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This paper establishes that caustics of inertial particles are decided by early alignment with the compressional direction of the background strain, not by strain magnitude alone, and that finitely dense particles require stronger strain th

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2026-08-03 09:19 UTC pith:7Q3QFEUO

load-bearing objection Solid finite-density extension of caustics theory with a clean point-vortex scaling; the turbulence mechanism is plausible but needs a prospective test. the 3 major comments →

arxiv 2601.14179 v2 pith:7Q3QFEUO submitted 2026-01-20 physics.flu-dyn cond-mat.stat-mechnlin.CD

Caustics of finitely dense inertial particles

C. Rajarshi , Rama Govindarajan This is my paper
classification physics.flu-dyn cond-mat.stat-mechnlin.CD
keywords causticsinertial particlespreferential clusteringcompressional strain alignmentfinite density ratioStokes numberpoint-vortex flowtwo-dimensional turbulence
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper asks why some inertial particles in a turbulent flow undergo caustics — extreme clustering events in which the particle number density diverges — while others that visit the same high-strain regions do not. Its answer is that early alignment of the particle velocity with the local strain field decides the outcome: particles that stay aligned with the compressional direction slow down, follow curved paths, and remain long enough in high-strain regions to collapse, while 'survivor' particles that briefly align with the extensional direction shoot through quickly and escape. The paper extends the analysis to particles denser than the fluid but finitely dense, showing that lighter particles must sample stronger background strain to form caustics, with a parameter-free point-vortex prediction that the caustics time scales as t_c = r0²/√(3α−2) and diverges for neutrally buoyant particles. If correct, this gives a practical local criterion for identifying caustics from particle velocity-gradient tracking, relevant to collision-rate estimates in rain formation, planetesimal growth, and plankton encounters.

Core claim

Caustics of inertial particles form not merely because a particle visits a high-strain region, but because the particle approaches that region aligned with the compressional eigendirection of the background strain and with moderate velocity. Such C particles experience a minimum in the strain-vorticity invariant Q several time units before the caustic, exit the high-strain region, and then collapse in a region of only moderate strain; survivor S particles, which reach comparable strain levels, first pass through extensional strain, which straightens and accelerates them so that they transit the high-strain region too quickly to collapse. This distinction holds across all density ratios studi

What carries the argument

The central object is the particle velocity-gradient tensor Z = St∇v, whose trace δ = Tr(Z) is tracked for every particle; a caustic is declared when δ crosses a large negative threshold, since the density diverges as δ → −∞. The decisive mechanism is the alignment angle θ_v between the particle velocity and the two eigendirections of the background strain, ê_+ (extensional) and ê_− (compressional): staying aligned with ê_− at moderate speed is what keeps a particle in a high-strain region long enough to collapse. In point-vortex flow the argument is carried by a parameter-free inner-layer solution R(T) = √(T²/R0² + R0²), which yields the caustics-time and radius scalings t_c = r0²/√(3α−2) a

Load-bearing premise

The central claim rests on the simplified equation of motion for a small spherical particle (Eq. 1) with the history force and Faxén corrections neglected, so the quantitative strain thresholds and near-neutral-density behavior could shift if those forces are significant.

What would settle it

Run the same 2D-turbulence setup with the history force included and compare the ⟨Q⟩_min(α) curve and the C/S alignment distributions: if near-neutrally buoyant particles (α≈0.7–0.8) no longer need a more negative Q than infinite-density ones, or no longer show the compressional-versus-extensional alignment split, the central claim is refuted.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • A local, Lagrangian criterion based on early particle–strain alignment (θ_v near 0 at moderate speed) can identify particles destined for caustics, potentially replacing expensive pair-collision statistics in collision-rate estimates.
  • The point-vortex scaling t_c = r0²/√(3α−2) predicts that lighter particles take longer to form caustics and that neutrally buoyant particles (α=2/3) never do, suppressing caustics-mediated collisions near neutral buoyancy.
  • Finite particle density raises the strain threshold for caustics; the measured ⟨Q⟩_min(α) relation quantifies how much stronger background strain lighter particles need, implying a density-dependent correction to collision rates.
  • The frozen-particle caustics condition (Eq. 6) is an analytic sufficient condition for finitely dense particles, but the moving-particle results show it should not be used as a predictor in real flows because particles leave the high-strain region before collapsing.
  • Because the C/S distinction is observed across all density ratios in both point-vortex flow and 2D turbulence, a single alignment-based mechanism is claimed to govern caustics formation for all particles denser than the fluid.

