{"id":"33fc8d93-dfe8-4cc3-ab53-1cfd4388b9f2","arxiv_id":"2607.12734","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":6.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"Homogenization of 3D NSCH with phase-dependent viscosity/mobility and holes of size ε^α (α>3) yields NSCH when capillary strength stays positive, or Stokes–Cahn–Hilliard when it vanishes.","lead":"This paper proves that 3D multiphase Navier–Stokes–Cahn–Hilliard flows through many tiny solid holes homogenize to either the original NSCH system or a Stokes–Cahn–Hilliard system, depending on capillary strength. It claims the first rigorous homogenization for evolutionary NSCH with phase-dependent viscosity and mobility under subcritical hole sizes.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.5","headline":"No significant objection identified beyond the abstract-only limitation already reflected in the UNVERDICTED verdict.","rationale":"The Reader's weakest_assumption correctly isolates the subcritical geometric condition that distinguishes the claimed pure limits from the critical/supercritical regimes that typically produce extra terms. The abstract presents a standard, well-posed homogenization statement for this literature; nothing in the available text contradicts it or reveals a circularity. With only the abstract, soundness cannot be confirmed, so the UNVERDICTED / LOW-confidence posture is appropriate and should not be altered. The concrete test above is the natural first check once the full proofs are accessible; until then no stronger objection can be substantiated.","tokens_in":2012,"tokens_out":454,"duration_ms":4805,"concrete_test":"When the full preprint becomes available, verify that the a-priori estimates and the passage to the limit in the convective and capillary terms (especially the phase-dependent viscosity and mobility) close without residual surface measures or friction terms precisely when α > 3; if residual terms appear for any α > 3, the pure-limit claim fails.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The Reader correctly flags that the geometric restriction α > 3 is load-bearing for the claimed pure NSCH / SCH limits (no extra friction or capacity terms). That restriction is stated explicitly in the abstract as the setting of the whole analysis, and the two-regime claim (NSCH when λ_ε \to λ > 0; SCH when λ_ε \to 0) is a coherent, non-circular statement of the expected type for subcritical homogenization of evolutionary NSCH with phase-dependent viscosity and mobility. Because the full text is unavailable, no internal inconsistency, hidden assumption, or gap in the estimates can be located; the only genuine limitation is the absence of the proofs themselves. Manufacturing a further technical concern from the abstract alone would be speculative rather than load-bearing.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.5","summary":"The manuscript studies homogenization of the three-dimensional Navier–Stokes–Cahn–Hilliard (NSCH) system in a domain perforated by many solid holes of diameter O(ε^α) with α>3 (subcritical scaling). Viscosity and mobility depend on the phase field. Two regimes are claimed: if the capillary strength λ_ε → λ > 0, the limit coincides with the original NSCH system; if λ_ε → 0, the scaled velocity, phase field and chemical potential converge to a weak solution of a Stokes–Cahn–Hilliard (SCH) system. The authors present this as the first rigorous homogenization analysis for evolutionary NSCH with phase-dependent coefficients under subcritical hole scaling.","tokens_in":2147,"tokens_out":520,"duration_ms":14169,"significance":"If the analysis is correct, the result fills a clear gap: evolutionary NSCH homogenization with phase-dependent viscosity and mobility in the subcritical regime, producing pure NSCH or SCH limits without extra friction or capacity terms. The two-regime structure controlled by capillary strength is coherent and of interest for multiphase flow in porous media and for the theory of diffuse-interface models in complex geometries. The geometric restriction α>3 is load-bearing and is stated explicitly as the setting of the whole analysis.","major_comments":[],"minor_comments":[{"comment":"Only the abstract is available for this review. Notation for the perforated domain, the precise weak formulations of the limit systems, and the scaling of the velocity in the λ_ε → 0 regime are not visible; these should be stated clearly in the introduction once the full text is assessed.","section":null},{"comment":"The abstract asserts that the work is the first of its kind; a short comparison paragraph with existing homogenization results for NS, Cahn–Hilliard, or NSCH (constant or phase-dependent coefficients, critical vs subcritical holes) would help the reader locate the contribution.","section":null}],"recommendation":"uncertain","confidential_remarks":"Full text was not available; only the abstract could be inspected. Energy estimates, compactness, treatment of phase-dependent coefficients, and passage to the limit cannot be checked. The abstract statement is coherent and non-circular, and the load-bearing geometric restriction α>3 is explicit. I cannot responsibly recommend accept/minor/major/reject without the manuscript. Please supply the full text for a proper report."},"author_rebuttal":null,"desk_editor":{"model":"grok-4.5","letter":"The one thing to know is that this claims the first rigorous homogenization of the evolutionary 3D NSCH system with phase-dependent viscosity and mobility, in the subcritical small-hole regime (diameter O(ε^α), α>3). They get two regimes: if capillary strength λ_ε stays positive the limit is just the original NSCH; if it goes to zero you get a Stokes–Cahn–Hilliard system after scaling. That combination of system, coefficient dependence, and geometry is presented as new relative to earlier Stokes/NS or constant-coefficient CH homogenization.