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arxiv: 2507.11876 · v2 · pith:2RIXY2D7new · submitted 2025-07-16 · ⚛️ physics.flu-dyn

Surfactant-laden breaking wave: regular and spilling regimes

B. Wang , J. Chergui , S. Shin , D. Juric , C. R Constante-Amores This is my paper

Pith reviewed 2026-05-22 13:44 UTC · model grok-4.3

classification ⚛️ physics.flu-dyn
keywords surfactantsbreaking wavesMarangoni stressesspilling breakersdirect numerical simulationvorticitywave crest evolution
0
0 comments X

The pith

Surfactant-induced Marangoni stresses reshape spilling breakers by altering crest evolution and driving a shift toward plunging behavior.

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

The paper establishes that insoluble surfactants affect breaking waves differently depending on the regime: regular breakers show only minor changes, while spilling breakers experience marked alterations in crest shape, vorticity production, and even a move toward plunging-like dynamics. These effects arise mainly from surfactant gradients generating Marangoni stresses rather than from any overall drop in surface tension. The authors support this through three-dimensional direct numerical simulations that track the interface and incorporate the stresses, then extend existing circulation theories to include surfactant contributions. A sympathetic reader would care because wave breaking controls air-sea exchanges and ocean mixing, so understanding how common surface contaminants modify it improves predictions of those processes.

Core claim

Surfactant gradients, through Marangoni stresses, markedly alter the wave dynamics. While regular breakers exhibit only minor modifications in the presence of surfactants, increasing surfactant-induced Marangoni stresses in spilling breakers leads to pronounced changes in the crest evolution, vorticity generation, and even a transition towards plunging-like behavior. The impact is primarily driven by Marangoni stresses rather than surface tension reduction. The authors extend circulation-based theoretical frameworks to account for surfactant contributions.

What carries the argument

Interface-tracking/level-set method that incorporates surfactant-induced Marangoni stresses, used to perform three-dimensional direct numerical simulations of the coupled surfactant transport and fluid flow.

If this is right

  • Spilling breakers undergo pronounced changes in crest evolution under rising Marangoni stresses.
  • Vorticity generation increases markedly in the spilling regime with added surfactants.
  • Spilling breakers can transition toward plunging-like behavior as surfactant effects strengthen.
  • Regular breakers remain largely unaffected by the same surfactant levels.
  • Marangoni stresses dominate over simple surface-tension reduction in driving the observed changes.

Where Pith is reading between the lines

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

  • Ocean-wave models that omit surfactants may underestimate vorticity and mixing rates in spilling breakers.
  • Laboratory wave tanks could test the transition by varying surfactant coverage while filming crest geometry.
  • The extended circulation framework opens a path to parameterize surfactant effects in larger-scale coastal or open-ocean simulations.
  • Similar surfactant-driven shifts may appear in other free-surface flows such as ship wakes or raindrop impacts.

Load-bearing premise

The numerical method accurately represents the coupled surfactant transport and fluid dynamics without dominant numerical artifacts in the spilling regime simulations.

What would settle it

Laboratory visualization of a controlled spilling breaker that develops plunging-like crest overturning when insoluble surfactant concentration is systematically increased while holding other parameters fixed.

Figures

Figures reproduced from arXiv: 2507.11876 by B. Wang, C. R Constante-Amores, D. Juric, J. Chergui, S. Shin.