Where Pith is reading between the lines

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

  • A natural out-of-sample test: classify particles as C or S at an early time using only θ_v and speed (before the strain minimum), then check against later caustics; if early alignment predicts the outcome better than strain magnitude, the causal reading is confirmed.
  • Since the point-vortex scaling is parameter-free, a single-vortex laboratory experiment with particles of different density ratios could directly test the divergence of caustics time near α = 2/3.
  • The paper's own caveat suggests the near-neutral regime is the fragile one: if history forces diminish clustering at O(1) Stokes number as earlier studies find, the density-dependence of the strain threshold may be weaker than reported.
  • If the alignment mechanism extends to three-dimensional turbulence, it would give a frame-independent geometric predictor usable in subgrid models of particle-laden flows.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 6 minor

Summary. The paper develops a Lagrangian framework for detecting caustics of inertial particles with finite density (2/3<α≤1) in two-dimensional flows. It derives an evolution equation for the particle velocity-gradient tensor Z (Eq. 3) and a sufficient frozen-particle caustics condition (Eq. 6) extending Ref. [18]. In a single point-vortex flow, an inner asymptotic analysis yields a parameter-free scaling for the caustics time tc = r0^2/√(3α−2) and radius rc = √2 r0, confirmed by numerical integration (Fig. 2). In 2D turbulence, conditional statistics on caustics (C) and survivor (S) particles show that C particles align with the compressional eigendirection of the background strain before caustics, while S particles align with the extensional direction, have straighter and faster trajectories, and spend less time in high-strain regions. The paper claims these differences are universal across particle density. The authors acknowledge the neglect of Basset–Boussinesq history and Faxén corrections as limitations.

Significance. If the central claims hold, this paper extends caustics theory from infinitely dense to finitely dense particles and offers a mechanistic explanation for why only a fraction of particles visiting high strain actually form caustics. The point-vortex scaling is a clean, parameter-free prediction that is verified numerically, and the derivation of Eq. (3) is internally consistent. The work makes its code and data publicly available, which is a strength. However, the strain-alignment mechanism is inferred from retrospective statistics on classes defined by the outcome, and the quantitative α-dependence rests on a model that neglects history forces known to affect near-neutral clustering. These concerns need to be addressed before the universality claim is fully accepted.

major comments (3)
  1. [§IV B 2, Figs. 6 and 7] The central causal claim—that early alignment with the compressional versus extensional strain eigendirection determines whether a particle forms a caustics—is not established by the retrospective conditional statistics. The classes C and S are defined by survival outcome, and all comparisons are made on the same data used to define them. A particle that exits a high-strain region quickly will mechanically have its velocity aligned with the local extensional eigenvector at the exit, regardless of whether that alignment caused its survival. The time-shift to t_m does not remove this selection effect. To support the causal mechanism, the authors should provide a prospective test—for example, use the initial alignment as a predictor of C versus S on an independent subsample or simulation, or perturb the alignment in a controlled model—and report error bars on the conditional averages.
  2. [§II and §V, Eq. (1)] The quantitative claim that finite-density particles require stronger background strain (Fig. 4g) and the universality across α are based on the Maxey–Riley equation without the Basset–Boussinesq history force. The authors themselves cite Refs. [32–34] showing that history forces reduce clustering for near-neutrally-buoyant particles at O(1) Stokes numbers. Since the paper emphasizes near-neutral densities (α=0.7–0.8), the neglect could shift the measured ⟨Q⟩_min(α) relation and weaken the universality claim. Please add a sensitivity estimate—for instance, a comparison with the history force included for a subset of parameters, or an order-of-magnitude bound—so readers can gauge the robustness of the α-dependence.
  3. [§III and §IV A, Eqs. (6) and (14)] The frozen-particle caustics condition (Eq. 6) is presented as a key extension, but the point-vortex analysis shows it gives a "vastly different" and unreliable prediction for moving particles (compare Eq. (14) with the actual threshold from Eq. (13)). To avoid overclaiming, the paper should clarify the limited applicability of Eq. (6) upfront and justify why it is presented as a central result, or frame it purely as a pedagogical/limiting-case analysis. As written, the paper gives equal weight to an approach it later deems unreliable, which obscures the main contribution.
minor comments (6)
  1. [§II, Eq. (8)] The trace equation could be explicitly reduced to the known α=1 case, which would help readers appreciate the effect of finite density.
  2. [§IV B, after Eq. (16)] The threshold δ=−20 for caustics detection is arbitrary; the sensitivity of the reported statistics to this threshold should be discussed or demonstrated.
  3. [§IV A, Eq. (11)] The statement "valid for all Stokes numbers and all particle densities at the lowest order" is slightly confusing given that the scaling ℓ² involves α; clarify that the inner solution is parameter-free after rescaling, not that α disappears.
  4. [§IV B 2] The definition of survivor particles depends on Q_m, the mean of the minimum Q for C particles. The text should specify how Q_m is computed in practice and whether the results are sensitive to this definition.
  5. [Figure 7 caption] Typo: "inifinitely dense" should be "infinitely dense." Also, "extensive" in Fig. 9 caption should likely be "extensional".
  6. [§V, Conclusions] The sentence "regions of large compressive strain, characterized by regions of negative Q and R, do not promote caustics" seems to contradict the earlier finding that high strain is essential for caustics. This should be rephrased to clarify that for finitely dense particles, the nature of the strain (compressive vs. extensive) and the finite-density correction alter the simple expectation, but high strain regions are still necessary.

