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REVIEW 2 major objections 6 minor 41 references

Heavy Particle Motion in Rotational Vortices

T0 review · 2 major / 6 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read Rotational lift forces, usually neglected for spherical particles, dominate at moderate Stokes numbers and determine whether particles settle into stable periodic orbits or escape the vortex.

desk verdict Plausible extension of the Rypina–Pratt vortex model, but a sign error in Eq. (10) likely shifts the bifurcation branches. read the letter →

arxiv 2506.09821 v1 pith:W6X4UDTB submitted 2025-06-11 physics.flu-dyn

Orr Avni , Alok Kumar , Yuval Dagan This is my paper · ORCID
classification physics.flu-dyn
keywords inertialparticlesrotationalliftMagnusforcevortexdynamicsparticletrappingStokesnumbersaddle-nodebifurcationLagrangiantracking
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

Spherical particles moving inside a steady rotating vortex do not just follow the fluid: depending on their size, density, and the vortex spin, they can be trapped on stable periodic orbits, drift to the boundary, or be ejected. This paper extends the standard particle equations of motion to include the spin of the sphere itself, and shows that the resulting Magnus lift, which is usually neglected for spheres, becomes a leading effect at moderate Stokes numbers ($\mathrm{St}\sim0.1$–$1$) and drives slow vertical oscillations in both position and spin. The central result is that the force balance yields equilibrium orbits that appear, merge, and vanish through saddle-node bifurcations as particle density and vortex rotation rate change, so heavy particles can be trapped while slightly buoyant ones escape. If correct, this gives a mechanism-based way to predict which particle classes aggregate in vortical flows, with direct consequences for microplastic clustering in oceanic eddies and for industrial vortex separators.

What carries the argument

The load-bearing object is a nine-dimensional nonlinear dynamical system for a particle's position, linear velocity, and angular velocity, assembled from a generalized small-sphere momentum balance plus a torque equation for the particle's spin. The key addition is the Magnus lift term, which couples spin to the velocity slip between particle and fluid and produces the slow vertical oscillations. The equilibrium reduction sets the radial and axial velocities and all spin time-derivatives to zero, collapsing the nine equations into five algebraic equations for the steady orbit $(r_0,z_0,v_{\theta,0},\omega_{r,0},\omega_{\theta,0})$. Linearizing around those equilibria produces eigenvalues that mark stable versus unstable orbits, and parameter continuation traces how the equilibria shift and merge, revealing the saddle-node bifurcations.

What would settle it

In a laboratory or high-resolution numerical vortex, add a small periodic perturbation to the flow and track particles of Stokes number near 1; if the predicted steady orbits and saddle-node bifurcation structure disappear or become chaotic, the steady-vortex picture fails.

Watch

Extended reading notes

Core claim

The paper claims that the asymptotic fate of a spherical particle in a rotating vortex is set by the competition among drag, buoyancy, virtual mass, Coriolis, and Magnus lift. At small Stokes numbers the magnitude of rotational lift is negligible and particles behave nearly as tracers; at $\mathrm{St}\sim0.1$–$1$ the Magnus force dominates the slow vertical dynamics, introducing low-frequency oscillations in the axial position and in the particle's spin that would be absent if the sphere's rotation were ignored. The system possesses fixed points of the form $x(t)=(r_0,\theta(t),z_0)$ with constant radial and axial coordinates, obtained by solving a five-equation algebraic system. These equilibria move non-monotonically with the fluid-to-particle density ratio and with the vortex rotation frequency, and the stable and unstable branches meet at saddle-node bifurcations: below a critical density ratio no equilibrium exists and particles sink out, while near neutral buoyancy an unstable branch can coexist with a stable orbit. The conclusion is that for $\mathrm{St}>0.1$, rotational dynamics must be modeled even for perfectly spherical particles, and equilibrium orbital stability is what selects which particles stay inside the vortex.

Load-bearing premise

The whole analysis rests on treating the vortex as a fixed, steady, symmetric flow and leaving out time-dependent disturbances, which earlier studies of the same flow showed can make trajectories chaotic.