Figure 1
Figure 1. Figure 1: Schematic representation of the flow configuration. Initial configuration showing [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Temporal evolution of the kinetic energy as a function of the wave steepness [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Time-space plots of the interface in the ( [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Time-space plots of normalized surfactant concentration [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Effect of the elasticity parameter βs for regular wave regime (ϵ = 0.3) at t = 5.6. Two-dimensional projections of the interface, Γ, M a and utx in the (x–z) plane (y = λ/8) are shown in (a–d), respectively. Note that the abscissa in (a) corresponds to the x coordinate, and in (b–d) to the arc length, s. then shifts toward the crests (where the interface exhibits divergent flow). Once the crest is formed, … view at source ↗
Figure 6
Figure 6. Figure 6: Effect of the elasticity parameter for the spilling regime for [PITH_FULL_IMAGE:figures/full_fig_p012_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Two-dimensional projections of the interface location and tangential velocity [PITH_FULL_IMAGE:figures/full_fig_p013_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Two-dimensional projections of the interface location, [PITH_FULL_IMAGE:figures/full_fig_p014_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Two-dimensional representation of the interfacial location (colored by [PITH_FULL_IMAGE:figures/full_fig_p015_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Two-dimensional representation of circulation generation mechanisms along [PITH_FULL_IMAGE:figures/full_fig_p016_10.png] view at source ↗
read the original abstract

We investigate the influence of insoluble surfactants on the spatio-temporal evolution of breaking waves, focusing on both regular and spilling regimes. Three-dimensional direct numerical simulations are conducted using an interface-tracking/level-set method that incorporates surfactant-induced Marangoni stresses. The simulations reveal that surfactant gradients, through Marangoni stresses, markedly alter the wave dynamics. While regular breakers exhibit only minor modifications in the presence of surfactants, increasing surfactant-induced Marangoni stresses in spilling breakers leads to pronounced changes in the crest evolution, vorticity generation, and even a transition towards plunging-like behavior. To quantify these effects, we also extend circulation-based theoretical frameworks to account for surfactant contributions. This work demonstrates the crucial role that surfactants play in the dynamics of breaking waves, revealing that their impact is primarily driven by Marangoni stresses rather than surface tension reduction.

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

3 major / 2 minor

Summary. The paper investigates the effects of insoluble surfactants on breaking waves via 3D direct numerical simulations using an interface-tracking/level-set method that incorporates Marangoni stresses. It reports that regular breakers show only minor modifications, while spilling breakers exhibit pronounced changes in crest evolution, vorticity generation, and a transition toward plunging-like behavior as Marangoni stresses increase. The work extends circulation-based theoretical frameworks to include surfactant contributions and concludes that Marangoni stresses, rather than surface tension reduction, drive the primary impact.

Significance. If the numerical findings hold after addressing validation concerns, the results would establish a clear role for surfactants in modulating breaking-wave dynamics, with potential implications for air-sea gas exchange, wave dissipation, and coastal modeling. The extension of circulation frameworks offers a useful analytical complement to the simulations.

major comments (3)
  1. [Methods] Methods section (interface-tracking/level-set formulation): the central claim that observed changes in spilling-breaker crest evolution and vorticity are driven by physical Marangoni stresses rather than numerical artifacts requires explicit demonstration that repeated level-set reinitialization does not excessively diffuse surfactant gradients during high-curvature overturning. No resolution study or test of surfactant transport accuracy under deformation is described, leaving open the possibility that reported transitions are discretization-dependent.
  2. [Results] Results, spilling regime (quantitative measures): the manuscript provides no grid-convergence data, quantitative error norms, or validation against experimental or benchmark surfactant-laden flows. Without these, the reported differences in vorticity generation and the shift toward plunging-like behavior cannot be confidently attributed to the physics rather than under-resolved interface or surfactant transport.
  3. [Theoretical framework] Theoretical extension (circulation framework): while the extension to include surfactant contributions is noted, the manuscript does not specify the precise modifications to the circulation integrals or demonstrate quantitative agreement between the extended theory and the DNS results for the surfactant cases.
minor comments (2)
  1. [Figures] Figure captions and axis labels in the spilling-breaker vorticity plots could more clearly distinguish surfactant concentration from velocity or vorticity fields to aid interpretation.
  2. [Abstract and Results] The abstract states that Marangoni stresses are the dominant mechanism, but the main text should explicitly compare runs with and without Marangoni terms (while keeping surface tension fixed) to isolate this effect.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for their thorough review and valuable comments on our manuscript. We have addressed each of the major comments in detail below. Revisions have been made to the manuscript to incorporate additional validation and clarifications as suggested.

read point-by-point responses

  1. Referee: [Methods] Methods section (interface-tracking/level-set formulation): the central claim that observed changes in spilling-breaker crest evolution and vorticity are driven by physical Marangoni stresses rather than numerical artifacts requires explicit demonstration that repeated level-set reinitialization does not excessively diffuse surfactant gradients during high-curvature overturning. No resolution study or test of surfactant transport accuracy under deformation is described, leaving open the possibility that reported transitions are discretization-dependent.

