Pith. sign in

REVIEW 4 major objections 6 minor 126 references

Dynamical regimes of thermally convective emulsions

T0 review · 4 major / 6 minor · reviewed 2026-08-12 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read Thermally convecting emulsions settle into two regimes with a shared Ra^{-1/3} droplet-size scaling.

desk verdict The first systematic phase diagram for thermally convective emulsions is a solid contribution, but the r_m ~ Ra^{-1/3} scaling needs a free-exponent fit and a direct test of the large-scale shear closure before it is sold as universal. read the letter →

arxiv 2411.11553 v2 pith:OMXEYK2R submitted 2024-11-18 physics.flu-dyn

Francesca Pelusi , Andrea Scagliarini , Mauro Sbragaglia , Massimo Bernaschi , Roberto Benzi This is my paper · ORCID
classification physics.flu-dyn
keywords emulsionsRayleigh-Bénardconvectiondropletbreakupphaseinversionyield-stressfluidslatticeBoltzmannmethodsizescalingheattransport
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

This paper maps, for the first time, the dynamical regimes of stabilized oil-in-water emulsions in Rayleigh-Bénard thermal convection, using a large set of mesoscale lattice-Boltzmann simulations that sweep the dispersed-phase volume fraction $\phi$ from dilute (0.16) to jammed (0.84) and the Rayleigh number $\mathrm{Ra}$ from about $4\times10^3$ to $8\times10^6$. It establishes that at sufficiently high Rayleigh numbers only two steady convective states survive: a breakup-dominated state for low and moderate $\phi$, where the emulsion remains Newtonian while the droplet number increases, and a phase-inverted state for high $\phi$, where the yield-stress oil-in-water emulsion irreversibly becomes a dilute Newtonian water-in-oil emulsion. In both states the statistically steady mean droplet radius follows the same scaling law, $r_m \sim \mathrm{Ra}^{-1/3}$, obtained by balancing the shear stress of the large-scale convective flow against the Laplace pressure at the droplet surface. The paper also reports $\mathrm{Nu} \sim \mathrm{Ra}^{2/7}$ for heat transport at high $\mathrm{Ra}$ and a droplet-size distribution tail $\sim r^{-10/3}$ in regimes where coalescence is frequent enough to supply large droplets.

What carries the argument

The load-bearing mechanism is a force balance at the droplet scale: the squared velocity fluctuation across a droplet of radius $r_m$, $(\delta_{r_m} u)^2$, is equated to the Laplace pressure $\Sigma/r_m$, the curvature pressure resisting breakup. Estimating the velocity fluctuation from the mean large-scale convective shear, $\delta_{r_m} u \sim (U_{FF}/H) r_m$ with $U_{FF} \sim \sqrt{\alpha g \Delta T H}$ the free-fall velocity, converts this balance into $r_m \sim (\Sigma H)^{1/3} (\alpha g \Delta T)^{-1/3}$, i.e., $r_m \propto \mathrm{Ra}^{-1/3}$ once $\mathrm{Ra} = \alpha g \Delta T H^3/(\nu \kappa)$ is substituted. The numerical side is a GPU-based two-component lattice-Boltzmann solver with a disjoining pressure between droplet interfaces that mimics surfactant stabilization, run in a 2D Rayleigh-Bénard cell of aspect ratio 2; the regime classification tracks the time-dependent Nusselt number and the droplet count relative to the initial value over about 300 simulations spanning $\phi \in [0, 0.84]$ and $\mathrm{Ra} \in [4\times10^3, 8\times10^6]$.

What would settle it

Measure the velocity fluctuation across droplets of size $r_m$ at fixed $\mathrm{Ra}$, or measure $r_m$ at Rayleigh numbers high enough for a turbulent cascade: if $\delta_{r_m} u$ does not grow linearly with $r_m$, or if $r_m$ deviates from $\mathrm{Ra}^{-1/3}$ once inertial-range eddies dominate the droplet scale, the central scaling and the regime classification built on it would need revision.

Watch

Extended reading notes

Core claim

The paper's central claim is that a thermally convecting emulsion driven above a strong enough buoyancy forcing forgets its initial droplet configuration and settles into one of two structurally distinct convective states determined by the dispersed-phase volume fraction $\phi$. For Newtonian emulsions at low-to-moderate $\phi$, the steady state is breakup-dominated: convective shear tears droplets apart faster than they coalesce, the droplet count increases, and the emulsion remains Newtonian with a slightly higher viscosity due to the extra interfacial area. For non-Newtonian, yield-stress emulsions at high $\phi$, the steady state is phase-inverted: intermittent coalescence events during the transient produce a water-in-oil emulsion that is dilute and Newtonian, and this inversion is irreversible, appearing as a discontinuous jump in the Nusselt number with hysteresis when $\mathrm{Ra}$ is ramped down. In both scenarios the mean droplet size obeys $r_m \sim \mathrm{Ra}^{-1/3}$ over nearly a decade of $\mathrm{Ra}$, and the droplet-radius probability distribution develops a power-law tail $\propto r^{-10/3}$ whenever coalescence can replenish the large-droplet population. Around the jamming point the sequence of regimes with increasing $\mathrm{Ra}$ is non-monotonic—stable convection, then coalescence-dominated convection, then stability again, then phase inversion—so the regime diagram is not a simple function of $\phi$.

Load-bearing premise

The scaling law depends on assuming that the velocity difference across a droplet is set by the large-scale convective shear, $\delta_{r_m} u \sim (U_{FF}/H) r_m$, rather than by turbulent eddies or by local yield-stress shear in jammed regions; this assumption is used to derive $\mathrm{Ra}^{-1/3}$ and is not independently measured in the simulations.