Editorial extensions

If this is right

  • For moderate Stokes numbers ($\mathrm{St}\gtrsim0.1$), particle spin and translational motion can no longer be treated separately; rotational lift must be included even for perfectly spherical particles.
  • Particles heavier than the fluid can remain trapped in a vortex on stable periodic orbits, stabilized by Coriolis and virtual mass effects rather than sinking immediately.
  • Equilibrium orbits are non-unique: stable and unstable branches can coexist and annihilate at saddle-node bifurcations, so small changes in density ratio or vortex rotation rate can abruptly switch trapping on or off.
  • Particles lighter than the fluid are destabilized by the Magnus force at moderate Stokes numbers, making escape more likely than the tracer-like behavior of low-Stokes particles.
  • The model identifies particle classes prone to aggregation in vortical flows, giving a parameter-based route toward predicting microplastic hotspots and designing vortex-based separation.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • A natural test of the paper's mechanism is to repeat the same Lagrangian simulations with a weak, time-periodic perturbation to the vortex; earlier work on the same base flow suggests chaotic advection can appear, and if the periodic orbits and bifurcations do not survive, the steady-vortex picture would not carry over to real eddies.
  • Because the Magnus effect is spin-induced, the predicted trapping thresholds should change for nonspherical particles, rough particles, or particles with initial spin; varying initial spin in the model would quantify how robust the equilibria are.
  • Applied to microplastics, the results imply that eddies may act as size-selective traps, retaining particles in a specific range of Stokes numbers while expelling others; this is testable with size-resolved sampling inside and outside ocean eddies.
  • For industrial vortex mixers, the bifurcation structure suggests a design principle: tune the vortex circulation so the targeted particle class sits on the stable branch, and eject unwanted particles by pushing them past the saddle-node.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 6 minor

Summary. The paper studies the coupled translational and rotational dynamics of spherical inertial particles in a steady, axisymmetric analytical vortex, using a Maxey–Riley-type momentum equation augmented by a Magnus lift force and a separate torque equation for particle spin. The authors integrate the resulting nine-dimensional ODE system over Stokes numbers St = 0.001–1, density ratios ρbar = 0.01–1.05, and vortex rotation frequencies φz = 0.02π–2π. They report Stokes-dependent particle trapping, Magnus-induced slow vertical oscillations, and equilibrium positions that exhibit saddle-node bifurcations. Section IV formulates algebraic equilibrium equations and uses them to produce the bifurcation diagrams in Figs. 6 and 7.

Significance. If the equilibrium analysis is correct, the paper offers a simple reduced-order picture of how inertia and density control aggregation in coherent vortices, with falsifiable predictions about limit points in the equilibrium locus. The work is entirely model-based, using standard closures (Maxey–Riley, Magnus lift, torque equation) and an analytic vortex taken from prior literature, with no fitted parameters. The new ingredient—coupling spin to translation—is physically motivated, and the numerical observation of slow vertical oscillations at St ~ 1 is interesting. However, the central quantitative claim (the bifurcation structure) rests on an algebraic equation that currently contains a sign error, so the main quantitative conclusion is not yet supported.

