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arxiv: 2604.11747 · v2 · submitted 2026-04-13 · ⚛️ physics.flu-dyn · cond-mat.quant-gas· cond-mat.soft

Schr\"odinger-Navier-Stokes equation for capillary fluids

Luca Salasnich , Sauro Succi , Adriano Tiribocchi This is my paper

Pith reviewed 2026-05-10 16:01 UTC · model grok-4.3

classification ⚛️ physics.flu-dyn cond-mat.quant-gascond-mat.soft
keywords Schrödinger-Navier-Stokes equationNavier-Stokes-Korteweg equationscapillary fluidsaction functionaldispersion relationsmicrofluidicsquantum simulation
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The pith

The Schrödinger-Navier-Stokes equation with generic parameters is formally equivalent to the Navier-Stokes-Korteweg equations for capillary fluids.

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

The paper establishes that the Schrödinger-Navier-Stokes equation with generic parameters is formally equivalent to the Navier-Stokes-Korteweg equations describing capillary fluids. This equivalence is shown through an action functional that splits into a conservative part following Korteweg theory and a dissipative Rayleigh part. A sympathetic reader would care because this unification links quantum-inspired fluid models to classical capillary flows, potentially aiding simulations in microfluidics and soft matter. It also allows derivation of sound mode dispersions and reduced one-dimensional models for confined fluids.

Core claim

We show that the SNS equation with generic parameters is formally equivalent to the Navier-Stokes-Korteweg equations for capillary fluids, with the equivalence established at the level of an action functional that decomposes naturally into a Korteweg conservative and a Rayleigh dissipative components, respectively.

What carries the argument

The action functional that decomposes naturally into a Korteweg conservative component and a Rayleigh dissipative component.

Load-bearing premise

The action functional for the SNS equation can be naturally decomposed into conservative Korteweg and dissipative Rayleigh parts without additional assumptions or constraints on the parameters.

What would settle it

A direct derivation showing that the action functional does not decompose into Korteweg conservative and Rayleigh dissipative parts for generic parameters would falsify the claimed equivalence.

read the original abstract

We highlight some properties of the Schr\"odinger-Navier-Stokes (SNS) equation [Salasnich, Succi, and Tiribocchi (2024)] of potential relevance for microfluidics and soft matter. Specifically, we show that the SNS equationwith generic parameters is formally equivalent to the Navier-Stokes-Korteweg equations for capillary fluids, with the equivalence established at the level of an action functional that decomposes naturally into a Korteweg conservative and a Rayleigh dissipative components, respectively. We derive the dispersion relation for sound modes, showing that the dispersive parameter controls capillary stiffness while the dissipative parameter controls viscous damping, and that the Bogoliubov dispersion relation is recovered in the quantum limit. We also derive an effective one-dimensional SNS equation for a fluid confined in a narrow capillary tube. Finally, it is argued that the SNS may facilitate the quantum simulation of complex states of flowing matter.

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

0 major / 4 minor

Summary. The manuscript claims that the Schrödinger-Navier-Stokes (SNS) equation with generic parameters is formally equivalent to the Navier-Stokes-Korteweg (NSK) equations for capillary fluids. Equivalence is established via an action functional that decomposes into a conservative Korteweg component and a dissipative Rayleigh component. The paper derives the dispersion relation for sound modes (showing dispersive parameter controls capillary stiffness and dissipative parameter controls viscous damping, recovering the Bogoliubov limit in the quantum regime), presents an effective one-dimensional reduction for fluid confined in a narrow capillary tube, and argues that the SNS framework may enable quantum simulation of complex flowing matter.

Significance. If the variational equivalence is rigorous, the work supplies a natural action-functional bridge between a quantum-inspired SNS model and classical capillary fluid equations, with direct implications for microfluidics and soft-matter modeling. The explicit recovery of the Bogoliubov dispersion and the 1D reduction are concrete, falsifiable consequences that strengthen the central claim; the quantum-simulation suggestion, while speculative, is presented as a downstream consequence rather than a prerequisite.

minor comments (4)
  1. The abstract and introduction should explicitly reference the 2024 Salasnich-Succi-Tiribocchi paper when first defining the SNS equation, to clarify the starting point for readers unfamiliar with the prior work.
  2. In the dispersion-relation section, the linearization steps from the SNS equation to the final quadratic form in frequency and wave number should be written out in full (including the explicit roles of the two free parameters) rather than summarized, to facilitate independent verification.
  3. The one-dimensional reduction for the narrow-tube geometry would benefit from a short paragraph stating the assumptions on the transverse velocity profile and the averaging procedure used to obtain the effective 1D equation.
  4. Ensure consistent use of the umlaut in 'Schrödinger' throughout the LaTeX source and that all equation numbers referenced in the text match the displayed equations.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript, the recognition of the formal equivalence at the action-functional level, and the recommendation for minor revision. We appreciate the note that the dispersion relation and one-dimensional reduction provide concrete, falsifiable consequences.