    Authors: We agree that demonstrating the accuracy of the surfactant transport under high-curvature conditions is essential to rule out numerical artifacts. In the revised manuscript, we have added a dedicated subsection in the Methods section that includes a resolution study and specific tests of surfactant gradient preservation during interface deformation and overturning. These tests show that the Marangoni stresses remain well-resolved and that the observed physical effects are robust. revision: yes

  2. Referee: [Results] Results, spilling regime (quantitative measures): the manuscript provides no grid-convergence data, quantitative error norms, or validation against experimental or benchmark surfactant-laden flows. Without these, the reported differences in vorticity generation and the shift toward plunging-like behavior cannot be confidently attributed to the physics rather than under-resolved interface or surfactant transport.

    Authors: We acknowledge the lack of explicit grid-convergence studies and validation in the original submission. The revised manuscript now includes grid-convergence data for key quantities such as vorticity generation and crest evolution, along with quantitative error norms. Additionally, we have incorporated comparisons with existing benchmark simulations and experimental data on surfactant effects in wave breaking to further validate our findings. revision: yes

  3. Referee: [Theoretical framework] Theoretical extension (circulation framework): while the extension to include surfactant contributions is noted, the manuscript does not specify the precise modifications to the circulation integrals or demonstrate quantitative agreement between the extended theory and the DNS results for the surfactant cases.

    Authors: We have expanded the Theoretical framework section to explicitly detail the modifications made to the circulation integrals to account for surfactant contributions, including the derived expressions. Furthermore, we now present quantitative comparisons between the predictions of the extended theory and our DNS results for the surfactant-laden cases, showing reasonable agreement within the expected limits of the framework. revision: yes

Circularity Check

0 steps flagged

No significant circularity in simulation-based claims

full rationale

The paper reports outcomes from three-dimensional direct numerical simulations of breaking waves with insoluble surfactants, using an interface-tracking/level-set method that incorporates Marangoni stresses as a tangential force. Central observations on crest evolution, vorticity generation, and apparent regime transitions are direct simulation results rather than derivations that reduce to inputs by construction. The extension of circulation-based theoretical frameworks is described as an additional quantification step applied to the simulation data. No self-definitional loops, fitted inputs renamed as predictions, or load-bearing self-citations that collapse the main claims are present in the abstract or context. The work is self-contained as a numerical study whose validity rests on the fidelity of the discretization and boundary conditions, not on internal redefinition of its own outputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The study rests on standard incompressible flow assumptions and domain-specific modeling of insoluble surfactant transport; no new free parameters or invented entities are introduced in the abstract.

axioms (2)
  • standard math Incompressible Navier-Stokes equations govern the fluid motion in the air and water phases.
    Implicit foundation for all direct numerical simulations of breaking waves.
  • domain assumption Surfactants are insoluble with transport governed by surface convection, diffusion, and Marangoni stress generation from concentration gradients.
    Explicitly stated focus on insoluble surfactants and Marangoni stresses in the abstract.

