Pith. sign in

REVIEW 3 major objections 6 minor 40 references

A semi-analytical model predicts the full shape and height of the free-standing sheet that rises when two drops collide on a dry surface, and shows that reported Weber-number scalings are a crossover, not a single law.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

T0 review · grok-4.5

2026-07-12 07:26 UTC pith:3SI4T5YF

load-bearing objection Clean, usable extension of Gordillo–Riboux to the two-drop rising sheet; L-stage is essentially closed-form and the We-exponent scatter is explained as a stage crossover, with the only real soft spot being the openly empirical post-lamella inlet. the 3 major comments →

arxiv 2607.02730 v1 pith:3SI4T5YF submitted 2026-07-02 physics.flu-dyn

Semi-analytical model for the rising sheet generated by droplet-pair impact

Shushan Hu , Liwu Fan , Nan Hu This is my paper
classification physics.flu-dyn
keywords droplet-pair impactrising sheetlamella collisionballistic characteristicsWeber number scalingRayleigh–Plateau pinch-offsemi-analytical modelthin-film flow
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.

When two identical low-viscosity drops hit a dry surface at the same time and close enough together, their spreading films meet and throw up a free-standing vertical sheet bounded by a retracting rim. Existing descriptions of how high that sheet climbs rest on empirical fits or energy balances calibrated to simulations. This paper extends an established single-drop lamella theory to the two-drop geometry, couples the thin-film flow inside the sheet to the colliding base and the capillary rim, and solves the interior along ballistic particle paths in two stages. In the early lamella-fed stage the velocity and thickness fields are closed-form; later they are continued with inlet data taken once from simulations. The resulting centreline formulae for apex height and thickness show that the different Weber-number exponents in the literature are successive regimes of the same continuous description rather than competing universal laws. At high Weber number the same fields feed a Rayleigh–Plateau calculation that caps the height once the apex pinches off. The model therefore supplies a predictive, largely analytical account of the sheet’s shape, rise, and breakup limit over the low-Ohnesorge regime of practical spray and coating flows.

Core claim

By matching the colliding single-drop lamellae to a free sheet whose fluid parcels follow ballistic characteristics and whose rim obeys mass–momentum balances, one obtains the three-dimensional velocity and thickness fields of the rising sheet and, on the centreline, explicit algebraic relations for apex height and thickness. Those relations demonstrate that the scattered Weber-number exponents reported experimentally arise from a smooth crossover between a lamella-fed regime and a post-lamella regime rather than from any single power law. A linear Rayleigh–Plateau analysis driven by the model’s own time-dependent jet diameter and deceleration then supplies an upper bound on attainable heigh

What carries the argument

Ballistic characteristic construction of the free-sheet interior: each fluid parcel injected at the collision line keeps constant velocity while thickness evolves by mass conservation; the construction is algebraic in the lamella-fed stage and continued numerically with an empirical post-lamella inlet, then closed by rim balances and a Rayleigh–Plateau cut-off.

Load-bearing premise

After the spreading film no longer reaches the collision line, the velocity and thickness that feed the sheet are taken from simple empirical fits calibrated to the same simulations the model is later compared against; outside that fitted window the post-lamella height and the pinch-off bound lose their quantitative footing.

What would settle it

Measure maximum sheet height versus Weber number at fixed half-spacing for a liquid whose Ohnesorge number lies inside the low-viscosity range but whose post-lamella inlet velocity and thickness deviate measurably from the fitted power-law and cubic forms; if the measured heights still follow the model’s inertial and Rayleigh–Plateau curves, the empirical inlet is robust, otherwise the post-lamella prediction fails.

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

If this is right

  • Different published Weber exponents for maximum sheet height can be reconciled as successive segments of one continuous centreline solution rather than as competing scalings.
  • The full three-dimensional sheet shape, not only apex height, becomes available from a single rim integration once the inlet is known.
  • At high Weber number the Rayleigh–Plateau cut-off supplies a parameter-free upper bound on height once the apex jet diameter and deceleration are taken from the sheet model.
  • The same characteristic-plus-rim construction can be reused for non-simultaneous or unequal-drop impacts once the appropriate collision-line inlet is supplied.

