Influence of Electrohydrodynamic on Droplet Stability in Leaky Dielectric Media

Bertrand Lecordier (CORIA); Franck Lefebvre (CORIA); IUF); Mostafa S Shadloo (CORIA; Saurabh Sharma (CORIA)

REVIEW 4 major objections 5 minor 2 references

For silicone-oil droplets suspended in castor oil, this paper claims that raising the DC field drives a sequence: oblate flattening, then a tilt-and-rotation instability from induced-electric-torque imbalance, then damped shape oscillations

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

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2026-08-01 14:12 UTC pith:TW43ZYUP

load-bearing objection Useful new high-field droplet data and an honest STB demonstration, but the 'Quincke rotation' claim is asserted, not evidenced, and the STB section contradicts itself on time-averaging. the 4 major comments →

arxiv 2607.21638 v1 pith:TW43ZYUP submitted 2026-07-21 physics.flu-dyn eess.SPphysics.class-ph

Influence of Electrohydrodynamic on Droplet Stability in Leaky Dielectric Media

Saurabh Sharma (CORIA) , Franck Lefebvre (CORIA) , Mostafa S Shadloo (CORIA , IUF) , Bertrand Lecordier (CORIA) This is my paper

classification physics.flu-dyn eess.SPphysics.class-ph PACS 47.55.Dr47.65.-d

keywords electrohydrodynamicsleaky dielectricdroplet deformationoblate deformationrotation instabilitythree-dimensional particle trackingaspect ratio dynamicsinternal circulation

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.

This paper tries to establish that a leaky-dielectric droplet pair with permittivity-to-conductivity ratio greater than one does not simply flatten and stop: once the electric field passes a critical value, the droplet tilts and rotates because the induced dipole opposes the applied field, so any misalignment is amplified. At even higher fields, the droplet's aspect ratio oscillates while the orientation angle grows, then relaxes to a transiently stable tilted configuration. The paper also claims that the internal and external flow reorganizes from a four-vortex quadrupolar pattern into two dominant circulation cells during the rotation stage. This matters because it maps the regimes of electric-field-driven droplet destabilization and shows that time-resolved 3D particle tracking can visualise flows inside a strongly deforming droplet.

Core claim

The central claim is that the classical steady-state leaky-dielectric deformation is only the first chapter. For a silicone oil droplet in castor oil, where the ratio of permittivity ratio to conductivity ratio exceeds unity, the induced dipole is anti-aligned with the applied field, making the zero-tilt configuration unstable. Above a critical field the droplet tilts and rotates; the tilt is not a separate deformation but the signature of that rotation instability. At still stronger fields the finite charge-relaxation time introduces a phase lag between the rotating surface charge and the droplet shape response, producing damped oscillations of the aspect ratio before the droplet locks into

What carries the argument

The load-bearing object is the leaky-dielectric droplet characterised by S/R > 1, where S is the ratio of permittivities and R the ratio of conductivities between droplet and suspending liquid. This condition makes the induced dipole oppose the applied field, so the electric torque p × E amplifies a small tilt instead of restoring it — the mechanism that turns the steady state into a rotation instability. A second mechanism, the finite charge-relaxation time, creates a phase lag between the rotating surface-charge distribution and the hydrodynamic/capillary response, which produces the damped oscillatory transient at high field. The main diagnostic machinery is shadow-imaging measurement of

Load-bearing premise

The interpretation relies on the two liquids behaving as clean leaky dielectrics with a uniform applied field equal to V/d — no significant charge injection, electrode polarization, or field non-uniformity — so that the S/R>1 induced-torque instability, rather than some other electrokinetic effect, is what tilts and oscillates the droplet.

What would settle it

Measure the actual electric field inside the droplet and the surface-charge distribution (or directly measure the torque) at 4.5–5.75 kV/cm; if the interior field is strongly non-uniform or injected space charge dominates, the rotation and oscillations are not the leaky-dielectric torque mechanism. A simpler comparative check: repeat the experiment with a fluid pair having S/R < 1 and otherwise similar properties — the tilt-and-wobble sequence should not appear if the S/R > 1 condition is the cause.