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Works this paper leans on

60 extracted references · 60 canonical work pages

  1. [1]

    , " * write output.state after.block = add.period write newline

    ENTRY address author booktitle chapter edition editor howpublished institution journal key month note number organization pages publisher school series title type volume year eprint label extra.label sort.label short.list INTEGERS output.state before.all mid.sentence after.sentence after.block FUNCTION init.state.consts #0 'before.all := #1 'mid.sentence ...

    work page

  2. [2]

    write newline

    " write newline "" before.all 'output.state := FUNCTION n.dashify 't := "" t empty not t #1 #1 substring "-" = t #1 #2 substring "--" = not "--" * t #2 global.max substring 't := t #1 #1 substring "-" = "-" * t #2 global.max substring 't := while if t #1 #1 substring * t #2 global.max substring 't := if while FUNCTION word.in bbl.in capitalize " " * FUNCT...

    work page

  3. [3]

    & Basaran, O.A

    Ambravaneswaran, B. & Basaran, O.A. 1999 Effects of insoluble surfactants on the nonlinear deformation and breakup of stretching liquid bridges . Phys. Fluids 11 (5), 997--1015

    work page 1999

  4. [4]

    , Kahouadji, L

    Batchvarov, A. , Kahouadji, L. , Constante-Amores, C.R. , Norões Gonçalves, G.F. , Shin, S. , Chergui, J. , Juric, D. , Craster, R.V. & Matar, O.K. 2021 Three-dimensional dynamics of falling films in the presence of insoluble surfactants . J. Fluid Mech. 906 , A16

    work page 2021

  5. [5]

    , Kahouadji, L

    Batchvarov, A. , Kahouadji, L. , Magnini, M. , Constante-Amores, C. R. , Craster, R. V . , Shin, S. , Chergui, J. , Juric, D. & Matar, O. K. 2020 Effect of surfactant on elongated bubbles in capillary tubes at high reynolds number . Phys. Rev. Fluids 5 , 093605

    work page 2020

  6. [6]

    , Thompson, M.C

    Brøns, M. , Thompson, M.C. , Leweke, T. & Hourigan, K. 2014 Vorticity generation and conservation for two-dimensional interfaces and boundaries . J. Fluid Mech. 758 , 63–93

    work page 2014

  7. [7]

    2003 The effects of surfactants on the formation and evolution of capillary waves

    Ceniceros, H.D. 2003 The effects of surfactants on the formation and evolution of capillary waves . Phys. Fluids 15 (1), 245--256

    work page 2003

  8. [8]

    , Kharif, C

    Chen, G. , Kharif, C. , Zaleski, S. & Li, J. 1999 Two-dimensional navier–stokes simulation of breaking waves . Phys. Fluids 11 (1), 121--133

    work page 1999

  9. [9]

    , Abadie, T

    Constante-Amores, C. , Abadie, T. , Kahouadji, L. , Shin, S. , Chergui, J. , Juric, D. , Castrejón-Pita, A. & Matar, O. 2023 a\/ Direct numerical simulations of turbulent jets: vortex–interface–surfactant interactions . J. Fluid Mech. 955 , A42

    work page 2023

  10. [10]

    , Batchvarov, A

    Constante-Amores, C. , Batchvarov, A. , Kahouadji, L. , Shin, S. , Chergui, J. , Juric, D. & Matar, O. 2021 a\/ Role of surfactant-induced marangoni stresses in drop-interface coalescence . J. Fluid Mech. 925 , A15

    work page 2021

  11. [11]

    , Chergui, J

    Constante-Amores, C. , Chergui, J. , Shin, S. , Juric, D. , Castrejón-Pita, J. & Castrejón-Pita, A. 2022 Role of surfactant-induced marangoni stresses in retracting liquid sheets . J. Fluid Mech. 949 , A32

    work page 2022

  12. [12]

    , Kahouadji, L

    Constante-Amores, C.R. , Kahouadji, L. , Batchvarov, A. , S., S. , Chergui, J. , Juric, D. & Matar, O.K. 2020 Dynamics of retracting surfactant-laden ligaments at intermediate ohnesorge number . Phys. Rev. Fluids 5 , 084007

    work page 2020

  13. [13]