Where Pith is reading between the lines

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

  • Because the post-lamella inlet is the sole empirical input, any future closed-form description of the residual bulk convergence after the lamella has passed would render the entire sheet dynamics analytical.
  • The framework already separates geometry (ballistic paths, rim balances) from feeding; the same separation should apply directly to multi-drop arrays once pairwise collision lines are identified.
  • The observed crossover in Weber scaling suggests that earlier energy-balance models may have been sampling different stages of the same continuous process rather than capturing distinct physics.

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

3 major / 6 minor

Summary. The paper develops a semi-analytical model for the free-standing vertical sheet formed when two low-Oh drops impact a dry substrate simultaneously. Extending the single-drop lamella theory of Gordillo et al. (2019), the authors couple thin-film sheet flow to colliding-lamella inlet conditions and a capillary-retarded rim, then solve the sheet interior along ballistic characteristics in a lamella-fed (L) stage (closed-form velocity and thickness) and a post-lamella (PL) stage (empirical inlet from DNS). The framework yields three-dimensional sheet fields and the full side-view shape; on the centreline it supplies explicit apex height and thickness, attributing literature scatter in We exponents to an L/PL crossover rather than a single power law. At high We a linear Rayleigh–Plateau analysis, driven by the model’s time-dependent jet diameter and deceleration, bounds the maximum height and closes the pinch-off regime. Predictions are compared with Goswami–Hardalupas experiments and the authors’ Basilisk DNS over We, a and modest Oh.

Significance. If the results hold, the work supplies the first predictive, largely first-principles description of the unsteady central sheet that has been the defining feature of simultaneous two-drop impact since Barnes et al. The L-stage construction is essentially parameter-free once the Gordillo lamella solution is accepted, yields closed-form centreline fields (Eqs. 20–25), and cleanly explains why reported We exponents differ. The full side-view profiles (Figs. 3–4) and H(t) trajectories (Fig. 5) match independent experiments as well as DNS, and the RP cut-off (Eqs. 31–33) is a falsifiable, literature-threshold closure with no additional tuning. These strengths—analytic L-stage fields, explicit crossover interpretation, and a parameter-light capillary bound—make the paper a useful advance for multi-drop impact, spray cooling and related applications, even though the PL inlet remains empirical.