major comments (2)
  1. [Section IIB, Eq. (10), fourth line] The azimuthal equilibrium equation contains a sign error. Substituting vr,0 = vz,0 = 0, ωz,0 = φz, uθ = φz r0 into the azimuthal component of Eq. (5) gives 0 = 2(φz r0 − vθ,0)/[(2+ρ)St] + 6ρφz U0/(2+ρ) + 3ρW0ωr,0/[2(2+ρ)]. Multiplying by (2+ρ)/2 yields (φz r0 − vθ,0)/St + 3ρφz U0 + (3ρ/4)W0ωr,0 = 0, i.e. (vθ,0 − φz r0)/St − 3ρ(U0φz + (1/4)W0ωr,0) = 0. The printed Eq. (10) has a plus sign before the bracket: (vθ,0 − φz r0)/St + 3ρbar(U0φz + (1/4)W0ωr,0) = 0. Since Figs. 6 and 7 are computed from Eq. (10), the equilibrium locations, their stability classification, and the claimed saddle-node bifurcation structure must be recomputed after correcting this sign.
  2. [Section IIB, Eq. (5)] The symbol ω0^2 appears in the radial component of Eq. (5) but is never defined in the manuscript. From comparison with Eq. (10) one can infer that ω0 is intended to equal φz, but this identification should be stated explicitly. As printed, Eq. (5) is not reproducible, and the derivation of the equilibrium system cannot be checked without guessing the meaning of ω0.
minor comments (6)
  1. [Section IIB, Eq. (2) and Eq. (10)] The notation for the density ratio is inconsistent: the text defines ρbar = ρf/ρp, but Eq. (2) and Eq. (5) use ρ without a bar, while Eq. (10) mixes ρbar and ρ, e.g., the third line contains both 3ρbar and 2(2+ρ). The authors should use a single symbol throughout.
  2. [Section III, page 11] The paragraph discussing Fig. 5 contains the literal placeholder '(rephrase)', and several sentences in Section III are grammatically incomplete. The manuscript needs a careful copyediting pass.
  3. [Section IIB, after Eq. (2)] The Froude number is misspelled as the 'Froud number'.
  4. [Section V, final sentence] The phrase 'mass biofueling processes' is unclear; it should presumably be 'biofouling' or 'biofuel production'. Please correct.
  5. [Section IV, Eq. (10)] The symbols U0, W0, and Φθ are used in Eq. (10) but are defined only in the following sentence. Define them immediately before presenting the equation.
  6. [Section III, Fig. 2 and Fig. 3] The text says that low-Stokes particles exhibit 'sustained low-frequency oscillatory motion' while high-Stokes oscillations 'dissipate completely within approximately five to ten turnover times'; please clarify which regimes have sustained versus damped oscillations, since the phrasing currently appears contradictory.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the derivation chain is self-contained and all load-bearing inputs come from external models and standard closures.

full rationale

I walked the derivation chain and found no circular step. The carrier flow, Eq. (1), is taken from prior external literature (Rypina et al. 2015, 2024; Pratt et al. 2014) and is not derived from, nor fitted to, the paper's own target results. The particle momentum equation, Eq. (5), and the rotational dynamics, Eq. (7), are standard Maxey-Riley and torque closures from external references [38, 40], with the rotating-frame transformation cited to [37, 39]; none of these cited results were produced by the present authors, and none depend on the paper's conclusions. The equilibrium system, Eq. (10), is obtained by substituting the stated steady-orbit conditions, Eqs. (8)-(9), into Eqs. (5) and (7); this is a direct algebraic reduction, not an assumption of the answer. The Stokes number, density ratio, and vortex frequency are scanned control parameters, not fitted quantities, and the reported equilibria, stability, and bifurcations are computed by solving the resulting algebraic system and linearizing the ODEs. No parameter is fitted to a subset of data and then renamed a prediction, no prior result by the same authors is invoked as a load-bearing uniqueness or existence theorem, and no known result is merely relabeled. The skeptical note concerning a possible sign discrepancy in Eq. (10) is a correctness concern about the algebraic reduction, not a circularity concern: even if the printed equation were erroneous, the derivation path from the governing equations is explicit and does not presuppose its own conclusion. Accordingly, the paper merits a circularity score of 0.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The paper introduces no new forces, particles, or mediators; it applies standard Maxey-Riley and Magnus closures to an existing vortex model from the literature. The scanned dimensionless parameters are control variables, not fitted coefficients, so the free-parameter ledger is empty. The main assumptions are modeling choices about the carrier flow, the force closures, and the neglect of unsteadiness and particle interactions.

assumptions (4)
  • domain assumption The analytical vortex flow field Eq. (1) accurately represents a steady rotating cylindrical vortex in the parameter range studied.
    Invoked in Section II A; the model is phenomenological and derived from prior work on rotating cylindrical flows, not from first principles for this study.
  • domain assumption Particle dynamics obey the Maxey-Riley equation augmented with a Magnus lift term and a Stokes torque relaxation law (Eqs. 5 and 7), with Basset history, Faxen corrections, and finite-Reynolds corrections neglected.
    Stated in Section II B; validity is assumed for solid spheres at Stokes numbers up to 1 and for the range of density ratios considered.
  • domain assumption The flow is one-way coupled and particle-particle interactions are negligible.
    Stated in Section II; the dilute suspension assumption means particles do not modify the carrier flow or interact with each other.
  • domain assumption Externally imposed perturbations that induce chaotic advection are neglected, so the steady vortex captures the essential transport physics.
    Stated in the Introduction (page 3); prior studies of the same vortex found chaotic advection when time-dependent perturbations are added, so this assumption is load-bearing for the clean periodic-orbit picture.