Circularity Check

1 steps flagged

Analytic results are self-contained; the C/S mechanism is partly constructed from the outcome-defined S class and a C-derived Qm threshold.

specific steps
  1. fitted input called prediction [Section IV B 2, around Fig. 5 and the paragraph on survivor S particles]
    "We define survivor S particles as those that sample Q≤Qm, but do not form caustics. ... The ⟨Q⟩... S particles explore larger negative Q regions — a consequence of our requirement that they visit a strain higher than Qm — and also spend shorter times in them. ... Therefore, we conclude that encountering a compressive strain with a moderate particle velocity is critical in staying in large strain regions long enough to form caustics."

    The threshold Qm is computed from C particles' minimum-Q histories, and the S class is defined by the combined criteria 'sample Q≤Qm' and 'do not form caustics'. The subsequent causal mechanism (extensional alignment, high speed, short residence) is read off averages over classes that were selected using the outcome and a C-derived threshold. The observation that S particles' δ recovers is essentially the statement that they did not caustics, and the shorter residence is a consequence of the Q≤Qm selection plus survival, not an independent test. Since no out-of-sample or prospective validation is given, the 'compressive strain is critical' conclusion is partially a restatement of the selection rule rather than a standalone prediction.

full rationale

The core mathematical derivation is not circular. Eq. (6) is obtained algebraically from Eq. (5) in Appendix A using the Cayley-Hamilton procedure, and it reduces to Ref. [18] at α=1. The point-vortex analysis is parameter-free: the inner scaling ℓ²=τ√(3α−2) is a dominant-balance choice, Eq. (11) follows from Eq. (10), and the caustics time tc=r0²/√(3α−2) is confirmed numerically. The turbulence simulations solve Eq. (1) and Eq. (3) without fitting the quantities they report. The paper's own limitations (neglect of Faxén and Basset-Boussinesq history forces, no 3D confirmation) are stated explicitly and do not constitute circularity. The only concerning step is the retrospective C/S classification in §IV B 2: S is defined by the absence of caustics and by a threshold Qm taken from C particles, so the mechanism inferred from C/S averages is partly selected by the outcome. This weakens the causal claim but does not affect the scaling laws or the frozen-caustics condition; hence a moderate score of 3 rather than a higher one.