major comments (3)
  1. [Appendix B, Eq. (1), §§IV B–V] Appendix B and Eq. (1): the post-lamella inlet (ū_PL = C/t^ζ with ζ = 1.2 fixed, and the four-coefficient cubic T_P(t)) is extracted from the same Basilisk DNS suite later used for validation of H_m and sheet shape in the L+PL regime (Figs. 5–7). Consequently part of the ‘prediction’ of H_m via Eqs. (28)–(30) and of H_RP via Eqs. (31)–(33) is conditioned on those data. The manuscript should (i) state explicitly which comparisons are independent of the PL fit (experiments of Goswami & Hardalupas; pure L-stage cases) versus which reuse the calibration DNS, (ii) report a sensitivity study of H_m and H_RP to modest variations of ζ and the T_P coefficients, and (iii) if possible hold out at least one (We, a) case from the fit and show it a posteriori. Without this, the quantitative claim for the PL and pinch-off regimes is weaker than the abstract suggests.
  2. [§IV, Figs. 6–7] §IV A, Eqs. (22)–(25) and §IV B, Eqs. (28)–(30): the central claim that ‘different Weber-number exponents reported in the literature arise from a crossover rather than from a single universal scaling law’ is attractive but only weakly demonstrated. The L-stage asymptote H_m ∼ We^{1/(1+ϕ_L)} is given, yet the paper never overlays the literature exponents (or the empirical scalings of Goswami & Hardalupas and Zhang et al.) on Fig. 6 or 7, nor does it show where each dataset sits relative to t_m ≶ t_ℓ,0. A short quantitative comparison—listing the published exponents against the model’s local effective slope d ln H_m / d ln We in the L-only, crossover and PL-dominated windows—would make the claim load-bearing rather than interpretive.
  3. [§V A, Eq. (32), Fig. 6] §V A, Eq. (32): the Rayleigh–Plateau cut-off adopts ln(r_jet/ε_0) ≈ 12 as a ‘standard value’. While the range 8–15 is cited, H_RP is exponentially sensitive to this threshold. The manuscript should either (a) show that H_RP remains within the experimental scatter for the conventional interval 8–15, or (b) calibrate the constant once against a single high-We pinch-off event and then predict the remaining cases. As written, the cut-off is not fully parameter-free and the high-We bound in Fig. 6a is less robust than claimed.
minor comments (6)
  1. [§II, Appendix A] §II and Appendix A: ϕ_L is introduced as a slowly varying function of (r,t) yet is thereafter treated as a single constant read from numerics for each (We, Oh). A one-sentence statement of the relative variation of ϕ across the lamella at fixed Oh (the text already says <10% in We) would clarify how much error the constant-ϕ_L approximation introduces into Eqs. (20) and (25).
  2. [Fig. 2c] Figure 2c: the regime map is drawn only for water. Adding a second Oh contour (or a brief remark that the L/PL boundary t_ℓ,0 = a²/3 is Oh-independent while s_max is not) would help readers place the glycerol–water cases of Appendix D.
  3. [§III A] §III A: gravity is neglected on the basis of We^{1/2} Fr^{-2} ≲ 0.1, yet late-time descent is visible in both experiment and DNS (Figs. 3–5). A short estimate of the cumulative gravitational deceleration over t ∼ 1–5 would quantify when the ballistic assumption begins to fail and would justify the rising-phase-only comparisons.
  4. [§III A, Eq. (1)] Notation: lower-case (ū, h, s, b) for lamella quantities and upper-case (W̄, T, H, B) for sheet quantities is helpful, but T_0, W̄_0, V̄_0 in Eq. (1) and T_b, T_r later are easy to confuse with the sheet thickness field T(y,z,t). A compact symbol table in §III A would reduce cognitive load.
  5. [Appendix D] Appendix D, Fig. 9: the Oh = 0.0141 comparison is valuable; stating explicitly that the same PL inlet coefficients (no re-fit) were used would strengthen the claim that the model is not retuned outside the water window.
  6. [Throughout / Acknowledgements] Typos / style: ‘§,III’ (p. 3) should be ‘§III’; ‘theycomponents’ / ‘they–zplane’ missing spaces appear in several places; ‘DeepSeek for assistance with spell-checking’ in the acknowledgements is unusual for a journal and may be better omitted or rephrased.

Circularity Check

1 steps flagged

Partial circularity confined to the post-lamella inlet: empirical forms fitted once to the authors' DNS are then propagated to 'predict' sheet shapes and Hm against the same DNS; L-stage closed forms and external experiments remain independent.

specific steps
  1. fitted input called prediction [§III.B Eq. (1); Appendix B (Fig. 8, Eq. B1); propagated in §IV.B Eqs. (26)–(30) and validated in Figs. 3–7]
    "This flow does not provide a closed-form inlet condition for the sheet, so we prescribe the PL inlet from simulations. Specifically, we use T(y,0,t)=T_P(r,t), … ū_PL(r,t)=C/t^ζ with ζ=1.2; the constant C is chosen to ensure continuity … These inlet forms are empirical … a given PL inlet fit is valid only over the range of conditions from which it was obtained. … T_P(t)=- 0.00396 t^3 + 0.01298 t^2 + 0.11063 t - 0.02427."

    The common cubic T_P(t) and power-law form (fixed ζ=1.2) are fitted to the authors' DNS. The identical DNS suite is then used as the primary validation target for the full sheet profiles, apex trajectories H(t) and maximum heights Hm obtained by integrating the ballistic characteristics and rim balances that are driven by this inlet. Quantitative success in the PL regime is therefore partly by construction of the inlet rather than an independent prediction, even though C is continuity-fixed and external experiments provide partial external corroboration.

full rationale

The lamella-fed stage is derived from the external single-drop solution of Gordillo et al. (ballistic characteristics, conserved quantity T t^{1+φ_L}, algebraic stall condition (24)), with only a weak, case-wise numerical read-off of the attenuation factor φ_L; this part is not circular. The post-lamella stage, by contrast, replaces the missing closed-form inlet with forms (ū_PL = C/t^ζ, ζ = 1.2 fixed; cubic T_P(t)) extracted and fitted from the authors' own DNS suite (Appendix B). Those same DNS later serve as the principal quantitative benchmark for the predicted three-dimensional profiles, centreline H(t) and Hm (Figs. 3–7 and Eqs. 28–30). Agreement in the PL-dominated and high-We regimes is therefore partly conditioned on the fitted inlet rather than being a pure first-principles prediction. The paper is transparent about the empiricism, C is fixed by continuity rather than free fitting, the product T ū_PL is robust across cases, and independent experiments of Goswami & Hardalupas supply additional non-circular support. The Rayleigh–Plateau cut-off uses only literature constants once the model jet is supplied. No self-definitional loops, load-bearing self-citations, uniqueness imports or renaming of known results appear. Overall circularity is therefore moderate and localized to the PL inlet, justifying score 4.