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

If this is right

The S/R > 1 criterion becomes a practical predictor: droplet–medium pairs with this ratio should show tilt-and-rotation above a threshold field, not merely prolate or oblate steady deformation.
Aspect-ratio and orientation-angle time series can be used to classify the regimes (I–V), giving experimental thresholds for the onset of rotation and for oscillatory behaviour.
The reorganisation of internal flow from four quadrupolar vortices into two dominant circulation cells can serve as an experimental marker that the droplet has entered the rotation-dominated regime.
The observed damped oscillations imply that viscous dissipation sets the relaxation time scale, so viscosity ratio should control how quickly a tilted equilibrium is reached and whether oscillations are visible.
Time-resolved 3D particle tracking inside a strongly deforming droplet is workable and captures topology changes even when the surface moves rapidly.

Where Pith is reading between the lines

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

Because the oscillation is attributed to charge-relaxation lag, a natural test is to vary the conductivities while keeping S/R fixed: the oscillation frequency should track the inverse charge-relaxation time if the mechanism is right.
The S/R > 1 condition suggests the instability should disappear (or reverse) for S/R < 1 pairs; a controlled two-fluid comparison would isolate the torque-imbalance mechanism from other high-field effects.
If the rotation instability is as robust as presented, the same electric-field protocol could be used as a contactless way to switch droplet orientation and internal mixing in microfluidic devices, without moving parts.
The transiently stable tilted state at high field implies an effective field-dependent torque balance that might let one control equilibrium orientation angle continuously by tuning field strength — a consequence the paper does not explore.

Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

Useful new high-field droplet data and an honest STB demonstration, but the 'Quincke rotation' claim is asserted, not evidenced, and the STB section contradicts itself on time-averaging.

read the letter

The useful thing in this paper is the data, not the story. You get AR/OA time series at four field strengths in a regime where leaky-dielectric asymptotic theory has nothing to say, plus an STB demonstration showing internal flow reorganizing from quadrupolar vortices toward two circulation zones. The regime sequence — oblate deformation, tilt, oscillations, quasi-steady equilibrium — is plausible and consistent with Salipante & Vlahovska and with the group's own Karp et al. work. The authors know their limits: they label the STB part preliminary and qualitative, and explicitly defer quantitative analysis to a later study. The observations are independent measurements, with no equation-level fitting.

Soft spots, in order of size.

First and largest: the central mechanism — Quincke-type rotation from induced torque imbalance — is asserted, not shown. The OA time series rises and then saturates at a constant orientation; there is no continuous advance of the major axis, no angular velocity, no full-revolution tracking. The paper slides between 'tilt,' 'rotation,' 'spinning,' and 'sustained Quincke rotation' as if they were the same thing. The data as presented are equally consistent with a static tilted equilibrium. The stress-test note has this right.

Second: the STB section contradicts itself. The methods say the velocity field was averaged over the complete 596-frame acquisition to improve convergence. The results describe a temporal evolution — quadrupolar flow transitioning to two circulation zones, then spinning. An average over the full sequence cannot show temporal evolution. Either time-resolved fields exist but aren't described, or the topology story rests on one averaged snapshot. That needs to be reconciled before the flow-visualization claims are usable.

Third: no repeated runs, no error bars, and the field is assumed uniform at V/d without validation. For a conference proceedings this is tolerable, but it caps how much weight the regime characterization can carry. Fourth, minor: the STB run used a 4.5 mm droplet at 3.75 kV/cm, compared qualitatively against 2.5 mm shadow-imaging runs at 4.5–5.75 kV/cm.

None of this sinks the paper. The observations look real, the STB-on-a-deforming-droplet demonstration is a useful methodological data point, and the authors are honest about scope. But the abstract and conclusions over-sell the mechanism relative to the evidence.

For whom: experimentalists in EHD droplet deformation, modelers needing benchmarks beyond the weak-field limit, and anyone interested in whether STB can survive strong interface motion. It deserves a serious referee; a desk reject would be wrong. The referee should require direct rotation evidence (OA unwrapping or angular velocity) or a downgrade of the claim to 'tilt,' plus a fix to the time-averaging inconsistency. With those changes it's a solid proceedings paper.

