{"id":"f2874436-414d-4146-9a7c-8e8fc9a209f5","arxiv_id":"1908.03767","paper_version":1,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":5.0,"correctness_risk":"low","formal_verification":"none","parameter_count":2,"one_line_summary":"In a gravitationally stretched capillary jet the optimal forcing frequency rises as the forcing amplitude falls, and a resolvent analysis with a simple gain threshold predicts both the optimal frequency and the breakup length.","lead":"Gravity-stretched liquid jets break into drops fastest at a forcing frequency that shifts when the forcing strength changes, breaking the usual rule that the best frequency is amplitude-independent. A linear response analysis plus a simple breakup rule reproduces the effect and predicts the breakup length where the standard slow-variation approximation fails.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The predicted l_c and omega_opt in Fig. 13 depend on the unspecified Q norm and the order-one threshold 1/epsilon in Eq. (4.34); a single parameter set is not enough to show this is not a coincidence.","rationale":"The reader's weakest assumption identifies the same load-bearing concern: the breakup-threshold premise of Eq. (4.34). I agree with that identification. The paper's nonlinear simulations are well validated and independently establish the qualitative amplitude dependence of omega_opt, including its absence for Boin = 0. The soft step is the resolvent-plus-threshold scheme that converts linear gains into quantitative predictions of l_c and omega_opt. The threshold constant is not derived, the energy norm Q is not specified, and the test is restricted to one parameter set, so the close agreement in Fig. 13 could be sensitive to choices that the paper does not expose. A direct check of whether the threshold holds at the simulated breakup points, plus a sensitivity sweep of the threshold constant, would settle whether the quantitative claim is robust. Since the reader's conditional verdict already reflects this uncertainty, my read does not require changing the verdict.","tokens_in":28440,"tokens_out":13189,"duration_ms":146209,"concrete_test":"For each epsilon in Fig. 13, take the simulated l_c and simulated omega_opt and compute the resolvent gain G_{h,fu}(omega_opt, l_c) with the paper's Q norm; check whether it is consistently close to 1/epsilon rather than, say, h_b(l_c)/epsilon or 2/epsilon. Then rerun the Fig. 12-13 prediction with thresholds C/epsilon for C = 0.5 and C = 2, or with Q replaced by an L-infinity norm of h' normalized by the local base radius h_b(z). If omega_opt shifts by more than the frequency sweep resolution, or l_c shifts by more than the quoted simulation mismatch, the quantitative agreement is not robust to the ad hoc criterion.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The quantitative core of the paper is Eq. (4.34): breakup occurs when the resolvent envelope G_opt(L) reaches 1/epsilon, with G_opt measured in the Chebyshev energy norm defined by Q and Qf in Section 4.2.1. This criterion is an order-one heuristic, as the authors themselves state when attributing residual differences to the ad-hoc definition of the required threshold. The gain is an integrated norm of the response over the whole domain, whereas the actual breakup condition is local: the interface radius h_b(z) + epsilon*h'(z) vanishes at z = l_c. Because h_b(l_c) < 1 for a gravitationally stretched jet, the local amplitude needed for breakup is h_b(l_c)/epsilon rather than 1/epsilon, and the integrated Q-norm is only indirectly related to that local condition. Neither Q nor Qf is specified in the text, so the threshold can be rescaled by a norm choice. The match in Fig. 13 therefore tests the combination of resolvent envelope, specific norm, and threshold 1/epsilon on a single parameter set (Ohin = 0.3, Wein = 1.75, Boin = 0.1); it does not establish that the threshold is universal. The qualitative trend that omega_opt increases as epsilon decreases is robust and independently supported by the nonlinear simulations, but the quantitative l_c and omega_opt predictions are the load-bearing claim that is least secure.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper investigates how the amplitude of an externally imposed harmonic inlet forcing selects the optimal frequency and breakup length of a gravitationally stretched capillary jet in the jetting regime. The authors use nonlinear simulations of the one-dimensional Eggers & Dupont equations for a jet with Ohin=0.3, Wein=1.75, Boin=0.1 and forcing amplitudes epsilon=10^-2 to 10^-8. The simulations show that lowering the forcing amplitude increases both the breakup length and the optimal forcing frequency, unlike the case Boin=0 where the optimal frequency is amplitude-independent. A local spatial and WKBJ stability analysis fails to reproduce the simulation trends quantitatively. The authors then perform a