Axiom & Free-Parameter Ledger

5 free parameters · 5 axioms · 0 invented entities

The model rests on the established single-drop lamella solution of Gordillo et al. (2019), standard thin-film and free-rim balances, and a small set of empirical constants needed only for the post-lamella inlet and the capillary cut-off. No new physical entities are postulated; the free parameters are ordinary fit coefficients whose limited range of validity is acknowledged.

free parameters (5)
  • ζ (post-lamella velocity decay exponent) = 1.2
    Fixed at 1.2 by inspection of DNS inlet velocity; not derived from first principles.
  • C (post-lamella velocity prefactor) = case-dependent continuity value
    Set by continuity with the lamella-fed velocity at t_ℓ rather than by a free fit, but still taken from the simulation suite.
  • T_P(t) cubic coefficients = -0.00396, 0.01298, 0.11063, -0.02427
    Four numerical coefficients of the cubic polynomial for post-lamella thickness, fitted once to the DNS inlet data (Appendix B).
  • ln(r_jet/ε_0) = 12
    Standard capillary-jet breakup threshold taken as ≈12 (literature range 8–15); not measured in the present experiments.
  • ϕ_L (viscous attenuation factor) = O(1) value <1, case-dependent
    Treated as a constant read from the single-drop numerical solution for each (We, Oh); small variation with We is neglected.
axioms (5)
  • domain assumption Early spreading of each drop is unaffected by the presence of the other until the rims collide; finite coalescence time is neglected.
    Stated in §III A; allows direct inheritance of the single-drop lamella solution as inlet.
  • domain assumption After collision the normal (x) momentum is redirected vertically while the tangential (y) component is preserved; fluid parcels inside the free sheet follow ballistic trajectories (DV/Dt = 0).
    §III A–B; standard for high-Re free sheets but excludes gravity and residual pressure gradients.
  • domain assumption Gravity is negligible during the rising phase (We^{1/2} Fr^{-2} ≲ 0.1).
    §III A; causes the late-time descent seen in experiment/DNS to be missed.
  • domain assumption The single-drop lamella velocity and thickness of Gordillo et al. (2019), including the first-order viscous correction ϕ, remain valid up to the collision line.
    §II and Appendix A; the entire L-stage solution is built on this inheritance.
  • domain assumption Quasi-steady Rayleigh–Plateau dispersion relation evaluated at the local jet radius supplies the breakup time once the integrated growth reaches ln(r/ε_0) ≈ 12.
    §V A; standard linear stability closure for capillary jets.

pith-pipeline@v1.1.0-grok45 · 25131 in / 3323 out tokens · 33374 ms · 2026-07-12T07:26:08.246067+00:00 · methodology

0 comments
read the original abstract

When two low-Ohnesorge-number drops impact a dry substrate simultaneously, their spreading lamellae collide and lift a free-standing vertical sheet. The sheet grows by inertial feeding from the spreading drops and is pulled back by capillary retraction at its rim. We develop a semi-analytical model for this rising sheet by extending the single-drop impact description of~\citet{Gordillo2019} to the two-drop geometry. The thin-film flow in the sheet is coupled at its base to the colliding lamellae and at its apex to a capillary-retarded rim. The sheet interior is then solved along ballistic characteristics in two stages: a lamella-fed stage, for which the velocity and thickness fields can be obtained in closed form, and a post-lamella stage, for which the inlet conditions are taken from simulations. The resulting framework gives the three-dimensional velocity and thickness fields and therefore the full sheet shape. On the centreline, the apex height and local thickness are obtained explicitly, showing that the different Weber-number exponents reported in the literature arise from a crossover rather than from a single universal scaling law. At sufficiently large Weber number, the apex pinches off. A linear Rayleigh--Plateau analysis, using the time-dependent jet diameter and deceleration predicted by the model, then bounds the maximum attainable height and closes the description of the pinch-off regime.