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Cite this review

Pith. "Pith review of Heavy Particle Motion in Rotational Vortices." pith.science (2026). https://pith.science/paper/W6X4UDTB

@misc{pith2026250609821,
  author       = {Pith},
  title        = {Pith review of: Heavy Particle Motion in Rotational Vortices},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/W6X4UDTB}},
  note         = {Machine review of arXiv:2506.09821}
}
read the original abstract

This study examines the motion of spherical inertial particles in a three-dimensional rotating cylindrical vortex - a simplified model of geophysical flow structures such as oceanic eddies. The analytical vortex formulation enables the isolation of the key mechanisms that govern particle transport in rotating flows. Using Lagrangian particle tracking simulations, we investigate the influence of drag, buoyancy, virtual mass, Coriolis, and Magnus lift forces across a range of particle sizes, densities, and vortex rotation rates. Results show that particle aggregation and periodic stability depend on both particle inertia and flow parameters. Rotational lift forces, though often neglected for spherical particles, become dominant at moderate particle Stokes numbers and introduce slow vertical oscillations in both particle position and spin. The balance of forces determines whether particles settle into stable periodic orbits or escape the vortex. Our analysis reveals unique equilibrium positions that exhibit bifurcations, with multiple or vanishing steady states depending on particle and flow characteristics. Our results demonstrate how particle aggregation and orbital stability stem from the complex coupling between rotational lift, Coriolis, drag, buoyancy, and virtual mass forces. This model may promote informed modeling of dispersed phase transport in marine flows and industrial mixing processes.

Figures

Figures reproduced from arXiv: 2506.09821 by the authors.

Figure 1
Figure 1. FIG. 1. Three-dimensional trajectories of neutrally buoyant [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Dynamic response of neutrally buoyant [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Rotational dynamics neutrally buoyant [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figures from the paper (4 more)
Figure 4
Figure 4. Figure 4: FIG. 4. Dynamic response of St [PITH_FULL_IMAGE:figures/full_fig_p010_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. Dynamic response of St [PITH_FULL_IMAGE:figures/full_fig_p011_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6. Particle equilibrium points as a function of its density ratio [PITH_FULL_IMAGE:figures/full_fig_p013_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7. Spatial translation of the particle equilibrium points [PITH_FULL_IMAGE:figures/full_fig_p014_7.png]

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Reference graph

Works this paper leans on

41 extracted references · 40 canonical work pages

  1. [1]

    G. I. Ogilvie, Astrophysical fluid dynamics, Journal of Plasma Physics82, 205820301 (2016)

    work page 2016

  2. [2]

    Corre- spondingly, we fix the Froude number at Fr−2 = 10, corresponding to a vortex depth and vertical velocity in the order of unity. Finally, the physicality of the analysis is maintained by limiting the computational domain to the cylindrical vortex domain; particles crossing out of it are eliminated from the simulation. To investigate the particle dyn...

    work page

  3. [3]

    Biferale, F

    L. Biferale, F. Bonaccorso, I. M. Mazzitelli, M. A. van Hinsberg, A. S. Lanotte, S. Musacchio, P. Per- lekar, and F. Toschi, Coherent structures and extreme events in rotating multiphase turbulent flows, Physical Review X6, 041036 (2016)

    work page 2016

  4. [4]

    H. P. Greenspan,The Theory of Rotating Fluids, facsim ed. with corr. and a ref. suppl. ed. (Breukelen press, Brookline, 1990)

    work page 1990

  5. [5]

    E. J. Hopfinger and G. J. F. Van Heijst, Vortices in Rotating Fluids, Annual Review of Fluid Mechanics 25, 241 (1993)

    work page 1993

  6. [6]