global resolvent analysis of the linearized operator of the stretched base state. They introduce the envelope G_opt(L) of the maximal resolvent gain over frequency as a function of domain length, and propose the breakup criterion G_opt(lc)=1/epsilon (Eq. 4.34). With this criterion, the resolvent analysis predicts both the optimal frequency and the breakup length as functions of epsilon, in good agreement with the nonlinear simulations (Fig. 13). The paper also compares the jet response under white-noise inlet forcing with the expected response at the optimal frequency.","tokens_in":28637,"tokens_out":3107,"duration_ms":34761,"significance":"If the quantitative predictions are robust, the paper makes a useful contribution by showing that resolvent analysis, supplemented by an amplitude-dependent breakup criterion, can capture the amplitude sensitivity of the preferred frequency in a non-parallel capillary jet, a result that goes beyond the constant-noise-level WKBJ approach of Le Dizès & Villermaux (2017). The strengths of the manuscript are the careful validation of the nonlinear code against van Hoeve et al. (2010) (Appendix 7.3), of the base state and global stability solver against Rubio-Rubio et al. (2013) (Appendices 7.1 and 7.6), and the direct nonlinear evidence for the amplitude dependence of the optimal frequency in Fig. 3. The white-noise analysis in Section 5 provides an additional, falsifiable characterization of the jet response. The central qualitative claim — that omega_opt increases as epsilon decreases in gravity-stretched jets — is convincingly supported by the nonlinear simulations, independent of the resolvent model.","major_comments":[{"comment":"The quantitative predictions of Fig. 13 rest on the gain norm through the weight matrices Q and Qf, but these matrices are never specified. Equation (4.29) defines the gain as ||s||^2 = s^† Q s and ||f||^2 = f^† Qf f, yet neither Q nor Qf is given in the text or appendices. Because the Chebyshev discretization and the choice of quadrature weights determine the numerical value of G_opt(L), the crossing condition G_opt(lc)=1/epsilon can be rescaled by an arbitrary norm choice. The authors should provide the explicit expressions for Q and Qf (or a precise reference to the convention used), and report the sensitivity of omega_opt and lc to this choice.","section":"§4.2.1 and Eq. (4.34)"},{"comment":"The breakup criterion is an order-one heuristic stated as 'nonlinearity enters the system when a small perturbation epsilon gives rise to a response of the order of 1.' However, the physically local breakup condition is h_b(lc) + epsilon h'(lc) = 0, which requires the local perturbation amplitude to reach h_b(lc)/epsilon, not 1/epsilon; h_b(lc) is smaller than unity in a gravitationally stretched jet. The integrated resolvent norm is only indirectly related to this local condition, so the threshold constant in Eq. (4.34) is not obviously universal. Since both lc and omega_opt are determined by the crossing of G_opt(L) with 1/epsilon, the authors should justify the order-one constant, test the sensitivity of the predicted lc and omega_opt to changing the constant by an O(1) factor, and preferably compare the integrated norm at lc with the local interface amplitude.","section":"§4.2.3, Eq. (4.34)"},{"comment":"The resolvent predictions are compared with nonlinear simulations for a single parameter set (Ohin=0.3, Wein=1.75, Boin=0.1) and for one combination of forcing and gain definitions (Gh,fu). The agreement in Fig. 13 therefore tests only one point in parameter space. The qualitative trend of rising omega_opt with decreasing epsilon is robust and independently supported, but the quantitative match could be coincidental given the unspecified norm and threshold. To establish the predictive claim, the authors should show at least one additional case with a different Bond or Weber number, or otherwise demonstrate that the threshold constant and the norm choice are not tuned to this single case.","section":"§4.2.3 and Fig. 13"},{"comment":"In solving the resolvent equation (4.28), the outlet boundary condition is replaced by the local spatial response k_+(omega,L), and the authors state that this is imposed on a larger domain L' > L so that the response over L is independent of the boundary condition. No convergence data or figure is provided to support this assertion. Because G_opt(L) is the central quantity in the breakup criterion, the independence of the computed gain from L' and from the specific outlet closure should be demonstrated explicitly.","section":"§4.2.1, outlet boundary condition"}],"minor_comments":[{"comment":"The caption contains the doubled article: 'the the breakup length is the minimum'; please correct.","section":"Fig. 1 caption"},{"comment":"The variables Ohin and Boin are introduced as controlling parameters, but the text later uses Ohz and Wez for local quantities; please