Axiom & Free-Parameter Ledger

5 free parameters · 5 axioms · 0 invented entities

Central claim rests on the Maxey–Riley model (with history forces neglected), the continuum-Z description, the frozen-particle idealization (later shown unreliable by the authors themselves), a correlational causal inference, and the custom 2D turbulence model. Five hand-chosen or data-derived parameters (δ threshold, γ, numerical parameters, Qm, inner scaling) enter the quantitative results. No new physical entities are postulated; the C/S/A classes are analytical classifications and ghost particles are inherited from Falkovich & Pumir [13].

free parameters (5)
  • Caustics detection threshold δ = Tr(Z) = −20 (Tr Z in DNS units)
    Hand-set threshold for declaring a caustics event; the authors argue blow-up is inevitable beyond it, but C/S/A counts (Table I) depend on its value. §IV B.
  • Linear drag coefficient γ = 10⁻²
    Added to the 2D turbulence equation (Eq. 15) to prevent inverse-cascade blow-up; modifies the flow statistics that particles sample. §IV B.
  • Numerical parameters (ν, kf, grid, run time) = ν=8×10⁻⁶, kf=4, 1024², 125τη
    Hand-chosen simulation parameters; no sensitivity study shows the results are robust to their variation. §IV B.
  • Survivor-defining threshold Qm = mean of per-particle minimum Q among C particles
    Data-derived threshold used to select S particles; makes the 'S particles visit comparable strain' claim true by construction. §IV B 2.
  • Inner scaling ℓ² = τ√(3α−2) = √(3α−2)
    Chosen distinguished-limit scaling in the point-vortex analysis (Eq. 11); the predicted tc ∝ (3α−2)^(−1/2) follows from this choice, though Fig. 2 confirms it numerically. §IV A.
axioms (5)
  • domain assumption Gatignol–Maxey–Riley equation (Eq. 1): one-way coupled spherical point particles with drag + added mass, no gravity, Faxén, lift, or Basset history terms
    Basis of Eqs. (3), (8), (10) and the DNS particle evolution. The paper states history forces "will be a factor for future evaluation" (§II) and that clustering of near-neutral particles at O(1) St is reduced by them ([32,33]).
  • domain assumption Continuum particle velocity field v exists until caustics; dn/dt = −n∇·v (Eq. 2); Z = St∇v evolves by Eq. (3)
    Justifies Lagrangian Z-tracking; after δ < −20 the particle is reinitialized with Z→−Z (ghost-particle picture of [13]). Singularities are removed by hand. §II, §IV B.
  • ad hoc to paper Frozen-particle analysis (constant A) is informative about moving-particle caustics
    The paper itself demonstrates the opposite near a vortex: Eq. (14) differs "vastly" from the moving-particle criterion, and the frozen approach is called "rather limited in reliability" (§IV A, §V).
  • domain assumption Causality runs from early strain alignment → path shape → caustics outcome
    The mechanism is inferred from time-aligned conditional statistics (θv, speed, curvature) within the same dataset that defines the C and S classes; no perturbation, intervention, or independent prediction tests the arrow. §IV B 2.
  • domain assumption The forced 2D turbulence with linear drag (Eq. 15) yields representative strain statistics for caustics
    Custom forcing fω = −νkf(cos(kfx)+sin(kfx)) and drag mitigate the inverse cascade; representativeness for real 2D and 3D flows is assumed. §IV B.

pith-pipeline@v1.3.0-alltime-deepseek · 13254 in / 31915 out tokens · 308748 ms · 2026-08-03T09:19:37.950757+00:00 · methodology

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read the original abstract

Estimating collision rates is of immense importance in particle-laden flows. An economical way of doing this is to directly identify incidences of caustics, or extreme clustering, by tracking particle velocity gradients in the neighborhoods of individual particles. The objective of this work is two-fold. (i) We find conditions under which caustics form, in point-vortex flow and in two-dimensional turbulence. While caustics are known to form in regions of strain, we show that the velocity alignment with strain directions is key. Particles must remain in compressional strain throughout the process to form caustics, whereas survivor particles: which visit high strain but do not form caustics, briefly go through extensional strain during the early part of the process. This enables survivor particles to attain significantly straighter paths, and to move faster, whereas caustics particles follow paths of high curvature and move slower. As a result, caustics particles stay longer in high-strain regions than survivors. (ii) We ask about the effect of finite particle density, where the particle is denser than the background fluid. We show that finite-density particles need to sample stronger background strain than infinite-density ones to trigger caustics, but our other findings are universal across particle density.

Figures

Figures reproduced from arXiv: 2601.14179 by C. Rajarshi, Rama Govindarajan.