Referee Report

4 major / 5 minor

Summary. The paper reports high-resolution shadow-imaging experiments on silicone oil droplets suspended in castor oil under DC electric fields of 4.5–5.75 kV/cm. The droplet aspect ratio and orientation angle are tracked as functions of time, and the authors partition the response into regimes: a sharp deformation increase, a quasi-steady gradual deformation, a decrease after a deformation peak, and, at higher fields, an oscillatory transient before a settled configuration. The paper additionally presents Shake-the-Box 3D particle-tracking measurements of the flow inside and around a 4.5 mm droplet at 3.75 kV/cm, and claims that the internal circulation evolves from a quadrupolar pattern to two dominant coherent zones during a claimed Quincke-type rotation regime.

Significance. If the central mechanism claim were supported, the work would be a useful addition to the experimental literature on leaky-dielectric droplet dynamics above the stability threshold, where oblate deformation, tilting, and oscillatory relaxation are less documented than the prolate/Quincke cases. The authors should be credited for measuring and reporting fluid properties, for using a high-resolution shadow-imaging setup, and for openly labeling the STB measurements as preliminary and qualitative. However, the paper's principal scientific claim—that the observed tilt and oscillations are caused by Quincke-type rotation driven by an imbalance in induced electric torque—is not directly evidenced by the data presented. The observations themselves may stand, but the mechanistic interpretation is currently asserted rather than demonstrated.

major comments (4)

[Abstract; §3, Fig. 3] The central claim of Quincke-type rotation is not supported by the orientation-angle data. Fig. 3 shows that the orientation angle reaches a constant value in the reported time window, and §3 explicitly states 'the droplet doesn't keep tilting indefinitely but instead reaches a constant orientation.' A droplet undergoing sustained torque-driven rotation would show a continuously advancing orientation angle (modulo 180°) with a measurable angular velocity. A static tilted equilibrium is also consistent with field non-uniformity or charge-injection electroconvection. The authors should either provide direct measurements of droplet angular position/velocity or surface charge/torque, or reframe the manuscript as reporting a tilted-oblate equilibrium rather than a rotation-driven instability. This is load-bearing because the mechanism is the main claimed contribution.
[§2 STB paragraph; §3, Fig. 5] The methods state that for the STB data 'the velocity field was further averaged over the complete acquisition sequence (596 frames).' Such a time-averaged field cannot support the temporal evolution described in §3: 'initially quadrupolar flow structure transitions into two dominant coherent circulation zones,' followed by 'spinning motion characteristic of the Quincke rotation regime,' and eventually 'sustained Quincke rotation.' Figures 5a–e are presented as a time sequence, but the text says the reconstructed field is an average over the full acquisition. In addition, the STB experiment used a different droplet diameter (4.5 mm) and a different field strength (3.75 kV/cm) from the shadow-imaging experiments (2.5 mm; 4.5–5.75 kV/cm), so the flow topology cannot be directly mapped onto the regime classification in Fig. 3 without an explicit statement of how the cases correspond.
[§3, Fig. 3] The regime classification (Regimes I–V) is based on what appears to be a single realization at each field strength. No repeated runs, error bars, or uncertainty quantification are provided for the aspect ratio or orientation angle. Since the oscillatory transients at high fields are central to the claimed regime structure, at least a few repeated trials are needed to establish that the observed peaks and oscillations are reproducible features rather than run-specific transients. This is a load-bearing issue for any quantitative claim about 'distinct regimes.'
[§3, first paragraph] The electric field is assumed to be uniform and equal to V/d with no validation. The S/R > 1 mechanism invoked in §1 and §3 is derived for a uniform applied field and ohmic leaky-dielectric response. If electrode polarization, charge injection, or field non-uniformity are significant at 4.5–5.75 kV/cm, the observed tilt and oscillations could arise from electroconvection or other mechanisms. The authors should either justify the uniform-field assumption with a direct field measurement or a control experiment (e.g., varying gap or electrode material), or temper the mechanistic claims and discuss these alternative explanations. This concern is not a circularity objection but a correctness-risk assessment of the central causal attribution.

minor comments (5)

[Abstract] The phrase 'due to the imbalance in induce electric torque' contains a typo ('induce' should be 'induced') and the causal wording is stronger than the evidence warrants.
[Table 1] The unit for electrical permittivity is listed as 'F/s'; it should be F/m. This is likely a typo, but it matters for the reproducibility of the property values.
[§3, first paragraph] The electric capillary number definition appears as a garbled sequence of special characters in the text. Please provide a clear, typeset equation defining E0, epsilon, gamma, etc.
[§3, Fig. 3] The regimes (I–V) are described in the text but are not annotated on any of the panels in Fig. 3. Adding shaded regions or labels directly on the figure would greatly improve readability.
[Conclusions] The text says 'silicon oil droplet' while the rest of the paper uses 'silicone oil.' Please make the terminology consistent.