Editorial extensions

If this is right

  • Any surfactant-stabilized emulsion in a buoyancy-driven convective flow that reaches sufficiently high $\mathrm{Ra}$ should show its mean droplet size decreasing as $\mathrm{Ra}^{-1/3}$, regardless of whether the steady state is breakup-dominated or phase-inverted.
  • For $\mathrm{Ra}$ above about $5\times10^5$ the heat transport follows $\mathrm{Nu} \sim \mathrm{Ra}^{2/7}$ for all volume fractions, meaning the convective steady states of emulsions transport heat like single-phase turbulent convection.
  • The phase-inverted state is irreversible: once a jammed oil-in-water emulsion inverts to a dilute water-in-oil emulsion, reducing $\mathrm{Ra}$ does not restore the original structure, producing hysteresis in the $\mathrm{Nu}$ versus $\mathrm{Ra}$ curve.
  • The droplet-size distribution exhibits a $r^{-10/3}$ tail at high $\mathrm{Ra}$ only when coalescence is frequent enough to refill the large-droplet population, as in semi-dilute and phase-inverted cases, but not in the very dilute case $\phi = 0.16$.

Reading between the lines

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

  • This suggests a direct experimental test: in a surfactant-stabilized emulsion in a Rayleigh-Bénard cell, the mean droplet size should shrink with the cube root of the imposed temperature difference; a deviation would indicate that convective shear is not the controlling breakup mechanism.
  • Because $\mathrm{Ra}$ is defined with the bare viscosity, the prefactor of the $r_m \sim \mathrm{Ra}^{-1/3}$ law should carry all the $\phi$-dependence through an effective critical Weber number; measuring that prefactor versus $\phi$ would separate rheological from purely convective effects.
  • If the coalescence-dominated regime near jamming is a genuine precursor to phase inversion, the intermittent heat bursts seen in the transient should show critical-like statistics (e.g., power-law waiting times) that sharpen as $\phi$ approaches the jamming point—an analysis not reported in the paper.
  • If the flow is driven to high enough $\mathrm{Ra}$ that a turbulent cascade establishes at the droplet scale, the same force balance with $\epsilon \sim U_{FF}^3/H$ would change the exponent from $\mathrm{Ra}^{-1/3}$ to $\mathrm{Ra}^{-3/5}$; searching for that crossover would test whether the large-scale-shear assumption, rather than the balance itself, is the regime-dependent element.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 6 minor

Summary. The paper reports a large set of two-dimensional lattice Boltzmann simulations of surfactant-stabilized oil-in-water emulsions in Rayleigh-Bénard convection, scanning droplet volume fraction phi from dilute to jammed (0.16 to 0.84) and Rayleigh number Ra from ~4e3 to 8e6. It classifies statistically steady states into conduction, stable convection, breakup-dominated convection, coalescence-dominated convection, and phase-inverted convection, and it documents the associated rheological changes, Nusselt-number behavior, and hysteresis. The central quantitative claim is a parameter-free scaling for the mean droplet radius, r_m ~ Ra^{-1/3}, derived in Sec. IV from a balance between convective shear stress and Laplace pressure, and verified against simulated average droplet sizes in Fig. 9. The paper also reports an r^{-10/3} tail in the droplet-size PDF for semi-dilute emulsions, with an explanation for its absence at low phi.

Significance. If the scaling and regime classification hold, the paper provides a useful first characterization of thermally convective emulsions, a parameter-free prediction for the average droplet size, and a connection between phase inversion and non-Newtonian-to-Newtonian rheological transitions. The strengths are the breadth of the simulation campaign (~300 runs), the explicit force-balance derivation of Eq. (16) without a fitted exponent, the rheological characterization with a Herschel-Bulkley fit, and the use of the open-source TLBfind code. The main weakness is that the central scaling rests on an unverified closure for the droplet-scale velocity fluctuation, and the supporting fit fixes the exponent to the predicted value over a narrow Ra range. These issues are local and addressable, so the paper is a candidate for major revision rather than rejection.