Figures

Figures reproduced from arXiv: 2607.02730 by Liwu Fan, Nan Hu, Shushan Hu.

Figure 1
Figure 1. Figure 1: FIG. 1. Schematic of droplet impact. ( [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. ( [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Side-view profiles of the central sheet at four Weber numbers [PITH_FULL_IMAGE:figures/full_fig_p014_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Side-view profiles of the central sheet at four half-spacings [PITH_FULL_IMAGE:figures/full_fig_p015_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. All results in this figure correspond to water ( [PITH_FULL_IMAGE:figures/full_fig_p020_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Maximum central-sheet height [PITH_FULL_IMAGE:figures/full_fig_p021_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Maximum central-sheet height [PITH_FULL_IMAGE:figures/full_fig_p023_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. Post-lamella inlet for four representative cases ( [PITH_FULL_IMAGE:figures/full_fig_p028_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. Validation against the glycerol–water dataset of Goswami and Hardalupas [ [PITH_FULL_IMAGE:figures/full_fig_p031_9.png] view at source ↗

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

40 extracted references · 7 canonical work pages

  1. [1]

    Goswami and Hardalupas [19] subsequently extended this characterization over nearly two decades in liquid viscosity, corresponding to Ohnesorge numbers from 0.002 to 0.177

    measured the time-dependent sheet height, width, and thickness over a range of Weber numbers and drop spacings, and proposed an empirical scaling for the maximum height. Goswami and Hardalupas [19] subsequently extended this characterization over nearly two decades in liquid viscosity, corresponding to Ohnesorge numbers from 0.002 to 0.177. They reported ...

  2. [2]

    The sheet is fed through the planez= 0, where the values ofT, ¯V, and ¯Ware prescribed by the stage-dependent conditions (1)

    Sheet domain We first describe the velocity and thickness fields in the sheet interior, away from the bounding rim. The sheet is fed through the planez= 0, where the values ofT, ¯V, and ¯Ware prescribed by the stage-dependent conditions (1). After entering the free sheet, the fluid is assumed to experience neither viscous friction nor gravitational decele...

  3. [3]

    We describe the rim by following material elements labelled by their source coordinatey∈[−S, S]

    Rim domain The sheet solution provides the velocity and thickness incident on the rim. We describe the rim by following material elements labelled by their source coordinatey∈[−S, S]. The element labelled byyhas position (Y r, Zr), velocityV r = (Vr, Wr), and cross-sectional diameterB(y, t). The derivative d/dtis taken at fixedy. For a giveny, the directi...

  4. [4]

    Comparison with experiment and simulation The sheet fields of§III C 1 and the rim balances of§III C 2 together close the model. The sheet solution of§III C 1 is obtained separately and enters the rim problem as a known field, so the variables advanced in time are the rim position, velocity and cross-section, governed by (13)–(15) with d(Y r, Zr)/dt=V r; i...

  5. [5]

    At this lower Weber number, the central sheet is relatively shallow, and the experimental side view is often partially obscured by the drop bulk

    Figure 4 compares the predicted profiles with the simulations and with the experiments of Goswami and Hardalupas [18]. At this lower Weber number, the central sheet is relatively shallow, and the experimental side view is often partially obscured by the drop bulk. The comparison therefore relies mainly on the simulations, while the experimental contour is...

    2000

  6. [6]

    viscous dissipation during the rise

    (blue for water, grey for the mixture); red lines are the present theory. viscous dissipation during the rise. The predicted contours nonetheless track the measured ones in both rows, so the model captures this viscous reshaping without any retuning of its parameters. Figure 9c–efollows the apex heightH(t) for the mixture across the full rise. The model r...

