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Rise regimes of freely rising droplets with a moderate viscosity ratio

T0 review · 1 major / 2 minor · reviewed 2026-06-27 · grok-4.3

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Pith's one-line read Moderate-viscosity droplets initiate axisymmetry breaking from internal flow instability.

desk verdict The paper gives a regime map for rising droplets at viscosity ratio 0.62 where internal m=2 instability leads the symmetry breaking, but the DNS methods need close checking for artifacts. read the letter →

arxiv 2606.08575 v1 pith:N65GNI3C submitted 2026-06-07 physics.flu-dyn

Pengyu Shi , Dirk Lucas , Jie Zhang , Éric Climent , Dominique Legendre This is my paper · ORCID
classification physics.flu-dyn
keywords risingdropletspathinstabilityviscosityratiodirectnumericalsimulationsaxisymmetrybreakinginternalflowwakedynamicsmultistableregimes
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 studies the rise of freely buoyant droplets with moderate viscosity ratio using direct numerical simulations. It identifies a series of regimes as droplet size grows, starting from steady vertical rise, then internal instability with m=2 mode, oblique paths, rotating waves, and chaotic motion. The key finding is that axisymmetry breaking starts inside the droplet, shaping the wake and dynamics differently than for bubbles or particles. This distinction matters for accurately modeling droplet motion in liquids.

What carries the argument

Internal flow instability with azimuthal mode m=2 that initiates axisymmetry breaking

What would settle it

An experiment that measures the velocity field inside the rising droplet and shows whether the m=2 azimuthal mode develops prior to any external wake asymmetry would confirm or refute the internal initiation of instability.

Watch

Extended reading notes

Core claim

As the droplet radius increases, the system undergoes a sequence of rise regimes beginning with steady axisymmetric rise, followed by an internal flow instability of azimuthal mode m=2 leading to biplanar-symmetric wake and reduced speed, then a steady oblique regime with coexisting m=1 and m=2 modes, an m=2 rotating-wave regime, and finally chaotic paths with shape oscillations and vortex shedding. Multistable states exist in certain size ranges. This demonstrates that axisymmetry breaking is initiated within the droplet due to internal flow instability, fundamentally differing from bubbles and solid particles.

Load-bearing premise

Direct numerical simulations accurately capture the physical internal flow instability sequence without significant numerical or modeling artifacts.

Editorial extensions

If this is right

  • Reduced terminal rise speed accompanies the biplanar-symmetric wake state.
  • The oblique regime features coexistence of m=1 and m=2 modes.
  • Multistable terminal states coexist in certain radius ranges depending on initial conditions.
  • The rotating-wave regime has the wake drifting azimuthally at constant angular velocity.
  • Chaotic paths emerge at larger sizes from persistent shape oscillations and vortex shedding.

Reading between the lines

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

  • Models of droplet dynamics in engineering applications may need to prioritize internal circulation effects over wake dynamics alone.
  • Similar internal instabilities could appear in other immiscible liquid systems with comparable viscosity ratios.
  • Experiments could test this by seeding the droplet with tracers to visualize internal flow patterns before wake asymmetry develops.
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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

1 major / 2 minor

Summary. The manuscript uses direct numerical simulations to study the rise of toluene droplets (viscosity ratio μ*=0.62) in water for radii 0.5–3 mm. It reports a sequence of regimes beginning with steady axisymmetric rise, followed by an internal m=2 azimuthal instability that produces a biplanar-symmetric wake and reduced terminal speed, then an oblique regime with coexisting m=1 and m=2 modes, a nearly vertical path, an m=2 rotating-wave regime, and finally chaotic paths with shape oscillations and vortex shedding. Multistable terminal states are identified depending on initial conditions. The central claim is that, unlike bubbles or solid particles, axisymmetry breaking is initiated inside the droplet by the internal flow instability.