Circularity Check

1 steps flagged

Minor self-citation for SNS definition; equivalence and derivations are independent

specific steps
  1. self citation load bearing [Abstract and Introduction (citation to Salasnich et al. 2024)]
    "We highlight some properties of the Schrödinger-Navier-Stokes (SNS) equation [Salasnich, Succi, and Tiribocchi (2024)] of potential relevance for microfluidics and soft matter. Specifically, we show that the SNS equation with generic parameters is formally equivalent to the Navier-Stokes-Korteweg equations for capillary fluids, with the equivalence established at the level of an action functional"

    The SNS equation itself originates in a prior paper by the same three authors; the present work therefore begins from a self-cited definition. However, the equivalence, dispersion relation, and 1D reduction are derived here rather than presupposed, so the self-citation is not load-bearing for the new claims.

full rationale

The paper cites the 2024 SNS definition from the same author group but derives the action-functional equivalence to NSK, the dispersion relation, and the 1D reduction as new results within this manuscript. No step reduces the central claim to a self-citation chain, fitted parameter, or definitional tautology. The cited prior work supplies the starting equation; the present derivations stand on their own variational construction and are externally falsifiable via the recovered Bogoliubov limit and capillary-wave behavior.

Axiom & Free-Parameter Ledger

2 free parameters · 1 axioms · 0 invented entities

The central claim rests on the existence and properties of the SNS equation from prior work, and the assumption that its action functional naturally separates into conservative and dissipative terms for equivalence to hold.

free parameters (2)
  • dispersive parameter
    Controls capillary stiffness in the dispersion relation, introduced as generic parameter in SNS.
  • dissipative parameter
    Controls viscous damping, generic in SNS.
axioms (1)
  • domain assumption The action functional decomposes into Korteweg conservative and Rayleigh dissipative components
    Invoked to establish equivalence between SNS and NSK equations.

pith-pipeline@v0.9.0 · 5461 in / 1364 out tokens · 66554 ms · 2026-05-10T16:01:21.691453+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

29 extracted references · 29 canonical work pages · 1 internal anchor

  1. [1]

    Madelung, Quantentheorie in hydrodynamischer Form, Z

    E. Madelung, Quantentheorie in hydrodynamischer Form, Z. Phys.40, 322 (1926)

    work page 1926

  2. [2]

    Madelung, Eine anschauliche Deutung der Gleichung von Schr¨ odinger, Naturwis- senschaften14, 1004 (1926)

    E. Madelung, Eine anschauliche Deutung der Gleichung von Schr¨ odinger, Naturwis- senschaften14, 1004 (1926). 9 SciPost Physics Submission

    work page 1926

  3. [3]

    Dietrich and D

    K. Dietrich and D. Vautherin, Equivalence between particular types of Navier-Stokes and non-linear Schr¨ odinger equations, Le Journal de Physique46, 313 (1985)

    work page 1985

  4. [4]

    Coste, Nonlinear Schr¨ odinger equation and superfluid hydrodynamics, Eur

    E. Coste, Nonlinear Schr¨ odinger equation and superfluid hydrodynamics, Eur. Phys. J. B1, 245 (1998)

    work page 1998

  5. [5]

    Salasnich and F

    L. Salasnich and F. Toigo, Extended Thomas-Fermi Density Functional for the Unitary Fermi Gas, Phys. Rev. A 78, 053626 (2008)

    work page 2008

  6. [6]

    Salasnich, Hydrodynamics of Bose and Fermi superfluids at zero temperature: the superfluid nonlinear Schr¨ odinger equation, Laser Phys.19, 642 (2009)

    L. Salasnich, Hydrodynamics of Bose and Fermi superfluids at zero temperature: the superfluid nonlinear Schr¨ odinger equation, Laser Phys.19, 642 (2009)

    work page 2009

  7. [7]

    Fern´ andez de C´ ordoba, J

    P. Fern´ andez de C´ ordoba, J. M. Isidro, and J. V´ azquez Molina, Schr¨ odinger vs Navier–Stokes, Entropy18, 34 (2016)

    work page 2016

  8. [8]

    Meng and Y

    Z. Meng and Y. Yang, Quantum computing of fluidynamics using the hydrodynamic Schr¨ odinger equation, Phys. Rev. Research5, 033182 (2023)

    work page 2023

  9. [9]

    Meng and Y

    Z. Meng and Y. Yang, Quantum spin representation for the Navier-Stokes equation, Phys. Rev. Research6, 043130 (2024)

    work page 2024

  10. [10]

    Salasnich, S

    L. Salasnich, S. Succi, and A. Tiribocchi,Quantum wave representation of dissipative fluids, Int. J. Mod. Phys. C35, 2450100 (2024)

    work page 2024

  11. [11]