major comments (4)
  1. [Sec. IV, between Eqs. (15) and (16)] The derivation of r_m ~ Ra^{-1/3} replaces the droplet-scale velocity fluctuation with the large-scale shear estimate delta_{r_m}u ~ (U_FF/H) r_m, justified by the statement that 'the Reynolds number is not exceptionally high.' This closure is load-bearing: if the droplet scale lies in an inertial range, the same force balance with delta u ~ (epsilon r)^{1/3} gives r_m ~ Ra^{-3/5}, the classical Hinze scaling. Since the simulations resolve the velocity field, please report a direct measurement of the velocity increment at r ~ r_m (for instance, the second-order structure function at the mean droplet radius) and test which closure actually holds at the relevant scales.
  2. [Sec. IV, Fig. 9] The caption states that the dashed lines are fits to a power law with exponent -1/3, and the data span roughly Ra ≈ 10^6 to 8×10^6, i.e., less than one decade, with only three or four points per volume fraction. A fit with the exponent fixed to the predicted value cannot by itself establish the scaling exponent. Please provide free-exponent fits with confidence intervals, and state whether the data are statistically distinguishable from alternatives such as Ra^{-3/5} or Ra^{-1/2}. If statistical uncertainty prevents a free-exponent fit, this should be stated explicitly rather than implying that Eq. (16) is quantitatively verified.
  3. [Sec. IV, phase-inverted case phi = 0.84] For the phase-inverted state, the droplets analyzed are water droplets in a dilute W/O emulsion, which have a different volume fraction (1 - phi), a different continuous phase, and a different effective rheology from the O/W cases. The same large-scale-shear closure leading to Eq. (16) is assumed there without independent verification. Please either validate the closure for the W/O case separately or soften the claim that a single universal scaling applies to both the breakup-dominated and phase-inverted regimes.
  4. [Secs. III and IV, Figs. 5, 8, 9, 10] The quantitative statements about Nu and droplet-size statistics are made without error bars or convergence checks. The time-averaging intervals are stated to range from 15 to 2000 free-fall times depending on Ra, but no test of statistical convergence is provided. Given that the scaling law in Fig. 9 is asserted to hold 'for almost a decade in Ra,' please add uncertainty estimates for r_m and Nu, and show that the reported averages are converged at the highest Ra values, where the statistics are the shortest.
minor comments (6)
  1. [Sec. IV, Eq. (15)] The balance is written as (delta_{r_m}u)^2 ~ Sigma/r_m; dimensionally a fluid density should appear on the left unless the authors are working in lattice units with rho ≈ 1. Please clarify this in the text.
  2. [Sec. III A, Fig. 2] The authors state that the transition lines are guides, which is an appropriate caveat, but the figure could avoid implying sharp boundaries by using shaded regions or by plotting only the simulation symbols.
  3. [Fig. 4] The axis labels in panels (a)-(f) are garbled in the manuscript (e.g., '˙∞[£10°5]'); please fix the LaTeX or font rendering.
  4. [Sec. II A] There is a typo: 'expertimental conditions' should be 'experimental conditions.'
  5. [Sec. IV, Figs. 8 and 10] Please state whether the PDFs are normalized and over what range of droplet radii the power-law fits are performed, since the log-log plots show only a limited scaling range.
  6. [Sec. IV, text after Eq. (16)] The phrase 'almost a decade in Ra' overstates the plotted range in Fig. 9, which extends from approximately 10^6 to 8×10^6; 'less than a decade' would be more accurate.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the r_m ~ Ra^{-1/3} scaling is derived from a stated force balance and checked against simulation data; self-citations are contextual, not load-bearing.

full rationale

The central quantitative claim of the paper, r_m ~ Ra^{-1/3}, is not circular. In Sec. IV the authors posit a force balance between turbulent stress and Laplace pressure (Eq. (15)) and close the droplet-scale velocity fluctuation with the large-scale convective shear, delta_{r_m}u ~ (U_FF/H)r_m; Eq. (16) then follows algebraically. The exponent -1/3 is a prediction of that stated closure, not a parameter fitted from the droplet-size data. The dashed lines in Fig. 9 are drawn at the predicted slope, so the figure is a consistency check of the exponent; it does not invert the derivation, because the data are free to scatter away from the line. The self-citations (Refs. [71]-[73], [88]) supply the open-source LBM code, the preparation protocol, and earlier reports of intermittency and phase inversion; but the present paper re-simulates and independently characterizes the breakup, coalescence, and phase-inverted regimes in its own phase diagram and rheology measurements, so no load-bearing conclusion rests solely on a self-citation. The paper's own stated limitations, namely that the Fig. 2 transition lines are guides, that phase-boundary locations will shift with surface tension and disjoining pressure, and that the disjoining-pressure-removal runs are preliminary, are transparent and do not mask a definitional circularity. The main correctness risk, not a circularity, is the unverified closure delta_{r_m}u ~ (U_FF/H)r_m: if the relevant fluctuations were inertial-range instead, a different exponent (e.g., Hinze-like) would result. This is an assumption about the flow, not an identity smuggled in as a prediction.

Assumptions & free parameters 2 free parameters · 7 assumptions · 0 invented entities

No new physical entities are introduced. The free parameters are the rheological fit and the LBM coupling constants. The axioms are mostly modeling choices and the shear estimate behind the scaling law.

free parameters (2)
  • Herschel-Bulkley parameters (sigma0, a, n) for phi=0.79 = sigma0 = 2.89e-5, a = 2.2e-2, n = 0.577
    Fitted to Couette rheology of the jammed emulsion in Fig. 3; establishes yield stress used to interpret phase inversion but not the central scaling.
  • LBM interaction constants (G_OW, G_a, G_r) and reference densities = G_OW=0.405, G_a_WW=-9.0, G_a_OO=-8.0, G_r_WW=8.1, G_r_OO=7.1, rho0=0.83
    Chosen by hand to set surface tension and disjoining pressure; the phase diagram boundaries and droplet statistics depend on them, as acknowledged in the conclusion.
assumptions (7)
  • domain assumption Two-dimensional RB convection captures the relevant phenomenology of 3D convection for emulsions.
    Sec. II A: '2D convection is different from 3D convection, but it still captures many of its relevant features [87]'; all results are from 2D runs.
  • domain assumption Equal kinematic viscosity and thermal diffusivity ratios, with Pr=1 and Gamma=2 fixed.
    Sec. II A: parameters reduced to Ra, Pr, Gamma, phi; Pr=1, Gamma=2 in all simulations.
  • domain assumption Surface tension and disjoining pressure are independent of temperature; Marangoni effects negligible.
    Sec. II A: 'we do not consider the temperature dependence of the surface tension', justified by small Marangoni in experiments.
  • standard math Boussinesq approximation holds for buoyancy forcing.
    Eq. (10): F_ext = -rho alpha g T; standard approximation in RB convection.
  • domain assumption Shan-Chen LBM with the given interaction parameters reproduces the Navier-Stokes and advection-diffusion equations with the intended rheology; spurious currents do not affect phenomenology.
    Sec. II B: LBM recovers NS and advection-diffusion in hydrodynamic limit; note 'spurious currents... do not affect the observed phenomenology'.
  • ad hoc to paper Droplet-scale velocity fluctuation is set by the large-scale shear: delta_{r_m} u ~ (U_FF/H) r_m.
    Sec. IV, before Eq. (16): used to derive r_m ~ Ra^{-1/3}; not directly measured.
  • domain assumption The -10/3 power-law tail requires coalescence to refill large droplets; coalescence rate is significant for phi=0.39 and 0.84.
    Sec. IV: explains why phi=0.16 does not show -10/3 tail; assumes coalescence refills the distribution.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Dynamical regimes of thermally convective emulsions." pith.science (2026). https://pith.science/paper/OMXEYK2R

