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arxiv: 2507.21602 · v3 · submitted 2025-07-29 · ⚛️ physics.flu-dyn · cond-mat.soft

On buoyancy in disperse two-phase flow and its impact on well-posedness of two-fluid models

Rui Zhu , Yulan Chen , Katharina Tholen , Zhiguo He , Thomas P\"ahtz This is my paper

Pith reviewed 2026-05-19 03:22 UTC · model grok-4.3

classification ⚛️ physics.flu-dyn cond-mat.soft
keywords buoyancy closuretwo-phase flowtwo-fluid modelsHadamard instabilityReynolds stressgeneralized buoyancydisperse flowwell-posedness
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The pith

The unique consistent buoyancy closure attributes all stresses except the Reynolds stress to the background flow and stabilizes two-fluid models via low-pass filtering.

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

The paper establishes the correct physical separation between the generalized-buoyancy force and other interfacial forces in disperse two-phase flow. Existing closures are shown to conflict with at least one of two simple thought experiments or with particle-resolving simulations. The derived closure requires no approximation and assigns every stress and pseudo-stress in the averaged fluid momentum equation, except the Reynolds stress, to the background flow that produces buoyancy. This assignment produces a low-pass filter that damps buoyancy contributions at short wavelengths, eliminating the source of Hadamard instabilities. As a direct result, even rudimentary two-fluid models become linearly well-posed when the closure is used.

Core claim

All existing buoyancy closures are inconsistent with particle-resolving numerical simulations and/or at least one of two simple thought experiments. The unique consistent closure requires no approximation and implies that all stresses and pseudo-stresses in the average fluid phase momentum balance, except the Reynolds stress, fully contribute to the background flow responsible for buoyancy. This closure exhibits a low-pass filter property that attenuates buoyancy at short wavelengths and thereby prevents Hadamard instabilities, so that even simplistic two-fluid models are linearly well-posed.

What carries the argument

The consistent buoyancy closure that assigns every stress except the Reynolds stress to the background flow driving buoyancy.

If this is right

  • Every prior buoyancy closure fails consistency checks with at least one of the two thought experiments or with particle-resolving simulations.
  • The new closure needs no small-particle approximation.
  • All stresses except the Reynolds stress act on the background flow that generates buoyancy.
  • The low-pass filter damps short-wavelength buoyancy contributions and thereby removes the Hadamard instability mechanism.
  • Simplistic two-fluid models become linearly well-posed once the closure is inserted.

Where Pith is reading between the lines

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

  • The same closure principle could be tested in three-dimensional or non-spherical particle configurations to check whether the Reynolds-stress exclusion survives.
  • Engineering codes for sediment transport or bubble columns could be rerun with the new closure to quantify changes in predicted mixing rates at resolved scales.
  • The low-pass filtering behavior suggests a natural cutoff scale set by particle diameter that might be compared with measured fluctuation spectra in experiments.

Load-bearing premise

The two thought experiments and the particle-resolving simulations correctly isolate the physical roles of pseudo-stresses and small-particle approximations.

What would settle it

A two-fluid simulation that remains linearly well-posed under the new closure at arbitrarily short wavelengths would be contradicted by direct observation of exponential growth of short-wavelength disturbances in the same setup.

Figures

Figures reproduced from arXiv: 2507.21602 by Katharina Tholen, Rui Zhu, Thomas P\"ahtz, Yulan Chen, Zhiguo He.

Figure 1
Figure 1. Figure 1: The Fourier transform 𝑀𝑘ˆ of the averaging operator M essentially acts like a low-pass filter, cutting off contributions from wavenumber vectors 𝒌 with ˆ𝑘 ≡ 𝑅|𝒌| ≳ 4. 5. Implications for two-fluid models 5.1. Well-posedness of two-fluid models To analyze whether two-fluid models are well-posed when employing our buoyancy closure, we use the form of the momentum balances in (4.17) and (4.18), combined with … view at source ↗
Figure 2
Figure 2. Figure 2: Snapshot of a DNS-DEM simulation of statistically steady, uniform sediment [PITH_FULL_IMAGE:figures/full_fig_p017_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: The DNS-DEM simulations of statistically steady, uniform sediment transport [PITH_FULL_IMAGE:figures/full_fig_p018_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Test of our (4.10) and existing (Anderson & Jackson 1967; Zhang et al. 2007; Chauchat 2018; Maurin et al. 2018) buoyancy closures against data from DNS-DEM simulations of statistically steady, uniform sediment transport. If lift forces are not significant, one expects 𝛽s⟨𝑓B𝑧 ⟩ s /(−𝛽s𝜌f𝑔𝑧 ) ≈ 1. However, close to the bed surface (𝑧 = 1 . . . 3), where significant positive lift forces can to occur (Chepil 1… view at source ↗
read the original abstract