\n\nWhat they do well, on the face of it, is organize the two capillary cases cleanly and stay inside the expected subcritical setting that avoids extra friction or capacity terms. The abstract states a coherent, non-circular limit theorem of the usual weak-convergence type; no free parameters or fitted entities appear. Self-citation is not an issue here because the novelty claim is scoped tightly to this specific multiphase evolutionary problem.\n\nThe soft spot is simply that we only have the abstract. Energy estimates, compactness for the phase-dependent coefficients, and the passage to the limit cannot be inspected, so soundness stays provisional. The geometric restriction α>3 is load-bearing and correctly flagged as such; if it were critical or supercritical the claimed pure NSCH/SCH limits would not hold in the same form. That is not a hidden flaw—it is the stated setting—but it does mean the result is regime-specific rather than universal.\n\nThis is for people who already work on mathematical homogenization of multiphase continuum fluids or NSCH systems. A reader who needs the first rigorous evolutionary result under phase-dependent mobility will get value from the statement and, if the proofs check out, from the estimates. It deserves a serious referee: the claim is important enough inside the subfield and formally grounded enough on paper to warrant full review rather than desk rejection. I would send it out.","headline":"Claimed first rigorous evolutionary NSCH homogenization with phase-dependent coeffs under subcritical holes; two clean capillary regimes, but abstract-only so proofs unchecked.","tokens_in":2751,"tokens_out":526,"would_cite":false,"duration_ms":10804,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35B27","76D05","35Q35","76T99"],"pacs":[],"model":"grok-4.5","headline":"In the small-hole regime, NSCH fluids with phase-dependent viscosity and mobility homogenize to either the full NSCH system or a Stokes–Cahn–Hilliard system according to capillary strength.","keywords":["homogenization","Navier-Stokes-Cahn-Hilliard","perforated domains","small-hole regime","phase-dependent viscosity","phase-dependent mobility","Stokes-Cahn-Hilliard","capillary strength"],"falsifier":"Construct a family of perforated domains with hole diameter exactly of order ε^3 (or larger) and check whether the same sequences still converge to solutions of the unmodified NSCH or Stokes–Cahn–Hilliard systems; the appearance of a nonzero Brinkman-type friction term would disprove the claimed limits.","tokens_in":2886,"feed_emoji":"🌊","tokens_out":713,"duration_ms":5856,"temperature":0.7,"pith_summary":"The paper studies what happens to a two-phase fluid flow when a three-dimensional domain is perforated by a large number of tiny solid obstacles whose diameter is much smaller than their separation. Viscosity and mobility both depend on the local phase, so the constitutive laws are nonlinear. Under a subcritical hole-size condition the authors prove that the perforated-domain Navier–Stokes–Cahn–Hilliard equations admit two clean continuum limits. When the capillary coefficient stays of order one the limit is simply the same NSCH system posed on the whole domain; when the capillary coefficient vanishes the inertial terms drop out and the limit becomes a Stokes–Cahn–Hilliard system. The result supplies the first rigorous homogenization theory for evolutionary NSCH models with phase-dependent coefficients in this geometric regime, giving a mathematical justification for replacing a highly complex perforated geometry by a simpler effective equation.","feed_headline":"Tiny holes leave NSCH fluids intact—or reduce them to Stokes","feed_subtitle":"Subcritical perforations and capillary strength decide whether inertia survives homogenization","key_machinery":"The subcritical geometric restriction α>3, which keeps the holes small enough that no additional friction or capacity terms appear; combined with a priori energy estimates and two-scale compactness adapted to phase-dependent coefficients, this restriction allows passage to the limit without residual obstacle effects.","core_discovery":"For the three-dimensional Navier–Stokes–Cahn–Hilliard system with phase-dependent viscosity and mobility, perforated by holes of diameter order ε^α with α>3, the homogenization limit is the original NSCH system when capillary strength λ_ε converges to a positive constant, and a Stokes–Cahn–Hilliard system when λ_ε tends to zero.","pith_inferences":[],"forward_implications":[],"fun_headline_variants":["Tiny holes keep NSCH intact or cut it to Stokes-Cahn-Hilliard","Capillary λ decides: NSCH survives holes or yields to SCH","Subcritical holes leave phase-dependent NSCH as is—or Stokes it","NSCH homogenizes to itself if λ holds, to SCH if λ→0","Holes of size ε^α (α>3) preserve NSCH or force Stokes limit"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The holes must shrink faster than the critical rate α=3; if they are larger, extra friction terms arise and the claimed pure NSCH or Stokes–Cahn–Hilliard limits no longer hold.","fun_headline_variants_meta":{"raw":{"variants":["Tiny holes keep NSCH intact or cut it to Stokes-Cahn-Hilliard","Capillary λ decides: NSCH survives holes or yields to SCH","Subcritical holes leave phase-dependent NSCH as is—or Stokes it","NSCH homogenizes to itself if λ holds, to SCH if λ→0","Holes of size ε^α (α>3) preserve NSCH or force Stokes limit"]},"model":"grok-4.5","effort":"low","cost_usd":0.009268,"raw_usage":{"total_tokens":2100,"prompt_tokens":737,"num_sources_used":0,"completion_tokens":109,"cost_in_usd_ticks":92680000,"prompt_tokens_details":{"text_tokens":737,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":1254,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":737,"tokens_out":109,"duration_ms":9582,"temperature":1.0,"reasoning_tokens":1254,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-15T03:45:19.218856+00:00","model_set":{"reader":"grok-4.5"},"falsifier":"Construct a family of perforated domains with hole diameter exactly of order ε^3 (or larger) and check whether the same sequences still converge to solutions of the unmodified NSCH or Stokes–Cahn–Hilliard systems; the appearance of a nonzero Brinkman-type friction term would disprove the claimed limits.","supporting_citations":[],"review_version":1}