    Dold and K

    P. Dold and K. Benz, Rotating magnetic fields: Fluid flow and crystal growth applications, Progress in Crystal Growth and Characterization of Materials38, 39 (1999)

    work page 1999

  7. [7]

    A. L. Andrady, Microplastics in the marine environment, Marine Pollution Bulletin62, 1596 (2011)

    work page 2011

  8. [8]

    B. R. Sutherland, M. DiBenedetto, A. Kaminski, and T. Van Den Bremer, Fluid dynamics challenges in predicting plastic pollution transport in the ocean: A perspective, Physical Review Fluids8, 070701 (2023)

    work page 2023

Show all 41 references
  1. [9]

    D. B. Chelton, P. Gaube, M. G. Schlax, J. J. Early, and R. M. Samelson, The influence of nonlinear mesoscale eddies on near-surface oceanic chlorophyll, Science334, 328 (2011)

    2011

  2. [10]

    Onink, D

    V. Onink, D. Wichmann, P. Delandmeter, and E. van Sebille, The Role of Ekman Currents, Geostrophy, andStokesDriftintheAccumulationofFloatingMicroplastic,JournalofGeophysicalResearch: Oceans 124, 1474 (2019)

    2019

  3. [11]

    Rahmani, A

    M. Rahmani, A. Gupta, and L. Jofre, Aggregation of microplastic and biogenic particles in upper-ocean turbulence, International Journal of Multiphase Flow157, 104253 (2022)

    2022

  4. [12]

    Balachandar and J

    S. Balachandar and J. K. Eaton, Turbulent dispersed multiphase flow, Annual Review of Fluid Me- chanics42, 111 (2010)

    2010

  5. [13]

    G. A. Voth and A. Soldati, Anisotropic particles in turbulence, Annual Review of Fluid Mechanics49, 249 (2017)

    2017

  6. [14]

    Brandt and F

    L. Brandt and F. Coletti, Particle-Laden Turbulence: Progress and Perspectives, Annual Review of Fluid Mechanics54, 159 (2022)

    2022

  7. [15]

    F. J. Beron-Vera, Nonlinear dynamics of inertial particles in the ocean: from drifters and floats to marine debris and Sargassum, Nonlinear Dynamics103, 1 (2021)

    2021

  8. [16]

    Marcu and E

    B. Marcu and E. Meiburg, The effect of streamwise braid vortices on the particle dispersion in a plane mixing layer. i. equilibrium points and their stability, Physics of Fluids8, 715 (1996)

    1996

  9. [17]

    Marcu, E

    B. Marcu, E. Meiburg, and N. Raju, The effect of streamwise braid vortices on the particle dispersion in a plane mixing layer. ii. nonlinear particle dynamics articles you may be interested in, Physics of Fluids8, 734 (1996)

    1996

  10. [18]

    Marcu and E

    B. Marcu and E. Meiburg, Three-dimensional features of particle dispersion in a nominally plane mixing 17 layer, Physics of Fluids8, 2266 (1996)

    1996

  11. [19]

    Dagan, Settling of particles in the vicinity of vortex flows, Atomization and Sprays31, 33 (2021)

    Y. Dagan, Settling of particles in the vicinity of vortex flows, Atomization and Sprays31, 33 (2021)

    2021

  12. [20]

    Avni and Y

    O. Avni and Y. Dagan, Dynamics of evaporating respiratory droplets in the vicinity of vortex dipoles, International Journal of Multiphase Flow148, 103901 (2022)

    2022

  13. [21]

    Marcu, E

    B. Marcu, E. Meiburg, and P. K. Newton, Dynamics of heavy particles in a burgers vortex, Physics of Fluids7, 400 (1995)

    1995

  14. [22]

    R. H. IJzermans and R. Hagmeijer, Accumulation of heavy particles in n-vortex flow on a disk, Physics of Fluids18, 063601 (2006)

    2006

  15. [23]

    Angilella, Dust trapping in vortex pairs, Physica D: Nonlinear Phenomena239, 1789 (2010)

    J.-R. Angilella, Dust trapping in vortex pairs, Physica D: Nonlinear Phenomena239, 1789 (2010)

    2010

  16. [24]