Significance. If the reported internal m=2 instability sequence is free of numerical artifacts, the work would be significant for multiphase flow dynamics by demonstrating the dominant role of internal circulation in shaping wake structure, rise speed, and path for moderate viscosity ratios. The identification of multistable regimes and the use of both axisymmetric and fully 3D simulations are strengths that could guide future modeling of droplet transport.

major comments (1)
  1. [Numerical Methods] Numerical Methods (and associated results sections): The claim that axisymmetry breaking originates inside the droplet via the m=2 mode before wake effects dominate (Abstract and main text) is load-bearing and requires explicit demonstration that internal-flow growth rates are not altered by interface regularization, spurious currents, or differential numerical diffusion inside versus outside the droplet. Grid-convergence checks focused on azimuthal mode amplitudes and growth rates within the droplet, together with validation against experimental terminal velocities for comparable μ* and Re, are needed to support the reported instability sequence and multistable states.
minor comments (2)
  1. The transition radii or equivalent dimensionless numbers (Re, Bo, etc.) between the reported regimes are not quantified in the abstract or early sections, making it difficult to compare with prior bubble and particle studies.
  2. Figure captions and text should explicitly state the grid resolution inside the droplet and the interface-capturing scheme parameters to allow assessment of internal-flow fidelity.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We appreciate the referee's detailed review and the emphasis on numerical validation for our central claim regarding the internal origin of the instability. We provide a point-by-point response below and will incorporate revisions as indicated.

read point-by-point responses

  1. Referee: [Numerical Methods] Numerical Methods (and associated results sections): The claim that axisymmetry breaking originates inside the droplet via the m=2 mode before wake effects dominate (Abstract and main text) is load-bearing and requires explicit demonstration that internal-flow growth rates are not altered by interface regularization, spurious currents, or differential numerical diffusion inside versus outside the droplet. Grid-convergence checks focused on azimuthal mode amplitudes and growth rates within the droplet, together with validation against experimental terminal velocities for comparable μ* and Re, are needed to support the reported instability sequence and multistable states.

    Authors: We thank the referee for this important comment. To address the concern about potential numerical artifacts affecting the internal m=2 instability, we will perform and report additional tests in the revised manuscript. Specifically, we will conduct grid-convergence studies with a focus on the growth rates and amplitudes of the m=2 mode inside the droplet, using multiple resolutions. We will also examine the sensitivity of these growth rates to the interface regularization parameters and the level of spurious currents. Furthermore, we will include a validation section comparing our simulated terminal velocities against available experimental data for droplets with similar viscosity ratios and Reynolds numbers. These revisions will provide stronger support for the claim that the axisymmetry breaking is initiated by the internal flow. revision: yes

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: observational DNS results with no derivation chain

full rationale

The paper presents results from direct numerical simulations of droplet rise regimes. No analytical derivations, parameter fittings, or self-citation chains are used to derive predictions or uniqueness claims. All reported behaviors (m=2 internal instability initiation, multistable states, regime transitions) are direct outputs of the simulations, with the central claim resting on the fidelity of the numerical method rather than any self-referential reduction. This matches the default expectation of no circularity for simulation-based observational studies.

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

The study is a numerical investigation relying on standard fluid mechanics assumptions rather than new postulates; the viscosity ratio is an input parameter taken from the physical system.

assumptions (2)
  • standard math Incompressible Navier-Stokes equations govern both phases
    Implicit in any DNS of immiscible liquid-liquid flow
  • domain assumption Clean interface with constant interfacial tension and no surfactants
    Explicitly stated for toluene droplets in clean water

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Cite this review

Pith. "Pith review of Rise regimes of freely rising droplets with a moderate viscosity ratio." pith.science (2026). https://pith.science/paper/N65GNI3C

@misc{pith2026260608575,
  author       = {Pith},
  title        = {Pith review of: Rise regimes of freely rising droplets with a moderate viscosity ratio},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/N65GNI3C}},
  note         = {Machine review of arXiv:2606.08575}
}
abstract