  7. [7]

    J. M. Gordillo, G. Riboux, and E. S. Quintero, A theory on the spreading of impacting droplets, Journal of Fluid Mechanics866, 298–315 (2019)

    2019

  8. [8]

    Yarin, Drop impact dynamics: Splashing, spreading, receding, bouncing

    A. Yarin, Drop impact dynamics: Splashing, spreading, receding, bouncing. . . , Annual Review 31 of Fluid Mechanics38, 159–192 (2006)

    2006

  9. [9]

    Josserand and S

    C. Josserand and S. Thoroddsen, Drop impact on a solid surface, Annual Review of Fluid Mechanics48, 365–391 (2016)

    2016

  10. [10]

    Cheng, T.-P

    X. Cheng, T.-P. Sun, and L. Gordillo, Drop impact dynamics: Impact force and stress distri- butions, Annual Review of Fluid Mechanics54, 57–81 (2022)

    2022

  11. [11]

    Clanet, C

    C. Clanet, C. B´ eguin, D. Richard, and D. Qu´ er´ e, Maximal deformation of an impacting drop, Journal of Fluid Mechanics517, 199 (2004)

    2004

  12. [12]

    I. V. Roisman, Inertia dominated drop collisions. ii. an analytical solution of the navier–stokes equations for a spreading viscous film, Physics of Fluids21, 10.1063/1.3129283 (2009)

    doi:10.1063/1.3129283 2009

  13. [13]

    Eggers, M

    J. Eggers, M. A. Fontelos, C. Josserand, and S. Zaleski, Drop dynamics after impact on a solid wall: Theory and simulations, Physics of Fluids22, 10.1063/1.3432498 (2010)

    work page doi:10.1063/1.3432498 2010

  14. [14]

    Riboux and J

    G. Riboux and J. M. Gordillo, Experiments of drops impacting a smooth solid surface: A model of the critical impact speed for drop splashing, Physical Review Letters113, 10.1103/phys- revlett.113.024507 (2014)

    doi:10.1103/phys- 2014

  15. [15]

    Wildeman, C

    S. Wildeman, C. W. Visser, C. Sun, and D. Lohse, On the spreading of impacting drops, Journal of Fluid Mechanics805, 636–655 (2016)

    2016

  16. [16]

    Sanjay and D

    V. Sanjay and D. Lohse, Unifying theory of scaling in drop impact: Forces and maximum spreading diameter, Physical Review Letters134, 10.1103/physrevlett.134.104003 (2025)

    doi:10.1103/physrevlett.134.104003 2025

  17. [17]

    Liang and I

    G. Liang and I. Mudawar, Review of mass and momentum interactions during drop impact on a liquid film, International Journal of Heat and Mass Transfer101, 577–599 (2016)

    2016

  18. [18]

    A. L. Yarin, I. V. Roisman, and C. Tropea,Collision Phenomena in Liquids and Solids(Cam- bridge University Press, Cambridge, 2017)

    2017

  19. [19]

    Breitenbach, I

    J. Breitenbach, I. V. Roisman, and C. Tropea, From drop impact physics to spray cooling models: a critical review, Experiments in Fluids59, 10.1007/s00348-018-2514-3 (2018)

    work page doi:10.1007/s00348-018-2514-3 2018

  20. [20]

    Moreira, A

    A. Moreira, A. Moita, and M. Pan˜ ao, Advances and challenges in explaining fuel spray im- pingement: How much of single droplet impact research is useful?, Progress in Energy and Combustion Science36, 554–580 (2010)

    2010

  21. [21]

    H. A. Barnes, Y. Hardalupas, A. M. K. P. Taylor, and J. H. Wilkins, An investigation of the interaction between two adjacent impinging droplets, inProceedings of the 15th Interna- tional Conference on Liquid Atomisation and Spray Systems (ILASS), edited by G. Lavergne (ONERA, Toulouse, France, 1999) pp. 1–7. 32

    1999

  22. [22]

    Roisman, B

    I. Roisman, B. Prunet-Foch, C. Tropea, and M. Vignes-Adler, Multiple drop impact onto a dry solid substrate, Journal of Colloid and Interface Science256, 396–410 (2002)

    2002

  23. [23]

    N. E. Ersoy and M. Eslamian, Central uprising sheet in simultaneous and near-simultaneous impact of two high kinetic energy droplets onto dry surface and thin liquid film, Physics of Fluids32, 10.1063/1.5135029 (2020)

    work page doi:10.1063/1.5135029 2020

  24. [24]

    Goswami and Y

    A. Goswami and Y. Hardalupas, Simultaneous impact of droplet pairs on solid surfaces, Jour- nal of Fluid Mechanics961, 10.1017/jfm.2023.249 (2023)

    work page doi:10.1017/jfm.2023.249 2023

  25. [25]