The Maxey-Riley-Gatignol equation for the flow around a sphere at low particle Reynolds number tells us that the fluid-particle interaction force decomposes into a contribution from the local flow disturbance caused by the particle's boundary -- consisting of the drag, virtual-mass, and history forces, and their Fax\'en corrections -- and another contribution from the stress of the background flow, termed generalized-buoyancy force. There is also a consensus that, for general disperse two-phase flow, the interfacial force density, coupling the average fluid phase and dispersed-phase momentum balances, decomposes in a likewise manner. However, there has been a long-standing controversy about the physical closure separating the generalized-buoyancy from the interfacial force density, especially whether or not pseudo-stresses, such as the Reynolds stress, should be attributed to the background flow. Furthermore, most existing propositions for this closure involve small-particle approximations. Here, we show that all existing buoyancy closures are inconsistent with particle-resolving numerical simulations and/or at least one of two simple thought experiments designed to determine the roles of pseudo-stresses and small-particle approximations. We then derive the unique consistent closure. It requires no approximation and implies that all stresses and pseudo-stresses in the average fluid phase momentum balance, except the Reynolds stress, fully contribute to the background flow responsible for buoyancy. Remarkably, it exhibits a low-pass filter property, attenuating buoyancy at short wavelengths, that prevents it from causing Hadamard instabilities, constituting a first-principle-based solution to the long-standing ill-posedness problem of two-fluid models. When employing the derived closure, even simplistic two-fluid models are linearly well-posed.

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

2 major / 2 minor

Summary. The manuscript claims that existing closures for the generalized-buoyancy term in averaged two-fluid models for disperse two-phase flow are inconsistent with particle-resolving simulations and two simple thought experiments that isolate the roles of pseudo-stresses and small-particle approximations. From this inconsistency analysis the authors derive a unique, approximation-free closure in which all stresses and pseudo-stresses in the averaged fluid-phase momentum balance except the Reynolds stress are attributed to the background flow. The resulting buoyancy term possesses a low-pass filter property that attenuates short-wavelength contributions and thereby eliminates the Hadamard instability, rendering even simple two-fluid models linearly well-posed.

Significance. If the uniqueness argument and the low-pass-filter property hold, the work supplies a first-principles resolution to a long-standing source of ill-posedness in two-fluid models. The absence of free parameters or small-particle approximations, together with direct consistency checks against independent simulations, would constitute a substantive advance for the modeling of particle-laden flows in engineering and geophysical applications.

major comments (2)
  1. [derivation section] The central uniqueness claim (abstract and derivation section) rests on the assertion that the two thought experiments plus the reported particle-resolving simulations jointly rule out every alternative buoyancy closure. Because the tests are described only at the level of isolating pseudo-stress roles and small-particle effects, it remains unclear whether they constrain the general functional form sufficiently to exclude other closures that agree on the tested cases yet differ for unsteady or inhomogeneous flows at finite particle Reynolds number; this gap directly affects the load-bearing step that links the closure to the low-pass filter and well-posedness result.
  2. [concluding section] The linear well-posedness statement (final paragraph of abstract and concluding section) is presented as a direct consequence of the low-pass filter property of the derived closure. A concrete dispersion-relation calculation or explicit eigenvalue analysis for the simplest two-fluid model employing the new closure would be required to confirm that the attenuation indeed removes the short-wavelength instability for the full range of wavenumbers and volume fractions of interest.
minor comments (2)
  1. Notation for the generalized-buoyancy term and the background-flow stress tensor should be introduced once with a clear table or equation reference to avoid ambiguity when comparing with prior closures.
  2. The manuscript would benefit from an explicit statement of the precise mathematical form of the two thought experiments (e.g., the imposed flow fields and boundary conditions) so that readers can reproduce the consistency checks independently.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thorough review and valuable feedback on our manuscript. The comments highlight important aspects of our uniqueness argument and the well-posedness demonstration. We provide point-by-point responses below and have revised the manuscript accordingly where appropriate.

read point-by-point responses

  1. Referee: [derivation section] The central uniqueness claim (abstract and derivation section) rests on the assertion that the two thought experiments plus the reported particle-resolving simulations jointly rule out every alternative buoyancy closure. Because the tests are described only at the level of isolating pseudo-stress roles and small-particle effects, it remains unclear whether they constrain the general functional form sufficiently to exclude other closures that agree on the tested cases yet differ for unsteady or inhomogeneous flows at finite particle Reynolds number; this gap directly affects the load-bearing step that links the closure to the low-pass filter and well-posedness result.

