for representative Ma and forcing frequencies. This comparison will be presented before the work-integral analysis, thereby confirming that the reduced model reproduces the linear predictions to within the expected truncation error and supporting the subsequent nonlinear interpretations. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The derivation relies on standard Floquet linear stability analysis applied to the base equations with Marangoni stress from insoluble surfactants, followed by a long-wave asymptotic factorization of the critical amplitude into capillary-gravity and elasto-inertial terms that follows directly from the linearized system under the stated assumptions. The weighted-residual reduced model is employed only for post-hoc work-integral dissection in nonlinear simulations, without any parameter fitting or renaming of outputs as predictions. No self-citations, ansatzes, or uniqueness theorems are invoked as load-bearing steps in the chain, and the reported mode suppression, merging, and transport reversal emerge from the tangential stress balance and phase relations without tautological reduction to inputs.

Axiom & Free-Parameter Ledger

1 free parameters · 3 axioms · 0 invented entities

The central claims rest on standard fluid assumptions plus the specific reduced model; no new entities are postulated.

free parameters (1)
  • Marangoni number Ma
    Dimensionless parameter varied to quantify surfactant strength; central to all reported trends.
axioms (3)
  • domain assumption Newtonian incompressible fluid
    Stated for the liquid film in the abstract.
  • domain assumption Insoluble surfactants
    Explicitly carried by the film; no desorption.
  • standard math Linear stability via Floquet theory
    Used to obtain tongues and thresholds.

pith-pipeline@v0.9.0 · 5585 in / 1343 out tokens · 36149 ms · 2026-05-10T02:27:03.772308+00:00 · methodology

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Reference graph

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