define once and keep notation consistent throughout.","section":"Section 1 and Eq. (1.3b)"},{"comment":"The word 'downstream' is misspelled as 'donwstream' in the sentence introducing the outlet boundary condition.","section":"Section 4.2.1, Eq. (4.28)"},{"comment":"The forcing frequency range is written as 'omega = [0.4− 3.2]'; please clarify whether this is the sampled discrete set or a continuous interval, and how many frequencies were simulated for each amplitude.","section":"Section 2.3"},{"comment":"The caption says 'The black circles represent the data from numerical simulations,' but the figure appears to show curves for each epsilon rather than isolated circles; please clarify the plotting conventions.","section":"Fig. 3 caption"},{"comment":"The reference to 'Mantiˇ c-Lugo & Gallaire (2016)' uses an unusual accent placement; please check the spelling and formatting against the journal style.","section":"References"}],"recommendation":"major_revision","confidential_remarks":null},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things to know upfront. The headline phenomenon — that the optimal forcing frequency for a gravity-stretched jet depends on forcing amplitude — was already reported in Lizzi 2016, and the authors say so. What is genuinely new is the systematic nonlinear simulation sweep from ε=10^-2 down to 10^-8, the quantitative demonstration that the WKBJ approximation fails for this strongly non-parallel jet, and the resolvent-based scheme that predicts both ω_opt and breakup length l_c from linear analysis combined with a simple breakup threshold.\n\nWhat the paper does well: the numerical work is careful. The 1D Eggers–Dupont solver is validated against van Hoeve et al., the base state against Rubio-Rubio et al., and the global stability code against known critical Weber numbers. The nonlinear simulations are clean and the amplitude trend is directly visible in Fig. 3: ω_opt rises from 1.38 at ε=10^-2 to around 1.68 at 10^-4 and keeps climbing as ε falls. The WKBJ failure is a useful cautionary result, and the white-noise section adds a nice point about the jet's dominant response being close to the optimal-frequency response.\n\nThe soft spot is exactly where the stress-test note points: the quantitative predictive scheme rests on Eq. (4.34), G_opt(l_c) = 1/ε, an order-one threshold that the authors themselves call ad hoc. The gain is an integrated norm over the whole domain, not the local interface amplitude at breakup, so the link between the norm hitting 1/ε and the actual pinch-off condition h_b(l_c)+εh'(l_c)=0 is indirect. More importantly for reproducibility, the Q and Q_f matrices that define the norm are not specified, so the gain values could be rescaled by a different norm choice. And everything quantitative is tested on one parameter set (Oh_in=0.3, We_in=1.75, Bo_in=0.1). The qualitative trend is robust — it appears directly in the nonlinear simulations and would survive any reasonable threshold change — but the quantitative match in Fig. 13 could be partly coincidental.\n\nMinor issues: Section 5 says ω_opt=1.65 for ε=10^-4 while the main text and Fig. 1 say 1.68, and the frequency sweep resolution in the simulations is never stated. The intro calls the resolvent analysis \"self-sufficient,\" which overstates it since the threshold criterion is external input.\n\nWho this is for: anyone working on capillary jet breakup, non-modal stability of stretched jets, or resolvent-based prediction of breakup. The nonlinear data and the WKBJ failure result are citable even if the threshold scheme needs more work. It deserves a serious referee — the central idea is sound and honest, but the norm and threshold need specification and at least one additional parameter set before the quantitative claim is fully convincing.","headline":"Solid nonlinear study of a stretched capillary jet with an appealing but under-specified resolvent threshold criterion; worth a serious referee.","tokens_in":29290,"tokens_out":2019,"would_cite":true,"duration_ms":22900,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":["47.20.Ma","47.55.D-"],"model":"deepseek-v4-flash","headline":"For a gravity-stretched capillary jet, the optimal forcing frequency depends strongly on forcing amplitude, and a linear resolvent analysis predicts both the frequency and the breakup length.","keywords":["capillary jet","gravity-stretched jet","jetting regime","resolvent analysis","optimal forcing frequency","breakup length","WKBJ approximation","forcing amplitude"],"falsifier":"Measure experimentally the breakup length and dominant drop frequency of a gravity-stretched jet with $Oh_{in}=0.3$, $We_{in}=1.75$, $Bo_{in}=0.1$ for amplitudes from $10^{-2}$ down to $10^{-8}$; the resolvent-plus-threshold predicts a monotone rise of $\\omega_{\\mathrm{opt}}$ and $l_c$ with no saturation, so observing a plateau or an abrupt deviation would directly falsify the