Figure 1
Figure 1. Figure 1: satisfy Eq. (6), whereas solid colors correspond to different numbers of real roots of −α (Z − A) − Z 2 + 3(1 − α)A 2 = 0, (7) [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5 [PITH_FULL_IMAGE:figures/full_fig_p010_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6 [PITH_FULL_IMAGE:figures/full_fig_p012_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7 [PITH_FULL_IMAGE:figures/full_fig_p013_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8 [PITH_FULL_IMAGE:figures/full_fig_p014_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9 [PITH_FULL_IMAGE:figures/full_fig_p014_9.png] view at source ↗

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Reference graph

Works this paper leans on

49 extracted references

  1. [1]

    Behavior of caustics or C particles In Fig. 4 (a-c) we show that C particles of all densities undergo a particular history of the Q field immediately before the caustics formation time tc, experiencing a maximum in strain or minimum Q at t =tm, which is 5− 8 time units before tc. Each particle’s time-series has been shifted so that their tc’s coincide. Th...

  2. [2]

    These experience similar strain levels as the C particles, but never form caustics

    Behavior of survivor S particles Apart from the C and A particles seen before in point-vortex flows, another behavior is seen in turbulent flow, displayed by survivor or ‘S’ particles. These experience similar strain levels as the C particles, but never form caustics. We investigate them below. B¨ atgeet al. [20] found that not all infinitely-dense partic...

    1970

  3. [3]

    Calzavarini, M

    E. Calzavarini, M. Cencini, D. Lohse, and F. Toschi, Quantifying Turbulence-Induced Segregation of Inertial Particles, Phys. Rev. Lett.101, 084504 (2008)

    2008

  4. [4]

    Falkovich, A

    G. Falkovich, A. Fouxon, and MG. Stepanov, Acceleration of rain initiation by cloud turbulence, Nature 419, 151 (2002)

    2002

  5. [5]

    C. T. Crowe, R. A. Gore, and T. R. Troutt, Particle dispersion by coherent structures in free shear flow, Particulate Science and Technology3, 149 (1985)

    1985

  6. [6]

    Balkovsky, G

    E. Balkovsky, G. Falkovich, and A. Fouxon, Intermittent distribution of inertial particles in turbulent flows, Physical Review Letters86, 2790 (2001)

    2001

  7. [7]

    Pumir and M

    A. Pumir and M. Wilkinson, Collisional aggregation due to turbulence, Annual Review of Condensed Matter Physics7, 141 (2016)

    2016

  8. [8]

    G¨ uttler, J

    C. G¨ uttler, J. Blum, A. Zsom, C. W. Ormel, and C. P. Dullemond, The outcome of protoplanetary dust growth: Pebbles, boulders, or planetesimals?-I. Mapping the zoo of laboratory collision experiments, Astronomy & Astrophysics513, A56 (2010)

    2010

  9. [9]

    R. A. Shaw, Particle-turbulence interactions in atmospheric clouds, Annual Review of Fluid Mechanics 35, 183 (2003)

    2003

  10. [10]

    Wilkinson, B

    M. Wilkinson, B. Mehlig, and V. Bezuglyy, Caustic activation of rain showers, Physical review letters 97, 048501 (2006)

    2006

  11. [11]

    Birnstiel, Dust growth and evolution in protoplanetary disks, Annual Review of Astronomy and Astrophysics62, 10.1146/annurev-astro-071221-052705 (2024)

    T. Birnstiel, Dust growth and evolution in protoplanetary disks, Annual Review of Astronomy and Astrophysics62, 10.1146/annurev-astro-071221-052705 (2024)

    doi:10.1146/annurev-astro-071221-052705 2024

  12. [12]

    Meibohm, V

    J. Meibohm, V. Pandey, A. Bhatnagar, K. Gustavsson, D. Mitra, P. Perlekar, and B. Mehlig, Paths to caustic formation in turbulent aerosols, Physical Review Fluids6, L062302 (2021)

    2021

  13. [13]

    Schr¨ apler and J

    R. Schr¨ apler and J. Blum, The physics of protoplanetesimal dust agglomerates. VI. Erosion of large aggregates as a source of micrometer-sized particles, The Astrophysical Journal734, 108 (2011)

    2011

  14. [14]