The dynamics of buoyant droplets rising freely in a large body of an immiscible liquid is investigated numerically for a moderate drop-to-fluid viscosity ratio $\mu^\ast$. We focus on toluene droplets rising in clean water, for which $\mu^\ast=0.62$, and vary the radius over $0.5\,\text{mm}\leq R\leq3.0\,\text{mm}$. Direct numerical simulations are performed in imposed axisymmetric and fully three-dimensional configurations. As $R$ increases, the system displays a rich sequence of rise regimes. Starting from steady vertical rise with an axisymmetric disturbance flow, it first undergoes an internal flow instability associated with an azimuthal mode $m=2$, leading to a biplanar-symmetric wake and reduced terminal speed. This state is followed by a steady oblique regime, in which the $m=1$ mode also becomes unstable and coexists with the $m=2$ mode. At larger radii, the path becomes nearly vertical again before the flow enters an $m=2$ rotating-wave regime, where the wake drifts azimuthally at an approximately constant angular velocity. For still larger droplets, persistent shape oscillations and vortex shedding lead to fully three-dimensional chaotic paths. Simulations initialised from finite-amplitude asymmetric states further reveal several multistable size ranges, in which distinct terminal states coexist depending on the initial condition. Taken together, these findings show that the path instability of moderate-viscosity-ratio droplets differs fundamentally from that of bubbles and solid particles: in most regimes encountered here, axisymmetry breaking is initiated within the droplet, highlighting the central role of the internal flow instability in shaping the subsequent wake structure, rise speed and droplet dynamics.

Figures

Figures reproduced from arXiv: 2606.08575 by the authors.