@misc{pith2026241111553,
  author       = {Pith},
  title        = {Pith review of: Dynamical regimes of thermally convective emulsions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/OMXEYK2R}},
  note         = {Machine review of arXiv:2411.11553}
}
abstract

Emulsions are paramount in various interdisciplinary topical areas, yet a satisfactory understanding of their behavior in buoyancy-driven thermal flows has not been established. In the present work, we unravel the dynamical regimes of thermal convection in emulsions by leveraging a large set of mesoscale numerical simulations. Emulsions are prepared with a given volume fraction of the initially dispersed phase, $\phi$, ranging from dilute (low values of $\phi$) to jammed emulsions (high values of $\phi$), resulting in different rheological responses of the emulsion, i.e., from Newtonian to non-Newtonian yield-stress behaviors, respectively. We then characterize the dynamics of the emulsions in the paradigmatic setup of the Rayleigh-B\'enard convection, i.e., when confined between two parallel walls at different temperatures under the effect of buoyancy forces, the latter encoded in the dimensionless Rayleigh number Ra. We thoroughly investigated the dynamics of the emulsion in the changing of $\phi$ and Ra. For a given $\phi$, at increasing Ra, we observe that the emulsion exhibits convection states, where structural changes may appear (i.e., droplet breakup, coalescence or phase inversion), which inevitably impact the emulsion rheology. For sufficiently high values of Ra, two states of convection are observed: for low/moderate values of $\phi$ (Newtonian emulsions), we observe breakup-dominated dynamics, whereas for high values of $\phi$ (non-Newtonian emulsions), we observe phase-inverted states. For both scenarios, the droplet size distribution depends on Ra, and scaling laws for the average droplet size are analyzed and quantified. Our results offer insights into the rich dynamics of emulsions under thermal convection, offering the first detailed characterization of the various dynamic regimes to be expected and their relation with structural changes occurring in such complex fluids.

Figures

Figures reproduced from arXiv: 2411.11553 by the authors.

Figure 1
Figure 1. FIG. 1. Characterization of dynamical regimes of thermally convective oil-in-water (O/W) emulsions in the setup of Rayleigh [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Phase diagram reporting a characterization of statistically steady states for different combinations of Ra and [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Rheological characterization of emulsions after the preparation step, before buoyancy forces are switched on. We report [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figures from the paper (7 more)
Figure 4
Figure 4. Figure 4: FIG. 4. Rheological characterization of emulsions in the statistically steady state for different combinations of [PITH_FULL_IMAGE:figures/full_fig_p008_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. Characterization of the heat flux in the statistically steady state in terms of the Nusselt number averaged in time [PITH_FULL_IMAGE:figures/full_fig_p009_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6. Transient dynamics and heat flux fluctuations in a Newtonian homogeneous fluid. We report the time evolution of [PITH_FULL_IMAGE:figures/full_fig_p010_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7. Transient dynamics and heat flux fluctuations in emulsions. We report the time evolution of the Nusselt number Nu [PITH_FULL_IMAGE:figures/full_fig_p010_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8. Probability distribution functions (PDFs) for the radius of droplets, [PITH_FULL_IMAGE:figures/full_fig_p011_8.png]
Figure 9
Figure 9. Figure 9: FIG. 9. Average value of the droplet radius, [PITH_FULL_IMAGE:figures/full_fig_p011_9.png]
Figure 10
Figure 10. Figure 10: FIG. 10. Log-log PDFs for the radius of droplets, [PITH_FULL_IMAGE:figures/full_fig_p012_10.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

126 extracted references · 77 canonical work pages

  1. [1]

    H. A. Barnes, Colloids and Surfaces A: Physicochemical and Engineering Aspects91, 89 (1994)

    1994

  2. [2]

    Mason, Current Opinion in Colloid & Interface Sci- ence4, 231 (1999)

    T. Mason, Current Opinion in Colloid & Interface Sci- ence4, 231 (1999)

    1999

  3. [3]

    S. R. Derkach, Advances in colloid and interface science 151, 1 (2009)

    2009

  4. [4]

    T. F. Tadros, Emulsion formation and stability1, 1 (2013)

    2013

  5. [5]

    E. E. Ferrero, K. Martens, and J.-L. Barrat, Physical Review Letters113, 1 (2014)

    2014

  6. [6]

    D. Bonn, M. M. Denn, L. Berthier, T. Divoux, and S. Manneville, Reviews of Modern Physics89, 035005 (2017)

    2017

  7. [7]

    Barrat, Physica A: Statistical Mechanics and its Applications504, 20 (2017)