    K. Wada, H. Tanaka, S. Okuzumi, H. Kobayashi, T. Suyama, H. Kimura, and T. Yamamoto, Growth efficiency of dust aggregates through collisions with high mass ratios, Astronomy & Astrophysics559, A62 (2013)

    2013

  15. [15]

    We note, however, that velocity gradients in turbulence are highly non-Gaussian [16]

    found that there are infinite ways to form caustics in the phase space of invariants of the particle velocity gradient tensor. We note, however, that velocity gradients in turbulence are highly non-Gaussian [16]. Studying collisions between individual infinitely-dense particles, Picardoet al

  16. [16]

    Meneveau, Lagrangian Dynamics and Models of the Velocity Gradient Tensor in Turbulent Flows, Annu

    C. Meneveau, Lagrangian Dynamics and Models of the Velocity Gradient Tensor in Turbulent Flows, Annu. Rev. Fluid Mech.43, 219 (2011)

    2011

  17. [17]

    The present study builds on two others

    showed that high-strain regions, just outside vortices, facilitate head-on collisions. The present study builds on two others. The first is the work of Meibohmet al. [12], who derive a caustics condition, expressed as an inequality, in two-dimensional flow for small infinitely-dense particles, and subsequently [ 18] a condition in three-dimensional flow. ...

  18. [18]

    Falkovich and A

    G. Falkovich and A. Pumir, Sling effect in collisions of water droplets in turbulent clouds, Journal of the Atmospheric Sciences64, 4497 (2007)

    2007

  19. [19]

    Bhatnagar, V

    A. Bhatnagar, V. Pandey, P. Perlekar, and D. Mitra, Rate of formation of caustics in heavy particles advected by turbulence, Philosophical Transactions of the Royal Society A380, 20210086 (2022)

    2022

  20. [20]

    Barta and J

    R. Barta and J. Vollmer, Caustics in turbulent aerosols: An excitable system approach, Journal of Fluid Mechanics949, A36 (2022)

    2022

  21. [21]

    J. R. Picardo, L. Agasthya, R. Govindarajan, and S. S. Ray, Flow structures govern particle collisions in turbulence, Physical Review Fluids4, 032601 (2019)

    2019

  22. [22]

    Meibohm, K

    J. Meibohm, K. Gustavsson, and B. Mehlig, Caustics in turbulent aerosols form along the Vieillefosse line at weak particle inertia, Physical Review Fluids8, 024305 (2023). 17

    2023

  23. [23]

    Meibohm, L

    J. Meibohm, L. Sundberg, B. Mehlig, and K. Gustavsson, Caustic formation in a non-Gaussian model for turbulent aerosols, Physical Review Fluids9, 024302 (2024)

    2024

  24. [24]

    B¨ atge, I

    T. B¨ atge, I. Fouxon, and M. Wilczek, Quantitative prediction of sling events in turbulence at high Reynolds numbers, Physical Review Letters131, 054001 (2023)

    2023

  25. [25]

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

    1983

  26. [26]

    Gatignol, The fax´ en formulae for a rigid particle in an unsteady non-uniform stokes flow, (1983)

    R. Gatignol, The fax´ en formulae for a rigid particle in an unsteady non-uniform stokes flow, (1983)

    1983

  27. [27]

    R. Volk, E. Calzavarini, G. Verhille, D. Lohse, N. Mordant, J.-F. Pinton, and F. Toschi, Acceleration of heavy and light particles in turbulence: Comparison between experiments and direct numerical simulations, Physica D: Nonlinear Phenomena237, 2084 (2008)

    2084

  28. [28]

    N. M. Qureshi, U. Arrieta, C. Baudet, A. Cartellier, Y. Gagne, and M. Bourgoin, Acceleration statistics of inertial particles in turbulent flow, The European Physical Journal B66, 531 (2008)

    2008

  29. [29]

    Fiabane, R

    L. Fiabane, R. Zimmermann, R. Volk, J.-F. Pinton, and M. Bourgoin, Clustering of finite-size particles in turbulence, Phys. Rev. E86, 035301 (2012)

    2012

  30. [30]

    Karchniwy, A

    E. Karchniwy, A. Klimanek, and N. E. L. Haugen, The effect of turbulence on mass transfer rates between inertial polydisperse particles and fluid, J. Fluid Mech.874, 1147 (2019)

    2019

  31. [31]