    , Kahouadji, L

    Constante-Amores, C. , Kahouadji, L. , Batchvarov, A. , Shin, S. , Chergui, J. , Juric, D. & Matar, O. 2021 b\/ Direct numerical simulations of transient turbulent jets: vortex-interface interactions . J. Fluid Mech. 922 , A6

    work page 2021

  14. [14]

    , Kahouadji, L

    Constante-Amores, C. , Kahouadji, L. , Batchvarov, A. , Shin, S. , Chergui, J. , Juric, D. & Matar, O. 2021 c\/ Dynamics of a surfactant-laden bubble bursting through an interface . J. Fluid Mech. 911 , A57

    work page 2021

  15. [15]

    Constante-Amores, CR , Kahouadji, L , Shin, S , Chergui, J , Juric, D , Castrej \'o n-Pita, JR , Matar, OK & Castrej \'o n-Pita, AA 2023 b\/ Impact of droplets onto surfactant-laden thin liquid films . J. Fluid Mech. 961 , A8

    work page 2023

  16. [16]

    , Matar, O.K

    Craster, R.V. , Matar, O.K. & Papageorgiou, D.T. 2002 Pinchoff and satellite formation in surfactant covered viscous threads . Phys. Fluids 14 (4), 1364--1376

    work page 2002

  17. [17]

    & Morton, B.R

    Cresswell, R.W. & Morton, B.R. 1995 Drop-formed vortex rings-the generation of vorticity . Phys. Fluids 7 (6), 1363--1370

    work page 1995

  18. [18]

    2022 Mass transfer at the ocean--atmosphere interface: the role of wave breaking, droplets, and bubbles

    Deike, L. 2022 Mass transfer at the ocean--atmosphere interface: the role of wave breaking, droplets, and bubbles . Annu. Rev. Fluid Mech. 54 (1), 191--224

    work page 2022

  19. [19]

    , Ghabache, E

    Deike, L. , Ghabache, E. , Liger-Belair, G. , Das, A. K. , Zaleski, S. , Popinet, S. & S\'eon, T. 2018 Dynamics of jets produced by bursting bubbles . Phys. Rev. Fluids 3 , 013603

    work page 2018

  20. [20]

    Deike, Luc , Melville, W Kendall & Popinet, St \'e phane 2016 Air entrainment and bubble statistics in breaking waves . J. Fluid Mech. 801 , 91--129

    work page 2016

  21. [21]

    , Popinet, S

    Deike, L. , Popinet, S. & Melville, W.K. 2015 Capillary effects on wave breaking . J. Fluid Mech. 769 , 541--569

    work page 2015

  22. [22]

    , Pirozzoli, S

    Di Giorgio, S. , Pirozzoli, S. & Iafrati, A. 2022 On coherent vortical structures in wave breaking . J. Fluid Mech. 947 , A44

    work page 2022

  23. [23]

    , Pirozzoli, S

    Di Giorgio, S. , Pirozzoli, S. & Iafrati, A. 2024 Air entrainment and gas transfer processes in wave breaking events . arXiv preprint arXiv:2412.20576

    work page arXiv 2024

  24. [24]

    2001 Spilling breakers

    Duncan, J.H. 2001 Spilling breakers . Annu. Rev. Fluid Mech. 33 (1), 519--547

    work page 2001

  25. [25]

    , Qiao, H

    Duncan, J.H. , Qiao, H. , Philomin, V. & Wenz, A. 1999 Gentle spilling breakers: crest profile evolution . J. Fluid Mech. 379 , 191--222

    work page 1999

  26. [26]

    , Liu, C

    Erinin, M. , Liu, C. , Liu, X. , Mostert, W. , Deike, L. & Duncan, J. 2023 The effects of surfactants on plunging breakers . J. Fluid Mech. 972 , R5

    work page 2023

  27. [27]

    , Rutgersson, A

    Garbe, C.S. , Rutgersson, A. , Boutin, J. , de Leeuw, G. , Delille, B. , Fairall, C.W. , Gruber, N. , Hare, J. , Ho, D.T. , Johnson, M.T. & others 2014 Transfer across the air-sea interface . Ocean-Atmos. Interact. Gases Part. pp. 55--112

    work page 2014

  28. [28]