    J.-L. Barrat, Physica A: Statistical Mechanics and its Applications504, 20 (2017)

    2017

  8. [8]

    Nicolas, E

    A. Nicolas, E. E. Ferrero, K. Martens, and J.-L. Barrat, Reviews of Modern Physics90, 045006 (2018)

    2018

Show all 126 references
  1. [9]

    Divoux, E

    T. Divoux, E. Agoritsas, S. Aime, C. Barentin, J.-L. Barrat, R. Benzi, L. Berthier, D. Bi, G. Biroli, D. Bonn, et al., Soft Matter20, 6868 (2024)

    2024

  2. [10]

    G. I. Taylor, Proceedings of the Royal Society of Lon- don. Series A, Containing Papers of a Mathematical and Physical Character138, 41 (1932)

    1932

  3. [11]

    Zinchenko, Journal of Applied Mathematics and Me- chanics48, 198 (1984)

    A. Zinchenko, Journal of Applied Mathematics and Me- chanics48, 198 (1984)

    1984

  4. [12]

    Ghigliotti, T

    G. Ghigliotti, T. Biben, and C. Misbah, Journal of Fluid Mechanics653, 489 (2010)

    2010

  5. [13]

    Pal, Journal of colloid and interface science225, 359 (2000)

    R. Pal, Journal of colloid and interface science225, 359 (2000)

    2000

  6. [14]

    N. J. Balmforth, I. A. Frigaard, and G. Ovarlez, Annual Review of Fluid Mechanics46, 121 (2014)

    2014

  7. [15]

    H. A. Stone, Annual Review of Fluid Mechanics26, 65 (1994)

    1994

  8. [16]

    Cristini, S

    V. Cristini, S. Guido, A. Alfani, J. B l awzdziewicz, and M. Loewenberg, Journal of Rheology47, 1283 (2003)

    2003

  9. [17]

    Vankova, S

    N. Vankova, S. Tcholakova, N. D. Denkov, I. B. Ivanov, V. D. Vulchev, and T. Danner, Journal of Colloid and Interface Science312, 363 (2007)

    2007

  10. [18]

    Vankova, S

    N. Vankova, S. Tcholakova, N. Denkov, V. Vulchev, and T. Danner, Journal of Colloid and Interface Science313, 612 (2007)

    2007

  11. [19]

    Tcholakova, N

    S. Tcholakova, N. Vankova, N. Denkov, and T. Dan- ner, Journal of Colloid and Interface Science310, 570 (2007)

    2007

  12. [20]

    Perlekar, L

    P. Perlekar, L. Biferale, M. Sbragaglia, S. Srivastava, and F. Toschi, Physics of Fluids24, 065101 (2012)

    2012

  13. [21]

    Scarbolo, F

    L. Scarbolo, F. Bianco, and A. Soldati, Physics of Fluids 27(2015)

    2015

  14. [22]

    Roccon, M

    A. Roccon, M. De Paoli, F. Zonta, and A. Soldati, Phys- ical Review Fluids2, 083603 (2017)

    2017

  15. [23]

    Soligo, A

    G. Soligo, A. Roccon, and A. Soldati, Journal of Fluid Mechanics881, 244 (2019)

    2019

  16. [24]

    H.-R. Liu, K. L. Chong, Q. Wang, C. S. Ng, R. Verz- icco, and D. Lohse, Journal of Fluid Mechanics913, A9 (2021)

    2021

  17. [25]

    Girotto, R

    I. Girotto, R. Benzi, G. Di Staso, A. Scagliarini, S. F. Schifano, and F. Toschi, Journal of Turbulence23, 265 (2022)

    2022

  18. [26]

    Girotto, A

    I. Girotto, A. Scagliarini, R. Benzi, and F. Toschi, Jour- nal of Fluid Mechanics986, A33 (2024)

    2024

  19. [27]

    Crialesi-Esposito, G

    M. Crialesi-Esposito, G. Boffetta, L. Brandt, S. Chib- baro, and S. Musacchio, Physical Review Fluids9, L072301 (2024)

    2024

  20. [28]

    B. A. Khan, N. Akhtar, H. M. S. Khan, K. Waseem, T. Mahmood, A. Rasul, M. Iqbal, and H. Khan, African Journal of Pharmacy and Pharmacology5, 2715 (2011)

    2011

  21. [29]

    D. J. McClements,Food emulsions: principles, prac- tices, and techniques(CRC press, 2015) p. 632

    2015

  22. [30]

    Goodarzi and S

    F. Goodarzi and S. Zendehboudi, The Canadian Journal of Chemical Engineering97, 281 (2019). 15

    2019

  23. [31]

    Gallegos and J

    C. Gallegos and J. Franco, Current Opinion in Colloid & Interface Science4, 288 (1999)

    1999

  24. [32]

    D. J. McClements, Critical Reviews in Food Science and Nutrition47, 611 (2007)

    2007

  25. [33]

    Benzi, M

    R. Benzi, M. Sbragaglia, S. Succi, M. Bernaschi, and S. Chibbaro, Journal of Chemical Physics131, 104903 (2009)

    2009

  26. [34]

    S. F. Roudsari, G. Turcotte, R. Dhib, and F. Ein- Mozaffari, Computers & Chemical Engineering45, 124 (2012)

    2012

  27. [35]

    M. E. Rosti, Z. Ge, S. S. Jain, M. S. Dodd, and L. Brandt, Journal of Fluid Mechanics876, 962 (2019)

    2019

  28. [36]