    A. J. Petersen, L. Baker, and F. Coletti, Experimental study of inertial particles clustering and settling in homogeneous turbulence, J. Fluid Mech.864, 925 (2019)

    2019

  32. [32]

    Motoori and S

    Y. Motoori and S. Goto, Multiscale clustering of heavy and light small particles in turbulent channel flow at high Reynolds numbers, International Journal of Heat and Fluid Flow102, 109166 (2023)

    2023

  33. [33]

    Calzavarini, R

    E. Calzavarini, R. Volk, M. Bourgoin, E. Leveque, J.-F. Pinton, and F. Toschi, Acceleration statistics of finite-sized particles in turbulent flow: The role of Fax´ en forces, Journal of Fluid Mechanics630, 179 (2009)

    2009

  34. [34]

    Mathai, D

    V. Mathai, D. Lohse, and C. Sun, Bubbly and Buoyant Particle–Laden Turbulent Flows, Annu. Rev. Condens. Matter Phys.11, 529 (2020)

    2020

  35. [35]

    I. M. Mazzitelli, D. Lohse, and F. Toschi, The effect of microbubbles on developed turbulence, Physics of Fluids15, L5 (2003)

    2003

  36. [36]

    Olivieri, F

    S. Olivieri, F. Picano, G. Sardina, D. Iudicone, and L. Brandt, The effect of the Basset history force on particle clustering in homogeneous and isotropic turbulence, Physics of Fluids26, 041704 (2014)

    2014

  37. [37]

    Daitche, On the role of the history force for inertial particles in turbulence, J

    A. Daitche, On the role of the history force for inertial particles in turbulence, J. Fluid Mech.782, 567 (2015)

    2015

  38. [38]

    Ferry and S

    J. Ferry and S. Balachandar, A fast Eulerian method for disperse two-phase flow, International journal of multiphase flow27, 1199 (2001)

    2001

  39. [39]

    Kapoor, D

    S. Kapoor, D. Jaganathan, and R. Govindarajan, Trapping and extreme clustering of finitely dense inertial particles near a rotating vortex pair, Journal of Fluid Mechanics996, A44 (2024)

    2024

  40. [40]

    M. S. Chong, A. E. Perry, and B. J. Cantwell, A general classification of three-dimensional flow fields, Physics of Fluids A: Fluid Dynamics2, 765 (1990)

    1990

  41. [41]

    Ravichandran and R

    S. Ravichandran and R. Govindarajan, Caustics and clustering in the vicinity of a vortex, Physics of Fluids27, 033305 (2015)

    2015

  42. [42]

    Ravichandran and R

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

    2022

  43. [43]

    Y. Ling, M. Parmar, and S. Balachandar, A scaling analysis of added-mass and history forces and their coupling in dispersed multiphase flows, International Journal of Multiphase Flow57, 102 (2013)

    2013

  44. [44]

    Deepu, S

    P. Deepu, S. Ravichandran, and R. Govindarajan, Caustics-induced coalescence of small droplets near a vortex, Physical Review Fluids2, 024305 (2017)

    2017

  45. [45]

    Pandey, P

    V. Pandey, P. Perlekar, and D. Mitra, Clustering and energy spectra in two-dimensional dusty gas turbulence, Physical Review E100, 013114 (2019)

    2019

  46. [46]

    M. A. T. van Hinsberg, J. H. M. Thije Boonkkamp, F. Toschi, and H. J. H. Clercx, On the Efficiency and Accuracy of Interpolation Methods for Spectral Codes, SIAM J. Sci. Comput.34, B479 (2012)

    2012

  47. [47]

    Calzavarini, M

    E. Calzavarini, M. Kerscher, D. Lohse, and F. Toschi, Dimensionality and morphology of particle and bubble clusters in turbulent flow, Journal of fluid mechanics607, 13 (2008)

    2008

  48. [48]

    Zhang and H

    Y. Zhang and H. Xu, Caustics of inertial particles observed along Lagrangian particle trajectories, Journal of Fluid Mechanics1014, A13 (2025)

    2025

  49. [49]

    Kronborg and J

    J. Kronborg and J. Hoffman, The triple decomposition of the velocity gradient tensor as a standardized real schur form, Physics of Fluids35, 031703 (2023)

    2023