    Goswami and Y

    A. Goswami and Y. Hardalupas, On the role of liquid viscosity during droplet-pair impacts on solid surfaces, Journal of Fluid Mechanics1033, 10.1017/jfm.2026.11437 (2026)

    work page doi:10.1017/jfm.2026.11437 2026

  26. [26]

    Zhang, A

    Z. Zhang, A. A. Castrejon-Pita, and W. Mostert, Numerical simulations of simultaneous pair- drop impacts and their energetics (2026)

    2026

  27. [27]

    H. Wagner, ¨Uber stoß- und gleitvorg¨ ange an der oberfl¨ ache von fl¨ ussigkeiten, ZAMM - Jour- nal of Applied Mathematics and Mechanics / Zeitschrift f¨ ur Angewandte Mathematik und Mechanik12, 193–215 (1932)

    1932

  28. [28]

    Riboux and J

    G. Riboux and J. M. Gordillo, Maximum drop radius and critical weber number for splashing in the dynamical leidenfrost regime, Journal of Fluid Mechanics803, 516–527 (2016)

    2016

  29. [29]

    Hasson and R

    D. Hasson and R. E. Peck, Thickness distribution in a sheet formed by impinging jets, AIChE Journal10, 752–754 (1964)

    1964

  30. [30]

    Bremond and E

    N. Bremond and E. Villermaux, Atomization by jet impact, Journal of Fluid Mechanics549, 273–306 (2006)

    2006

  31. [31]

    Rayleigh, On the instability of jets, Proceedings of the London Mathematical Societys1-10, 4–13 (1878)

    L. Rayleigh, On the instability of jets, Proceedings of the London Mathematical Societys1-10, 4–13 (1878)

  32. [32]

    Chandrasekhar,Hydrodynamic and Hydromagnetic Stability(Oxford University Press, 1961)

    S. Chandrasekhar,Hydrodynamic and Hydromagnetic Stability(Oxford University Press, 1961)

    1961

  33. [33]

    Eggers and E

    J. Eggers and E. Villermaux, Physics of liquid jets, Reports on Progress in Physics71, 036601 (2008)

    2008

  34. [34]

    Weber, Zum zerfall eines fl¨ ussigkeitsstrahles, ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift f¨ ur Angewandte Mathematik und Mechanik11, 136–154 (1931)

    C. Weber, Zum zerfall eines fl¨ ussigkeitsstrahles, ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift f¨ ur Angewandte Mathematik und Mechanik11, 136–154 (1931)

    1931

  35. [35]

    Riboux and J

    G. Riboux and J. M. Gordillo, The diameters and velocities of the droplets ejected after splashing, Journal of Fluid Mechanics772, 630–648 (2015)

    2015

  36. [36]

    Popinet, An accurate adaptive solver for surface-tension-driven interfacial flows, Journal of 33 Computational Physics228, 5838–5866 (2009)

    S. Popinet, An accurate adaptive solver for surface-tension-driven interfacial flows, Journal of 33 Computational Physics228, 5838–5866 (2009)

    2009

  37. [37]

    Popinet, Numerical models of surface tension, Annual Review of Fluid Mechanics50, 49–75 (2018)

    S. Popinet, Numerical models of surface tension, Annual Review of Fluid Mechanics50, 49–75 (2018)

    2018

  38. [38]

    Brackbill, D

    J. Brackbill, D. Kothe, and C. Zemach, A continuum method for modeling surface tension, Journal of Computational Physics100, 335–354 (1992)

    1992

  39. [39]

    Sanjay, P

    V. Sanjay, P. Chantelot, and D. Lohse, When does an impacting drop stop bouncing?, Journal of Fluid Mechanics958, 10.1017/jfm.2023.55 (2023)

    work page doi:10.1017/jfm.2023.55 2023

  40. [40]

    Sanjay, S

    V. Sanjay, S. Lakshman, P. Chantelot, J. H. Snoeijer, and D. Lohse, Drop impact on viscous liquid films, Journal of Fluid Mechanics958, 10.1017/jfm.2023.13 (2023). 34

    work page doi:10.1017/jfm.2023.13 2023