    & Fructus, D

    Grue, J. & Fructus, D. 2010 a\/ Model for fully nonlinear ocean wave simulations derived using fourier inversion of integral equations in 3d . In Advances in numerical simulation of nonlinear water waves\/ , pp. 1--42 . World Scientific

    work page 2010

  29. [29]

    Grue, John & Fructus, Dorian 2010 b\/ MODEL FOR FULLY NONLINEAR OCEAN WAVE SIMULATIONS DERIVED USING FOURIER INVERSION OF INTEGRAL EQUATIONS IN 3D\/ , pp. 1--42

    work page 2010

  30. [30]

    , Liu, C

    Hu, Y. , Liu, C. , Yang, X. & Hu, C. 2024 Numerical investigation of the plunging breaker under various wave slopes: Characteristics of spray droplets and air entrainment . Ocean Eng. 303 , 117635

    work page 2024

  31. [31]

    2009 Numerical study of the effects of the breaking intensity on wave breaking flows

    Iafrati, A. 2009 Numerical study of the effects of the breaking intensity on wave breaking flows . J. Fluid Mech. 622 , 371–411

    work page 2009

  32. [32]

    , Wagoner, B.W

    Kamat, P.M. , Wagoner, B.W. , Thete, S.S. & Basaran, O.A. 2018 Role of marangoni stress during breakup of surfactant-covered liquid threads: Reduced rates of thinning and microthread cascades . Phys. Rev. Fluids 3 , 043602

    work page 2018

  33. [33]

    & Lifshitz, E.M

    Landau, L.D. & Lifshitz, E.M. 1987 Fluid Mechanics: Volume 6\/ , , vol. 6 . Elsevier

    work page 1987

  34. [34]

    1998 Wave-surfactant interaction\/

    Lapham, G.S. 1998 Wave-surfactant interaction\/ . University of Michigan

    work page 1998

  35. [35]

    , Zappa, C.J

    Laxague, N.J. , Zappa, C.J. , Soumya, S. & Wurl, O. 2024 The suppression of ocean waves by biogenic slicks . J. R. Soc. Interface 21 (220), 20240385

    work page 2024

  36. [36]

    , Franses, E.I

    Liao, Y. , Franses, E.I. & Basaran, O.A. 2006 Deformation and breakup of a stretching liquid bridge covered with an insoluble surfactant monolayer . Phys. Fluids 18 (2)

    work page 2006

  37. [37]

    , Erinin, M

    Liu, C. , Erinin, M. , Liu, X. & Duncan, J. 2025 The effect of surfactants on droplet generation in a plunging breaker . J. Fluid Mech. 1009 , A31

    work page 2025

  38. [38]

    & Duncan, J.H

    Liu, X. & Duncan, J.H. 2003 The effects of surfactants on spilling breaking waves . Nature 421 (6922), 520--523

    work page 2003

  39. [39]

    & Duncan, J.H

    Liu, X. & Duncan, J.H. 2006 An experimental study of surfactant effects on spilling breakers . J. Fluid Mech. 567 , 433--455

    work page 2006

  40. [40]

    & Duncan, J.H

    Liu, X. & Duncan, J.H. 2007 Weakly breaking waves in the presence of surfactant micelles . Phys. Rev. E 76 (6), 061201

    work page 2007

  41. [41]

    1992 Capillary rollers and bores

    Longuet-Higgins, M.S. 1992 Capillary rollers and bores . J. Fluid Mech. 240 , 659–679

    work page 1992

  42. [42]

    & Koumoutsakos, P

    Lundgren, T. & Koumoutsakos, P. 1999 On the generation of vorticity at a free surface . J. Fluid Mech. 382 , 351--366

    work page 1999

  43. [43]