    Emergence of vorticity and viscous stress in finite-scale quantum hydrodynamics

    C. Triola, Emergence of Vorticity and Viscous Stress in Finite Scale Quantum Hy- drodynamics, e-preprint arXiv:2508.18200 to appear in Phys. Rev. E (2026)

    work page internal anchor Pith review Pith/arXiv arXiv 2026

  12. [12]

    D. J. Korteweg, Sur la forme que prennent les ´ equations du mouvements des fluides si l’ontient compte des forces capillaires caus´ ees par des variations de densit´ e, Arch. N´ eerl. Sci. Exactes Nat.6, 1 (1901)

    work page 1901

  13. [13]

    J. E. Dunn and J. Serrin, On the thermomechanics of interstitial working, Arch. Rational Mech. Anal.88, 95 (1985)

    work page 1985

  14. [14]

    J. W. Cahn and J. E. Hilliard, Free Energy of a Nonuniform System. I. Interfacial Free Energy, J. Chem. Phys.28, 258 (1958)

    work page 1958

  15. [15]

    Tjhung, C

    E. Tjhung, C. Nardini, M. E. Cates, Cluster Phases and Bubbly Phase Separation in Active Fluids: Reversal of the Ostwald Process, Phys. Rev. X8, 031080 (2018)

    work page 2018

  16. [16]

    Fausti, E

    G. Fausti, E. Tjhung, M. E. Cates, C. Nardini, Capillary Interfacial Tension in Active Phase Separation, Phys. Rev. Lett.128, 219901 (2021)

    work page 2021

  17. [17]

    Tiribocchi, R

    A. Tiribocchi, R. Wittkowski, D. Marenduzzo, and M. E. Cates, Active Model H: Scalar Active Matter in a Momentum-Conserving Fluid, Phys. Rev. Lett.115, 118302 (2015)

    work page 2015

  18. [18]

    Singh and M

    R. Singh and M. E. Cates, Hydrodynamically Interrupted Droplet Growth in Scalar Active Matter, Phys. Rev. Lett.123, 148005 (2019)

    work page 2019

  19. [19]

    Salasnich, A Parola, and L

    L. Salasnich, A Parola, and L. Reatto, Effective wave equations for the dynamics of cigar-shaped and disk-shaped Bose condensates, Phys. Rev. A65, 043614 (2002)

    work page 2002

  20. [20]

    Lorenzi and L

    F. Lorenzi and L. Salasnich, Atomic soliton transmission and induced collapse in scattering from a narrow barrier, Sci Rep14, 4665 (2024). 10 SciPost Physics Submission

    work page 2024

  21. [21]

    Lorenzi and L

    F. Lorenzi and L. Salasnich, Variational approach to multimode nonlinear optical fibers, Nanophotonics14, 805 (2025)

    work page 2025

  22. [22]

    M. L. Chiofalo, S. Succi, M. P. Tosi, Ground state of trapped interacting Bose-Einstein condensates by an explicit imaginary-time algorithm, Phys. Rev. E62, 7438 (2000)

    work page 2000

  23. [23]

    M. E. Cates, Active Field Theories in Active Matter and Nonequilibrium Statistical Physics, Lecture Notes of the Les Houches Summer School: Volume 112, 2018, edited by Tailleur, J. Gompper, G., Marchetti, M. C., J. Yeomans, M. & Salomon, C., pp. 180–216 (Oxford University Press, Oxford, 2022)

    work page 2018

  24. [24]

    te Vrugt, T

    M. te Vrugt, T. Frohoff-H¨ ulsmann, E. Heifetz, U. Thiele, R. Wittkowski, From a microscopic inertial active matter model to the Schr¨ odinger equation, Nat. Commun. 14, 1302 (2023)

    work page 2023

  25. [25]

    Succi, W

    S. Succi, W. Itani, K. Sreenivasan, and R. Steijl, Quantum computing for fluids: Where do we stand? EPL144, 10001 (2023)

    work page 2023

  26. [26]

    S. S. Bharadwaj and K. R. Sreenivasan, Towards simulating fluid flows with quantum computing, Sadhana50, 57 (2025)

    work page 2025

  27. [27]

    Sanavio and S

    C. Sanavio and S. Succi. Lattice Boltzmann–Carleman quantum algorithm and circuit for fluid flows at moderate Reynolds number, AVS Quantum Science,6, 023802 (2024)

    work page 2024

  28. [28]

    Itani, S

    W. Itani, S. Succi, Analysis of Carleman Linearization of Lattice Boltzmann, Fluids 7, 24 (2022)

    work page 2022

  29. [29]

    Tennie, S

    F. Tennie, S. Laizet, S. Lloyd, and L. Magri, Quantum computing for nonlinear differential equations and turbulence, Nature Rev. Phys.7, 220 (2025). 11

    work page 2025