    Ravichandran and R

    S. Ravichandran and R. Govindarajan, Waltz of tiny droplets and the flow they live in, Physical Review Fluids7, 110512 (2022)

    2022

  17. [25]

    Avni and Y

    O. Avni and Y. Dagan, Droplet dynamics in Burgers vortices. I. Mass transport, Physical Review Fluids 8, 083604 (2023)

    2023

  18. [26]

    Avni and Y

    O. Avni and Y. Dagan, Droplet dynamics in Burgers vortices. II. Heat transfer, Physical Review Fluids 8, 083605 (2023)

    2023

  19. [27]

    Wang and Y

    N. Wang and Y. Dagan, Brownian particle diffusion in generalized polynomial shear flows, Physical Review E110, 024117 (2024)

    2024

  20. [28]

    Dagan, J

    Y. Dagan, J. Greenberg, and D. Katoshevski, Similarity solutions for the evolution of polydisperse droplets in vortex flows, International Journal of Multiphase Flow97, 1 (2017)

    2017

  21. [29]

    Dagan, D

    Y. Dagan, D. Katoshevski, and J. B. Greenberg, Particle and droplet clustering in oscillatory vortical flows, Atomization and Sprays27, 629 (2017)

    2017

  22. [30]

    Dagan, D

    Y. Dagan, D. Katoshevski, and J. B. Greenberg, Similarity solutions for the evolution of unsteady spray diffusion flames in vortex flows, Combustion Science and Technology190, 1110 (2018)

    2018

  23. [31]

    Avni and Y

    O. Avni and Y. Dagan, Dispersion of free-falling saliva droplets by two-dimensional vortical flows, Theoretical and Computational Fluid Dynamics36, 993 (2022)

    2022

  24. [32]

    S. R. Yerasi, R. Govindarajan, and D. Vincenzi, Spirographic motion in a vortex, Physical Review Fluids7, 074402 (2022)

    2022

  25. [33]

    M. H. DiBenedetto, N. T. Ouellette, and J. R. Koseff, Transport of anisotropic particles under waves, Journal of Fluid Mechanics837, 320 (2018)

    2018

  26. [34]

    L.K.Clark, M.H.DiBenedetto, N.T.Ouellette,andJ.R.Koseff,Dispersionoffinite-size, non-spherical particles by waves and currents, Journal of Fluid Mechanics954, A3 (2023)

    2023

  27. [35]

    L. K. C. Sunberg, M. H. DiBenedetto, N. T. Ouellette, and J. R. Koseff, Parametric study of the dispersion of inertial ellipsoidal particles in a wave-current flow, Physical Review Fluids9, 034302 (2024)

    2024

  28. [36]

    I. I. Rypina, L. J. Pratt, P. Wang, T. M. Özgökmen, and I. Mezic, Resonance phenomena in a time- dependent, three-dimensional model of an idealized eddy, Chaos: An Interdisciplinary Journal of Non- linear Science25, 087401 (2015). 18

    2015

  29. [37]

    L. J. Pratt, I. I. Rypina, T. M. Özgökmen, P. Wang, H. Childs, and Y. Bebieva, Chaotic advection in a steady, three-dimensional, Ekman-driven eddy, Journal of Fluid Mechanics738, 143 (2014)

    2014

  30. [38]

    I. I. Rypina, L. J. Pratt, and M. Dotzel, Aggregation of slightly buoyant microplastics in 3D vortex flows, Nonlinear Processes in Geophysics31, 25 (2024)

    2024

  31. [39]

    M. R. Maxey and J. J. Riley, Equation of motion for a small rigid sphere in a nonuniform flow, Physics of Fluids26, 883 (1983)

    1983

  32. [40]

    F. J. Beron-Vera, M. J. Olascoaga, and P. Miron, Building a Maxey–Riley framework for surface ocean inertial particle dynamics, Physics of Fluids31, 096602 (2019)

    2019

  33. [41]

    C. T. Crowe, J. D. Schwarzkopf, M. Sommerfeld, and Y. Tsuji,Multiphase Flows with Droplets and Particles(CRC Press, Boza Raton, 2011) publication Title: CRC press

    2011

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Reviewed August 7, 2026 · model on record in the stance chip above.