    Authors: We thank the referee for raising this point. The thought experiments are constructed to probe the attribution of stresses to the background flow without relying on specific flow configurations. The first experiment isolates the role of pseudo-stresses by considering a uniform flow with superimposed fluctuations, showing that only the Reynolds stress should be excluded from buoyancy. The second tests the small-particle approximation by examining cases where particle size is not negligible. The particle-resolving simulations, which include unsteady and inhomogeneous flows at finite Re_p, confirm consistency only with our closure. Any other closure agreeing on these but differing elsewhere would contradict the decomposition from the Maxey-Riley-Gatignol equation applied to the averaged quantities. We have added a subsection in the derivation to explicitly demonstrate that alternative forms lead to inconsistencies with at least one of the tests or simulations. Thus, we maintain that the closure is unique under the stated physical principles. revision: partial

  2. Referee: [concluding section] The linear well-posedness statement (final paragraph of abstract and concluding section) is presented as a direct consequence of the low-pass filter property of the derived closure. A concrete dispersion-relation calculation or explicit eigenvalue analysis for the simplest two-fluid model employing the new closure would be required to confirm that the attenuation indeed removes the short-wavelength instability for the full range of wavenumbers and volume fractions of interest.

    Authors: We agree that an explicit calculation strengthens the well-posedness claim. In the revised manuscript, we have included a new appendix with the dispersion relation analysis for the simplest two-fluid model. This shows that the low-pass filter property of the buoyancy closure damps the growth rate for high wavenumbers, eliminating the Hadamard instability across the relevant range of volume fractions and wavenumbers. The analysis confirms linear well-posedness without additional assumptions. revision: yes

Circularity Check

0 steps flagged

Derivation relies on independent consistency checks rather than reducing to inputs by construction

full rationale

The paper demonstrates inconsistency of existing closures against particle-resolving simulations and two thought experiments, then identifies the unique form consistent with those external benchmarks. This chain uses independent physical tests and numerical data as constraints, without evidence of self-definitional equations, parameters fitted to the target result then relabeled as predictions, or load-bearing self-citations that substitute for derivation. The low-pass filter property and well-posedness follow from the derived expression rather than being presupposed in the inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The claim rests on the standard decomposition of interfacial force into disturbance and generalized-buoyancy parts, the validity of extending the Maxey-Riley-Gatignol picture to averaged flows, and the premise that thought experiments plus resolved simulations suffice to select the unique physical closure.

axioms (2)
  • domain assumption The interfacial force density in general disperse two-phase flow decomposes into a local disturbance contribution and a generalized-buoyancy contribution from background-flow stress.
    Stated as existing consensus in the abstract.
  • domain assumption Thought experiments can unambiguously determine whether pseudo-stresses belong to the background flow or not.
    Used to rule out existing closures.

pith-pipeline@v0.9.0 · 5857 in / 1512 out tokens · 40263 ms · 2026-05-19T03:22:04.682797+00:00 · methodology

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Reference graph

Works this paper leans on

36 extracted references · 36 canonical work pages

  1. [1]

    Anderson, T. B. & Jackson, R.1967 Fluid mechanical description of fluidized beds. Equations of motion. Industrial & Engineering Chemistry Fundamentals 6 (4), 527–539

    work page 1967

  2. [2]

    Bagnold, R. A. 1956 The flow of cohesionless grains in fluid. Philosophical Transactions of the Royal Society London A 249 (964), 235–297

    work page 1956

  3. [3]

    Batchelor, G. K. 2000 An Introduction to Fluid Dynamics. Cambridge University Press, Cambridge

    work page 2000

  4. [4]

    & Fraccarollo, L

    Berzi, D. & Fraccarollo, L. 2013 Inclined, collisional sediment transport. Physics of Fluids 25 (10), 106601

    work page 2013

  5. [5]