Figure 1
Figure 1. Fully developed state in the imposed axisymmetric configuration. [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 2
Figure 2. Drag coefficient and droplet radius as functions of the external Reynolds number [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. Base flow past droplets of different radius [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figures from the paper (23 more)
Figure 4
Figure 4. Figure 4: Terminal shapes of droplets of different radius [PITH_FULL_IMAGE:figures/full_fig_p010_4.png]
Figure 5
Figure 5. Figure 5: Time evolution and phase relation of droplets with [PITH_FULL_IMAGE:figures/full_fig_p010_5.png]
Figure 6
Figure 6. Figure 6: Terminal rise speed of the droplet as a function of the droplet radius [PITH_FULL_IMAGE:figures/full_fig_p012_6.png]
Figure 7
Figure 7. Figure 7: Wake structures associated with the five asymmetric regimes: [PITH_FULL_IMAGE:figures/full_fig_p013_7.png]
Figure 8
Figure 8. Figure 8: (𝑎) Rise speed as a function of time for 0.95 mm ⩽ 𝑅 ⩽ 1.1 mm. (𝑏) Isosurfaces of the axial vorticity, 𝜔𝑦 = ±0.5, at six selected time instants for 𝑅 = 1.1 mm, indicated by numbers in (𝑏) and marked by circles in (𝑎). Blue and red threads correspond to positive and neg…
Figure 9
Figure 9. Figure 9: Evolution of the normalised modal energy associated with azimuthal wavenumber [PITH_FULL_IMAGE:figures/full_fig_p016_9.png]
Figure 10
Figure 10. Figure 10: (𝑎) Rise paths of droplets for 1.10 mm ⩽ 𝑅 ⩽ 1.45 mm. (𝑏)–(𝑑) Evolution of several characteristics of the droplet motion during an SO scenario for 𝑅 = 1.4 mm. (𝑏) Rise speed 𝑉𝑦 (solid line, left axis) and aspect ratio χ (dashed line, right axis); (𝑐) horizontal veloci…
Figure 11
Figure 11. Figure 11: Isosurfaces of the axial vorticity, 𝜔𝑦 = ±0.5, at six selected time instants, indicated by numbers at the top of each panel and marked by circles in figure 10(𝑐). Blue and red threads correspond to positive and negative 𝜔𝑦, respectively. SV2 scenario, such as that obs…
Figure 12
Figure 12. Figure 12: (𝑎) Rise speed as a function of time for 1.5 mm ⩽ 𝑅 ⩽ 2.0 mm. (𝑏) Evolution of the normalised modal energy associated with azimuthal wavenumber 𝑚 during an SV𝑚 scenario for 𝑅 = 1.5 mm. (𝑐) Isosurfaces of the axial vorticity, 𝜔𝑦 = ±0.25, at selected time instants, indi…
Figure 13
Figure 13. Figure 13: Evolution of the rise speed of droplets with 2 [PITH_FULL_IMAGE:figures/full_fig_p020_13.png]
Figure 14
Figure 14. Figure 14: Evolution of quantities characterising the axisymmetry breaking for the droplet with [PITH_FULL_IMAGE:figures/full_fig_p020_14.png]
Figure 15
Figure 15. Figure 15: Isosurfaces of the axial vorticity, 𝜔𝑦 = ±0.75, for the droplet with 𝑅 = 2.1 mm at selected time instants, indicated by numbers at the top of each panel and, for the first five instants, marked by circles in figure 14(𝑎). First row: front view; second row: top view. T…
Figure 16
Figure 16. Figure 16: Three-dimensional paths in the 3DC regime for 2 [PITH_FULL_IMAGE:figures/full_fig_p022_16.png]
Figure 16
Figure 16. Figure 16: shows the trajectories in the 3DC regime for 𝑅 increasing from 2.3 to 3.0 mm. For 𝑅 ⩽ 2.6 mm, the trajectory is only weakly three-dimensional: its projection onto the horizontal plane extends preferentially along a single lateral direction. More specifically, within t…
Figure 17
Figure 17. Figure 17: Evolution of quantities characterising droplet motion and flow structure during a 3DC scenario [PITH_FULL_IMAGE:figures/full_fig_p024_17.png]
Figure 18
Figure 18. Figure 18: Isosurfaces of the axial vorticity, 𝜔𝑦 = ±0.75, for the droplet with 𝑅 = 2.4 mm at selected time instants, indicated by numbers at the top of each panel and marked by circles in figure 17. First row: front view; second row: top view. The plane (𝑥 ′ , 𝑧′ ) is horizonta…
Figure 11
Figure 11. Figure 11: figure 11. At this instant, the asymmetry is dominated by the azimuthal mode [PITH_FULL_IMAGE:figures/full_fig_p026_11.png]
Figure 19
Figure 19. Figure 19: Terminal rise speed of the droplet as a function of the droplet radius [PITH_FULL_IMAGE:figures/full_fig_p027_19.png]
Figure 20
Figure 20. Figure 20: Evolution of (𝑎) the vertical velocity and (𝑐) the horizontal velocity of the droplet for 1.5 mm ⩽ 𝑅 ⩽ 2.0 mm. All simulations are initialised from the 𝑚 = 2 initial state. In both panels, black lines show the evolution in the reference case, namely the case 𝑅 = 1.4 m…
Figure 21
Figure 21. Figure 21: Same as figure 20, but with all simulations initialised from the [PITH_FULL_IMAGE:figures/full_fig_p029_21.png]
Figure 22
Figure 22. Figure 22: Evolution of quantities characterising the response of droplets initialised from an [PITH_FULL_IMAGE:figures/full_fig_p030_22.png]
Figure 23
Figure 23. Figure 23: Three-dimensional paths for 2.3 mm ⩽ 𝑅 ⩽ 3.0 mm obtained from simulations initialised from an 𝑅𝑊 state. In all panels, black lines show the evolution of the reference case, namely 𝑅 = 2.2 mm initialised from the 𝑚 = 0 state; the black circle indicates the instant at w…
Figure 24
Figure 24. Figure 24: (𝑎) Evolution of the droplet rise speed for 2.3 mm ⩽ 𝑅 ⩽ 3.0 mm. (𝑏) Isosurfaces of the axial vorticity, 𝜔𝑦 = ±0.75, for 𝑅 = 2.4 mm at 𝑇 = 800, marked by a green circle in (𝑎). (𝑐) Evolution of the normalised modal energies for 𝑅 = 2.4 mm. (𝑑) Evolution of the global …

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