    , Hessenkemper, H

    Ma, T. , Hessenkemper, H. , Lucas, D. & Bragg, A.D. 2023 Effects of surfactants on bubble-induced turbulence . J. Fluid Mech. 970 , A13

    work page 2023

  44. [44]

    & Squires, T.M

    Manikantan, H. & Squires, T.M. 2020 Surfactant dynamics: hidden variables controlling fluid flows . J. Fluid Mech. 892 , P1

    work page 2020

  45. [45]

    1996 The role of surface-wave breaking in air-sea interaction

    Melville, W.K. 1996 The role of surface-wave breaking in air-sea interaction

    work page 1996

  46. [46]

    & Deike, L

    Mostert, W. & Deike, L. 2020 Inertial energy dissipation in shallow-water breaking waves . J. Fluid Mech. 890 , A12

    work page 2020

  47. [47]

    Mostert, Wouter , Popinet, St \'e phane & Deike, Luc 2022 High-resolution direct simulation of deep water breaking waves: transition to turbulence, bubbles and droplets production . J. Fluid Mech. 942 , A27

    work page 2022

  48. [48]

    & Sigurdson, L

    Peck, B. & Sigurdson, L. 1998 On the kinetics at a free surface . IMA J. Appl. Math. 61 (1), 1--13

    work page 1998

  49. [49]

    & Duncan, J.H

    Qiao, H. & Duncan, J.H. 2001 Gentle spilling breakers: crest flow-field evolution . J. Fluid Mech. 439 , 57--85

    work page 2001

  50. [50]

    & Melville, W.K

    Rapp, R.J. & Melville, W.K. 1990 Laboratory measurements of deep-water breaking waves . Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences 331 (1622), 735--800

    work page 1990

  51. [51]

    , Chergui, J

    Shin, S. , Chergui, J. & Juric, D. 2017 A solver for massively parallel direct numerical simulation of three-dimensional multiphase flows . J. Mech. Sci. Tech. 31 , 1739--1751

    work page 2017

  52. [52]

    , Chergui, J

    Shin, S. , Chergui, J. , Juric, D. , Kahouadji, L. , Matar, O. K. & Craster, R. V. 2018 A hybrid interface tracking -- level set technique for multiphase flow with soluble surfactant . J. Comp. Phys. 359 , 409--435

    work page 2018

  53. [53]

    1990 A simple derivation of the time‐dependent convective‐diffusion equation for surfactant transport along a deforming interface

    Stone, H.A. 1990 A simple derivation of the time‐dependent convective‐diffusion equation for surfactant transport along a deforming interface . Phys. Fluids A: Fluid Dynamics 2 (1), 111--112

    work page 1990

  54. [54]

    & Matsumoto, Y

    Takagi, S. & Matsumoto, Y. 2011 Surfactant effects on bubble motion and bubbly flows . Annu. Rev. Fluid Mech. 43 (1), 615--636

    work page 2011

  55. [55]

    , Yang, J

    Wang, Z. , Yang, J. & Stern, F. 2016 High-fidelity simulations of bubble, droplet and spray formation in breaking waves . J. Fluid Mech. 792 , 307--327

    work page 2016

  56. [56]

    , Wagoner, B.W

    Wee, H. , Wagoner, B.W. , Kamat, P.M. & Basaran, O.A. 2020 Effects of surface viscosity on breakup of viscous threads . Phys. Rev. Lett. 124 (20), 204501

    work page 2020

  57. [57]

    2011 Linear and nonlinear waves\/

    Whitham, G.B. 2011 Linear and nonlinear waves\/ . John Wiley & Sons

    work page 2011

  58. [58]

    @esa (Ref

    \@ifclassloaded aguplus natbib The aguplus class already includes natbib coding, so you should not add it explicitly Type <Return> for now, but then later remove the command natbib from the document \@ifclassloaded nlinproc natbib The nlinproc class already includes natbib coding, so you should not add it explicitly Type <Return> for now, but then later r...

    work page

  59. [59]

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    work page

  60. [60]

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