criterion.","tokens_in":28094,"feed_emoji":"💧","tokens_out":8744,"duration_ms":75827,"temperature":0.7,"pith_summary":"The paper studies a vertical capillary jet that gravity stretches and thins as it falls, in the regime where the jet is globally stable and acts as a noise amplifier. It claims that, contrary to what standard linear stability theory assumes, the forcing frequency that produces the shortest breakup length is strongly amplitude-dependent: as the forcing amplitude drops, both the optimal frequency and the intact jet length increase. The paper shows that a local WKBJ analysis misses the nonlinear simulation results quantitatively, whereas a global resolvent analysis of the linearized one-dimensional slender-jet equations, together with the simple criterion that breakup occurs when the linear gain reaches $1/\\epsilon$, reproduces the simulated optimal frequency and breakup length. The practical upshot is a route from the imposed forcing amplitude to the drop-formation frequency and jet length without running a nonlinear calculation.","feed_headline":"Gravity-stretched jets prefer higher frequencies when forcing weakens","feed_subtitle":"A linear resolvent analysis predicts both the optimal frequency and breakup length from the forcing amplitude alone","key_machinery":"The central object is the resolvent gain $G_{h,f_u}(\\omega,L)$ of the linearized one-dimensional slender-jet operator: the energy norm of the harmonic response in the interface height to unit inlet velocity forcing, measured in a Chebyshev-collocation energy norm with weight matrices $Q$ and $Q_f$. For each domain size $L$, the envelope $G_{\\mathrm{opt}}(L)=\\max_\\omega G_{h,f_u}(\\omega,L)$ of the dominant frequency is built; the breakup-threshold identity $G_{\\mathrm{opt}}(l_c)=1/\\epsilon$ then fixes the breakup length and, through the frequency at which the envelope is crossed, the optimal forcing frequency. The WKBJ amplitude equation serves as the comparison baseline: it improves the zeroth-order spatial gain but still underpredicts the gain magnitude, which is why the resolvent analysis is required for quantitative agreement.","core_discovery":"Using the one-dimensional slender-jet approximation for an axisymmetric jet with inlet parameters $Oh_{in}=0.3$, $We_{in}=1.75$, $Bo_{in}=0.1$, the authors find in nonlinear simulations that the optimal forcing frequency rises from about $1.38$ at amplitude $10^{-2}$ to about $1.68$ at $10^{-4}$, and keeps increasing without visible saturation down to $10^{-8}$, while the breakup length grows with decreasing amplitude. They then construct the dominant-frequency envelope $G_{\\mathrm{opt}}(L)$ of the resolvent gain as a function of domain size and impose the order-one breakup criterion $G_{\\mathrm{opt}}(l_c)=1/\\epsilon$. The crossing of the envelope with the horizontal line $1/\\epsilon$ selects both the breakup length $l_c$ and the amplitude-dependent optimal frequency $\\omega_{\\mathrm{opt}}$ without any input from nonlinear simulations, and the predicted values match the simulations. The amplitude dependence disappears when gravity is removed ($Bo_{in}=0$), where $\\omega_{\\mathrm{opt}}$ remains fixed as the amplitude varies, showing that the effect is caused by the gravitational stretching of the base flow. Under band-limited white-noise forcing, the jet's breakup statistics remain peaked near the quantities associated with the optimal frequency at the same amplitude.","pith_inferences":["The threshold constant in $G_{\\mathrm{opt}}(l_c)=1/\\epsilon$ is chosen ad hoc as order one; a different constant would shift $l_c$ and $\\omega_{\\mathrm{opt}}$ quantitatively, but any monotone threshold would preserve the qualitative rise of $\\omega_{\\mathrm{opt}}$ as $\\epsilon$ falls.","The same envelope-crossing construction could be applied to other spatially developing, convectively unstable flows—stretched viscous threads, electrically forced jets, or spinning fibres—wherever the base flow stretches perturbation wavelengths downstream; the amplitude-dependence of the preferred frequency would appear whenever the forcing level changes.","The lack of saturation at $10^{-8}$ is falsifiable: experiments or simulations at even smaller amplitudes should show the optimal frequency still climbing if the mechanism is right, and saturating if a competing nonlinear selection or noise floor takes over.","The oscillation of the breakup period between main and satellite drop groups suggests that the timing statistics, not just the mean period, carry information about the forcing frequency and could serve as a sensitive experimental probe."],"forward_implications":["For the jet with $Oh_{in}=0.3$, $We_{in}=1.75$, $Bo_{in}=0.1$, decreasing the forcing amplitude from $10^{-2}$ to $10^{-8}$ raises both the optimal forcing frequency and the breakup length, with no saturation observed at the smallest amplitude.","The resolvent envelope