    & Meiburg, E.2017 A collision model for grain-resolving simulations of flows over dense, mobile, polydisperse granular sediment beds

    Biegert, E., Vowinckel, B. & Meiburg, E.2017 A collision model for grain-resolving simulations of flows over dense, mobile, polydisperse granular sediment beds. Journal of Computational Physics 340, 105–127

    work page 2017

  6. [6]

    & Chauchat, J

    Chassagne, R., Bonamy, C. & Chauchat, J. 2023 A frictional–collisional model for bedload transport based on kinetic theory of granular flows: discrete and continuum approaches. Journal of Fluid Mechanics 964, A27

    work page 2023

  7. [7]

    2018 A comprehensive two-phase flow model for unidirectional sheet-flows

    Chauchat, J. 2018 A comprehensive two-phase flow model for unidirectional sheet-flows. Journal of Hydraulic Research 56 (1), 15–28

    work page 2018

  8. [8]

    Chepil, W. S. 1961 The use of spheres to measure lift and drag on wind-eroded soil grains. Soil Science Society of America Journal 25 (5), 343–345

    work page 1961

  9. [9]

    Clift, R., Seville, J. P. K., Moore, S. C. & Chavarie, C.1987 Comments on buoyancy in fluidised beds. Chemical Engineering Science 42 (1), 191–194

    work page 1987

  10. [10]

    T., Schwarzkopf, J

    Crowe, C. T., Schwarzkopf, J. D., Sommerfeld, M. & Tsuji, Y. 2012 Multiphase Flows with Droplets and Particles. Taylor & Francis Group, Boca Raton. Di Felice, R.1995 Hydrodynamics of liquid fluidisation.Chemical Engineering Science50 (8), 1213–1245

    work page 2012

  11. [11]

    Drew, D. A. 1983 Mathematical modeling of two-phase flow. Annual Review of Fluid Mechanics 15, 261–291

    work page 1983

  12. [12]

    & Kaper, H

    Ferziger, J. & Kaper, H. 1972 Mathematical Theory of Transport Processes in Gases . North-Holland,

    work page 1972

  13. [13]

    & Pierson, J

    Fintzi, N. & Pierson, J. L.2025 Averaged equations for disperse two-phase flow with interfacial properties and their closures for dilute suspension of droplets. https://doi.org/10.48550/arXiv.2410.10752

    work page doi:10.48550/arxiv.2410.10752 2025

  14. [14]

    Fox, R. O. 2019 A kinetic-based hyperbolic two-fluid model for binary hard-sphere mixtures. Journal of Fluid Mechanics 877, 282–329

    work page 2019

  15. [15]

    Fox, R. O. 2025 The particle-fluid-particle pressure tensor for ideal-fluid–particle flow. Journal of Fluid Mechanics 1010, A8

    work page 2025

  16. [16]

    1997 Locally averaged equations of motion for a mixture of identical spherical particles and a Newtonian fluid

    Jackson, R. 1997 Locally averaged equations of motion for a mixture of identical spherical particles and a Newtonian fluid. Chemical Engineering Science 52 (15), 2457–2469

    work page 1997

  17. [17]

    2000 The Dynamics of Fluidized Particles

    Jackson, R. 2000 The Dynamics of Fluidized Particles. Cambridge University Press, New York

    work page 2000

  18. [18]

    & Mazzei, L.2018 CFD modeling of fluidized beds.Reference Module in Chemistry, Molecular Sciences and Chemical Engineering

    Jamshidi, R. & Mazzei, L.2018 CFD modeling of fluidized beds.Reference Module in Chemistry, Molecular Sciences and Chemical Engineering

    work page 2018

  19. [19]

    Joseph, D. D. & Saut, J. C. 1990 Short-wave instabilities and ill-posed initial-value problems. Theoretical and Computational Fluid Dynamics 1, 191–227

    work page 1990

  20. [20]

    Keh, H. J. & Chen, S. H. 1996 The motion of a slip spherical particle in an arbitrary stokes flow.European Journal of Mechanics - B/Fluids 15 (6), 791–807

    work page 1996

  21. [21]

    & Fuller, B

    Lamb, Michael P., Brun, F. & Fuller, B. M. 2017 Direct measurements of lift and drag on shallowly submerged cobbles in steep streams: Implications for flow resistance and sediment transport. Water Resources Research 53 (9), 7607–7629

    work page 2017

  22. [22]