    Z. Wang, Y. Xu, Y. Liu, X. Liu, and Z. Rui, Arabian Journal for Science and Engineering45, 7161 (2020)

    2020

  29. [37]

    Lohse and X

    D. Lohse and X. Zhang, Nature Reviews Physics2, 426 (2020)

    2020

  30. [38]

    M. M. Abdulredha, H. S. Aslina, and C. A. Luqman, Arabian Journal of Chemistry13, 3403 (2020)

    2020

  31. [39]

    Jeffreys, St

    H. Jeffreys, St. John’s Coll. Camb. Sir Harold Jeffreys Pap.C17(1957)

    1957

  32. [40]

    Orowan, Philosophical Transactions of the Royal So- ciety A258, 284 (1965)

    E. Orowan, Philosophical Transactions of the Royal So- ciety A258, 284 (1965)

    1965

  33. [41]

    W. J. Morgan, Nature230, 42 (1971)

    1971

  34. [42]

    Montelli, G

    R. Montelli, G. Nolet, F. Dahlen, and G. Masters, Geo- chemistry, Geophysics, Geosystems7(2006)

    2006

  35. [43]

    French and B

    S. French and B. Romanowicz, Nature525, 95 (2015)

    2015

  36. [44]

    Davaille, P

    A. Davaille, P. Carrez, and P. Cordier, Geophysical Re- search Letters45, 1349 (2018)

    2018

  37. [45]

    Griffiths, Annual Review of Fluid Mechanics32, 477 (2000)

    R. Griffiths, Annual Review of Fluid Mechanics32, 477 (2000)

    2000

  38. [46]

    Lavall´ ee, D

    Y. Lavall´ ee, D. B. Dingwell, J. B. Johnson, C. Cimarelli, A. J. Hornby, J. E. Kendrick, F. W. Von Aulock, B. M. Kennedy, B. J. Andrews, F. B. Wadsworth,et al., Na- ture528, 544 (2015)

    2015

  39. [47]

    Schellart and V

    W. Schellart and V. Strak, Journal of Geodynamics 100, 7 (2016)

    2016

  40. [48]

    Di Giuseppe, F

    E. Di Giuseppe, F. Corbi, F. Funiciello, A. Massmeyer, T. Santimano, M. Rosenau, and A. Davaille, Tectono- physics642, 29 (2014)

    2014

  41. [49]

    Reber, M

    J. Reber, M. Cooke, and T. Dooley, Earth-Science Re- views202, 103107 (2020)

    2020

  42. [50]

    Oresta, R

    P. Oresta, R. Verzicco, D. Lohse, and A. Prosperetti, Physical Review E80, 026304 (2009)

    2009

  43. [51]

    Lakkaraju, R

    R. Lakkaraju, R. J. A. M. Stevens, P. Oresta, R. Verz- icco, D. Lohse, and A. Prosperetti, Proceedings of the National Academy of Sciences110, 9237 (2013)

    2013

  44. [52]

    Biferale, P

    L. Biferale, P. Perlekar, M. Sbragaglia, and F. Toschi, Physical Review Letters108, 104502 (2012)

    2012

  45. [53]

    Garoosi and T.-F

    F. Garoosi and T.-F. Mahdi, International Journal of Heat and Mass Transfer172, 121163 (2021)

    2021

  46. [54]

    P. R. Santos, A. Lugarini, S. L. Junqueira, and A. T. Franco, International Journal of Thermal Sciences166, 106991 (2021)

    2021

  47. [55]

    H.-R. Liu, K. L. Chong, C. S. Ng, R. Verzicco, and D. Lohse, Journal of Fluid Mechanics933, R1 (2022)

    2022

  48. [56]

    H.-R. Liu, K. L. Chong, R. Yang, R. Verzicco, and D. Lohse, Journal of Fluid Mechanics938, A31 (2022)

    2022

  49. [57]

    A. M. Bilondi, N. Scapin, L. Brandt, and P. Mirbod, Journal of Fluid Mechanics999, A4 (2024)

    2024

  50. [58]

    Mangani, A

    F. Mangani, A. Roccon, F. Zonta, and A. Soldati, Jour- nal of Fluid Mechanics978, A12 (2024)

    2024

  51. [59]

    Zhang, D

    J. Zhang, D. Vola, and I. A. Frigaard, Journal of Fluid Mechanics566, 389 (2006)

    2006

  52. [60]

    N. J. Balmforth and A. C. Rust, Journal of Non- Newtonian Fluid Mechanics158, 36 (2009)

    2009

  53. [61]

    Vikhansky, Physics of Fluids21(2009)

    A. Vikhansky, Physics of Fluids21(2009)

    2009

  54. [62]

    Vikhansky, Journal of Non-Newtonian Fluid Me- chanics165, 1713 (2010)

    A. Vikhansky, Journal of Non-Newtonian Fluid Me- chanics165, 1713 (2010)

    2010

  55. [63]

    Albaalbaki and R

    B. Albaalbaki and R. E. Khayat, Journal of Fluid Me- chanics668, 500 (2011)

    2011

  56. [64]

    Turan, N

    O. Turan, N. Chakraborty, and R. J. Poole, Journal of Non-Newtonian Fluid Mechanics171-172, 83 (2012)

    2012

  57. [65]

    Massmeyer, E

    A. Massmeyer, E. Di Giuseppe, A. Davaille, T. Rolf, and P. J. Tackley, Journal of Non-Newtonian Fluid Me- chanics195, 32 (2013)

    2013

  58. [66]