alone, with the $G_{\\mathrm{opt}}(l_c)=1/\\epsilon$ criterion, predicts both quantities without nonlinear simulation input; the remaining discrepancy with simulations is attributed to this threshold criterion.","When gravity is absent ($Bo_{in}=0$), the optimal frequency is independent of amplitude, so the amplitude dependence is a distinctive consequence of gravitational stretching of the base flow.","Under broadband white-noise forcing, the drop-size and breakup-length statistics remain centered near the values obtained with the optimal frequency at the same amplitude, implying the jet selects the amplifier-preferred mode even when forced with a broad spectrum.","Because the jet is globally stable in the jetting regime, no self-sustained oscillations contaminate the response, so the linear resolvent is the appropriate global description of the amplification."],"supporting_citations":[{"why":"Supplies the one-dimensional slender-jet equations that form the model for both simulations and stability analyses.","marker":"Eggers & Dupont 1994"},{"why":"Provides the steady gravity-stretched thread base states and the global stability framework the paper validates against.","marker":"Rubio-Rubio et al. 2013"},{"why":"The prior WKBJ noise-amplification theory for capillary jet breakup that this paper goes beyond by including forcing amplitude.","marker":"Le Dizès & Villermaux 2017"},{"why":"Supplies the singular-value-decomposition optimization used to compute the optimal resolvent gain.","marker":"Marquet & Sipp 2010"},{"why":"Supplies the resolvent eigenvalue formulation for the harmonic response of incompressible jets.","marker":"Garnaud et al. 2013"},{"why":"Provides the experimentally validated breakup simulation data used to validate the numerical scheme.","marker":"van Hoeve et al. 2010"},{"why":"Supplies the WKBJ weakly nonparallel formalism and spatial branch analysis used for comparison and for outlet closure.","marker":"Huerre & Rossi 1998"}],"fun_headline_variants":["Gravity-stretched jets shift optimal frequency with forcing strength","Forcing amplitude tunes unstable frequency in falling jets","Resolvent analysis predicts amplitude-dependent jet breakup frequency","Optimal jet frequency varies with forcing amplitude under gravity"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that breakup happens at the downstream location where the linear response gain first reaches $1/\\epsilon$; if a different threshold value or a different way of measuring the response were used, the predicted frequencies and breakup lengths would change, even though the overall trend would survive.","fun_headline_variants_meta":{"raw":{"variants":["Gravity-stretched jets shift optimal frequency with forcing strength","Forcing amplitude tunes unstable frequency in falling jets","Resolvent analysis predicts amplitude-dependent jet breakup frequency","Optimal jet frequency varies with forcing amplitude under gravity"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000626,"raw_usage":{"total_tokens":2914,"prompt_tokens":981,"completion_tokens":1933,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":597,"completion_tokens_details":{"reasoning_tokens":1870}},"tokens_in":597,"tokens_out":1933,"duration_ms":16916,"temperature":1.0,"reasoning_tokens":1870,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T14:04:01.809434+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Measure experimentally the breakup length and dominant drop frequency of a gravity-stretched jet with $Oh_{in}=0.3$, $We_{in}=1.75$, $Bo_{in}=0.1$ for amplitudes from $10^{-2}$ down to $10^{-8}$; the resolvent-plus-threshold predicts a monotone rise of $\\omega_{\\mathrm{opt}}$ and $l_c$ with no saturation, so observing a plateau or an abrupt deviation would directly falsify the criterion.","supporting_citations":[{"cited_title":"& Dupont, T","cited_arxiv_id":null,"evidence_quote":"Supplies the one-dimensional slender-jet equations that form the model for both simulations and stability analyses."},{"cited_title":", Sevilla, A","cited_arxiv_id":null,"evidence_quote":"Provides the steady gravity-stretched thread base states and the global stability framework the paper validates against."},{"cited_title":"& Sipp, D","cited_arxiv_id":null,"evidence_quote":"Supplies the singular-value-decomposition optimization used to compute the optimal resolvent gain."},{"cited_title":", Lesshafft, L","cited_arxiv_id":null,"evidence_quote":"Supplies the resolvent eigenvalue formulation for the harmonic response of incompressible jets."},{"cited_title":", Gekle, S","cited_arxiv_id":null,"evidence_quote":"Provides the experimentally validated breakup simulation data used to validate the numerical scheme."},{"cited_title":"& Rossi, M","cited_arxiv_id":null,"evidence_quote":"Supplies the WKBJ weakly nonparallel formalism and spatial branch analysis used for comparison and for outlet closure."}],"review_version":1}