    P., Johnson, C

    Langham, J., Meng, X., Webb, J. P., Johnson, C. G. & Gray, J.M.N.T. 2025 Ill posedness in shallow multi-phase debris-flow models. Journal of Fluid Mechanics 1015, A52

    work page 2025

  23. [23]

    Lhuillier, D., Chang, C. H. & Theofanous, T. G.2013 On the quest for a hyperbolic effective-field model of disperse flows. Journal of Fluid Mechanics 731, 184–194

    work page 2013

  24. [24]

    & Bai, B

    Li, X., Balachandar, S., Lee, H. & Bai, B. 2019 Fully resolved simulations of a stationary finite-sized particle in wall turbulence over a rough bed. Physical Review Fluids 4 (9), 094302

    work page 2019

  25. [25]

    & Dorgan, A

    Loth, E. & Dorgan, A. J. 2009 An equation of motion for particles of finite reynolds number and size. Environmental Fluid Mechanics 9, 187–206

    work page 2009

  26. [26]

    & Frey, P

    Maurin, R., Chauchat, J. & Frey, P. 2018 Revisiting slope influence in turbulent bedload transport: consequences for vertical flow structure and transport rate scaling.Journal of Fluid Mechanics 839, 135–156. 26

    work page 2018

  27. [27]

    Maxey, M. R. & Riley, J. J.1983 Equation of motion for a small rigid sphere in a nonuniform flow.Physics of Fluids 26 (4), 883–889. P¨ahtz, T., Chen, Y., Rui, Z., Tholen, K. & He, Z. 2025 Coarse-graining particulate two-phase flow. https://doi.org/10.48550/arXiv.2308.09661 . P¨ahtz, T. & Dur ´an, O. 2018 Universal friction law at granular solid-gas transi...

    work page doi:10.48550/arxiv.2308.09661 1983

  28. [28]

    & Fox, R.O.2018 On the hyperbolicity of the two-fluid model for gas-liquid bubbly flows.Applied Mathematical Modelling 57, 432–447

    Panicker, N., Passalacqua, A. & Fox, R.O.2018 On the hyperbolicity of the two-fluid model for gas-liquid bubbly flows.Applied Mathematical Modelling 57, 432–447

    work page 2018

  29. [29]

    & Pagliai, P

    Rotondi, M., Di Felice, R. & Pagliai, P. 2015 Validation of fluid–particle interaction force relationships in binary-solid suspensions. Particuology 23, 40–48

    work page 2015

  30. [30]

    & Rzehak, R.2019 Lift forces on solid spherical particles in unbounded flows.Chemical Engineering Science 208, 115145

    Shi, P. & Rzehak, R.2019 Lift forces on solid spherical particles in unbounded flows.Chemical Engineering Science 208, 115145

    work page 2019

  31. [31]

    & Wendroff, B

    Stewart, B. & Wendroff, B. 1984 Two-phase flow: Models and methods. Journal of Computational Physics 56 (3), 363–409

    work page 1984

  32. [32]

    Tholen, K., P¨ahtz, T., Kamath, S., Parteli, E. J. R. & Kroy, K.2023 Anomalous scaling of aeolian sand transport reveals coupling to bed rheology. Physical Review Letters 130 (5), 058204

    work page 2023

  33. [33]

    2005 An immersed boundary method with direct forcing for the simulation of particulate flows

    Uhlmann, M. 2005 An immersed boundary method with direct forcing for the simulation of particulate flows. Journal of Computational Physics 209, 448–476

    work page 2005

  34. [34]

    Zhang, D. Z. 2021 Ensemble average and nearest particle statistics in disperse multiphase flows.Journal of Fluid Mechanics 910, A16

    work page 2021

  35. [35]

    Z., VanderHeyden, W

    Zhang, D. Z., VanderHeyden, W. B., Zou, Q. & Padial-Collins, N. T. 2007 Pressure calculations in disperse and continuous multiphase flows.International Journal of Multiphase Flow33 (1), 86–100

    work page 2007

  36. [36]

    & Meiburg, E.2022 Grain-resolving simulations of submerged cohesive granular collapse

    Zhu, R., He, Z., Zhao, K., Vowinckel, B. & Meiburg, E.2022 Grain-resolving simulations of submerged cohesive granular collapse. Journal of Fluid Mechanics 942, A49

    work page 2022