    Hassan, M

    M. Hassan, M. Pathak, and M. K. Khan, Journal of Non-Newtonian Fluid Mechanics226, 32 (2015)

    2015

  59. [67]

    Karimfazli, I

    I. Karimfazli, I. Frigaard, and A. Wachs, Journal of Fluid Mechanics787, 474 (2016)

    2016

  60. [68]

    Goyon, A

    J. Goyon, A. Colin, G. Ovarlez, A. Ajdari, and L. Boc- quet, Nature454, 84 (2008)

    2008

  61. [69]

    Goyon, A

    J. Goyon, A. Colin, G. Ovarlez, A. Ajdari, and L. Boc- quet, Soft Matter6, 2668 (2010)

    2010

  62. [70]

    Davaille, B

    A. Davaille, B. Gueslin, A. Massmeyer, and E. Di Giuseppe, Journal of Non-Newtonian Fluid Mechanics193, 144 (2013)

    2013

  63. [71]

    Pelusi, M

    F. Pelusi, M. Sbragaglia, R. Benzi, A. Scagliarini, M. Bernaschi, and S. Succi, Soft Matter17, 3709 (2021)

    2021

  64. [72]

    Pelusi, S

    F. Pelusi, S. Ascione, M. Sbragaglia, and M. Bernaschi, Soft Matter19, 7192 (2023)

    2023

  65. [73]

    Pelusi, A

    F. Pelusi, A. Scagliarini, M. Sbragaglia, M. Bernaschi, and R. Benzi, Physical Review Letters133, 244001 (2024)

    2024

  66. [74]

    Jadhav, P

    K. Jadhav, P. Rossi, and I. Karimfazli, Journal of Fluid Mechanics912, A10 (2021)

    2021

  67. [75]

    Groeneweg, W

    F. Groeneweg, W. Agterof, P. Jaeger, J. Janssen, J. Wieringa, and J. Klahn, Chemical Engineering Re- search and Design76, 55 (1998)

    1998

  68. [76]

    Kumar, S

    A. Kumar, S. Li, C.-M. Cheng, and D. Lee, Industrial & Engineering Chemistry Research54, 8375 (2015)

    2015

  69. [77]

    Bouchama, G

    F. Bouchama, G. Van Aken, A. Autin, and G. Koper, Colloids and Surfaces A: Physicochemical and engineer- ing aspects231, 11 (2003)

    2003

  70. [78]

    Perazzo, V

    A. Perazzo, V. Preziosi, and S. Guido, Advances in Col- loid and Interface Science222, 581 (2015)

    2015

  71. [79]

    Fernandez, V

    P. Fernandez, V. Andr´ e, J. Rieger, and A. K¨ uhnle, Col- loids and Surfaces A: Physicochemical and Engineering Aspects251, 53 (2004)

    2004

  72. [80]

    T. R. Borrin, E. L. Georges, I. C. Moraes, and S. C. Pinho, Journal of Food Engineering169, 1 (2016)

    2016

  73. [81]

    L. Yeo, O. Matar, E. de Ortiz, and G. Hewitt, Chemical Engineering Science57, 3505 (2002)

    2002

  74. [82]

    Bakhuis, R

    D. Bakhuis, R. Ezeta, P. A. Bullee, A. Marin, D. Lohse, C. Sun, and S. G. Huisman, Physical Review Letters 126, 064501 (2021)

    2021

  75. [83]

    L. Yi, I. Girotto, F. Toschi, and C. Sun, Physical Review Letters133, 134001 (2024)

    2024

  76. [84]

    Grossmann and D

    S. Grossmann and D. Lohse, Physical Review Letters 86, 3316 (2001)

    2001

  77. [85]

    Ahlers, S

    G. Ahlers, S. Grossmann, and D. Lohse, Reviews of Modern Physics81, 503 (2009)

    2009

  78. [86]

    Chill` a and J

    F. Chill` a and J. Schumacher, The European Physical Journal E35, 1 (2012)

    2012

  79. [87]

    E. P. Van Der Poel, R. J. Stevens, and D. Lohse, Journal of Fluid Mechanics736, 177 (2013)

    2013

  80. [88]

    Pelusi, M

    F. Pelusi, M. Lulli, M. Sbragaglia, and M. Bernaschi, Computer Physics Communications273, 108259 16 (2022)

    2022

  81. [89]

    Chandrasekhar,Hydrodynamic and hydromagnetic stability(Oxford University Press, 1961)

    S. Chandrasekhar,Hydrodynamic and hydromagnetic stability(Oxford University Press, 1961)

    1961

  82. [90]

    Bodenschatz, W

    E. Bodenschatz, W. Pesch, and G. Ahlers, Annual Re- view of Fluid Mechanics32, 709 (2000)

    2000

  83. [91]

    Lohse and K.-Q

    D. Lohse and K.-Q. Xia, Annual Review of Fluid Me- chanics42, 335 (2010)

    2010

  84. [92]

    B. I. Shraiman and E. D. Siggia, Physical Review A42, 3650 (1990)

    1990

  85. [93]

    Stevens, R

    R. Stevens, R. Verzicco, and D. Lohse, Journal of Fluid Mechanics643, 495 (2010)

    2010

  86. [94]

    Benzi, S

    R. Benzi, S. Succi, and M. Vergassola, Physics Reports 222, 145 (1992)

    1992

  87. [95]

    Kr¨ uger, H

    T. Kr¨ uger, H. Kusumaatmaja, A. Kuzmin, O. Shardt, G. Silva, and E. M. Viggen, Springer International Pub- lishing10, 4 (2017)

    2017

  88. [96]

    Succi,The lattice Boltzmann Equation(Oxford Uni- versity Press, 2018)

    S. Succi,The lattice Boltzmann Equation(Oxford Uni- versity Press, 2018)

    2018

  89. [97]

    Benzi, M

    R. Benzi, M. Sbragaglia, P. Perlekar, M. Bernaschi, S. Succi, and F. Toschi, Soft Matter10, 4615 (2014)

    2014

  90. [98]

    Dollet, A

    B. Dollet, A. Scagliarini, and M. Sbragaglia, Journal of Fluid Mechanics766, 556 (2015)

    2015

  91. [99]

    Lulli, R

    M. Lulli, R. Benzi, and M. Sbragaglia, Physical Review X8, 021031 (2018)

    2018

  92. [100]

    Negro, L

    G. Negro, L. N. Carenza, G. Gonnella, F. Mackay, A. Morozov, and D. Marenduzzo, Science Advances9, eadf8106 (2023)

    2023

  93. [101]

    Pelusi, D

    F. Pelusi, D. Filippi, L. Derzsi, M. Pierno, and M. Sbra- gaglia, Soft Matter (2024)

    2024

  94. [102]

    Tiribocchi, M

    A. Tiribocchi, M. Durve, M. Lauricella, A. Montessori, J.-M. Tucny, and S. Succi, Physics Reports1105, 1 (2025)

    2025

  95. [103]

    Shan and H

    X. Shan and H. Chen, Physical Review E47, 1815 (1993)

    1993

  96. [104]

    Sbragaglia, R

    M. Sbragaglia, R. Benzi, L. Biferale, S. Succi, K. Sugiyama, and F. Toschi, Physical Review E75 (2007)

    2007

  97. [105]

    Sbragaglia, R

    M. Sbragaglia, R. Benzi, M. Bernaschi, and S. Succi, Soft Matter8, 10773 (2012)

    2012

  98. [106]

    P. L. Bhatnagar, E. P. Gross, and M. Krook, Physical Review94, 511 (1954)

    1954

  99. [107]

    D. P. Silva, R. C. Coelho, M. M. T. da Gama, and N. A. Ara´ ujo, Physical Review E107, 035106 (2023)

    2023

  100. [108]

    D. P. Silva, R. C. Coelho, I. Pagonabarraga, S. Succi, M. M. T. da Gama, and N. A. Ara´ ujo, Soft Matter (2024)

    2024

  101. [109]

    E. A. Spiegel and G. Veronis, Astrophysical Journal 131, 442 (1960)

    1960

  102. [110]

    Hinze, A.I.Ch.E

    J. Hinze, A.I.Ch.E. Journal1, 289 (1955)

    1955

  103. [111]

    Andersson and B

    R. Andersson and B. Andersson, A.I.Ch.E. Journal52, 2020 (2006)

    2006

  104. [112]

    Cioni, S

    S. Cioni, S. Ciliberto, and J. Sommeria, Journal of Fluid Mechanics335, 111 (1997)

    1997

  105. [113]

    Castaing, G

    B. Castaing, G. Gunaratne, F. Heslot, L. Kadanoff, A. Libchaber, S. Thomae, X.-Z. Wu, S. Zaleski, and G. Zanetti, Journal of Fluid Mechanics204, 1 (1989)

    1989

  106. [114]

    Ciliberto, S

    S. Ciliberto, S. Cioni, and C. Laroche, Physical Review E54, R5901 (1996)

    1996

  107. [115]

    Benzi, F

    R. Benzi, F. Toschi, and R. Tripiccione, Journal of Sta- tistical Physics93, 901 (1998)

    1998

  108. [116]

    Grossmann and D

    S. Grossmann and D. Lohse, Journal of Fluid Mechanics 407, 27 (2000)

    2000

  109. [117]

    Stevens, E

    R. Stevens, E. van der Poel, S. Grossmann, and D. Lohse, Journal of Fluid Mechanics730, 295 (2013)

    2013

  110. [118]

    R. E. Ecke, H. Haucke, Y. Maeno, and J. C. Wheatley, Physical Review A33, 1870 (1986)

    1986

  111. [119]

    Heslot, B

    F. Heslot, B. Castaing, and A. Libchaber, Physical Re- view A36, 5870 (1987)

    1987

  112. [120]

    G. B. Deane and M. D. Stokes, Nature418, 839 (2002)

    2002

  113. [121]

    Mukherjee, A

    S. Mukherjee, A. Safdari, O. Shardt, S. Kenjereˇ s, and H. E. Van den Akker, Journal of Fluid Mechanics878, 221 (2019)

    2019

  114. [122]

    Crialesi-Esposito, S

    M. Crialesi-Esposito, S. Chibbaro, and L. Brandt, Com- munications Physics6, 5 (2023)

    2023

  115. [123]

    Roccon, F

    A. Roccon, F. Zonta, and A. Soldati, Physical Review Fluids8, 090501 (2023)

    2023

  116. [124]

    Garrett, M

    C. Garrett, M. Li, and D. Farmer, Journal of Physical Oceanography30, 2163 (2000)

    2000

  117. [125]

    Benzi, T

    R. Benzi, T. Divoux, C. Barentin, S. Manneville, M. Sbragaglia, and F. Toschi, Physical Review E104, 034612 (2021)

    2021

  118. [126]

    Benzi, T

    R. Benzi, T. Divoux, C. Barentin, S. Manneville, M. Sbragaglia, and F. Toschi, Physical Review Letters 127, 148003 (2021)

    2021

Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.