Circularity Check

0 steps flagged

No significant circularity: experimental study with independent measurements; only minor same-group citation used for qualitative comparison.

full rationale

The paper is an experimental characterization of droplet deformation and internal flow, not a derivation. The main quantities—aspect ratio and orientation angle—are measured directly from shadow imaging, and the internal flow is measured with 3D particle tracking. No parameter is fitted and then renamed as a prediction. The only same-group reference is Karp et al. (2024), which shares two co-authors with the present work; it is cited to state that silicone-in-castor oil with S/R > 1 gives oblate deformation and that quadrupolar vortical structures appear at lower fields. These citations are used as background and as a qualitative comparison for the observed flow topology, not as the logical basis for the measured deformation or the reported regimes. The mechanism invoked (Quincke-type rotation due to induced electric torque imbalance) is imported from external references (Salipante & Vlahovska 2010, 2013) and Taylor's leaky-dielectric model; even if the evidence for sustained rotation is indirect, that is an evidentiary gap, not circularity. No equation in the paper is shown to be equivalent to its inputs by construction. The result therefore does not reduce to a fitted parameter or to a self-citation chain.

Axiom & Free-Parameter Ledger

3 free parameters · 4 axioms · 0 invented entities

The paper introduces no new physical entities. Its central claim depends on untested experimental and modeling assumptions about field uniformity, ohmic conduction, and measurement fidelity, rather than on fitted theory parameters.

free parameters (3)

Interface extraction intensity threshold
Section 2 applies a pixel intensity threshold to extract the droplet interface; the threshold value is not stated, and the measured aspect ratio and orientation angle depend on it.
VIC# reconstruction parameters = denoising factor 0.001, 40 iterations, time projection ±1
Section 2 specifies these hand-chosen regularization parameters for the VIC# velocity reconstruction; they influence the flow topology reported in Figure 5.
Acquisition frame rates = 50 Hz shadow imaging; 20 Hz STB
Section 2 asserts these rates are 'sufficient' without spectral or validation evidence; the oscillatory-deformation claim could alias if the dynamics are faster than half the frame rate.

axioms (4)

domain assumption The liquids behave as leaky dielectrics: conduction is ohmic, with negligible charge injection, electrochemical reactions, or surface conduction.
Invoked in Section 1 via Garton & Krasucki and Taylor's leaky dielectric model; the interpretation of all observed deformation and rotation as electrohydrodynamic rests on this. Strong fields near 5.75 kV/cm may violate it.
domain assumption The applied electric field is uniform and equal to V/d between the electrodes.
Section 2 states E0 was determined by dividing the applied potential by the electrode separation 'assuming a uniform electric field'; no field measurement or correction for electrode or droplet field distortion is provided.
domain assumption The instability is Quincke-like, driven by induced electric torque for S/R > 1.
Section 3 attributes tilt and rotation to an imbalance in induced electric torque, citing Salipante & Vlahovska; the paper does not measure surface charge or torque directly.
domain assumption Tracer particles faithfully follow the flow and STB/VIC# reconstruction resolves the relevant scales.
Section 2 seeds 20 μm Rhodamine-B particles and uses STB/VIC#; no Stokes-number check, spatial-resolution check, or convergence validation is reported.

pith-pipeline@v1.3.0-alltime-deepseek · 6929 in / 15461 out tokens · 138808 ms · 2026-08-01T14:12:18.399152+00:00 · methodology

0 comments

read the original abstract

This work examines the deformation dynamics of dielectric liquid droplet when exposed to a uniform electric field. The experimental investigation involves two-phase configurations, here as silicone oil droplets suspended in castor oil. The droplet dynamics including deformation, elongation, oscillation, and rotation are investigated over a range of electric field strengths using shadow-imaging. In parallel, fluorescent tracer particles were employed to perform threedimensional Lagrangian Particle Tracking (3D-LPT) using the Shake-The-Box (STB) technique, allowing for a detailed characterization of the internal electrohydrodynamic flow within the droplet subjected to a uniform electric field. For silicone oil droplets in castor oil medium (S/R > 1), the droplets initially deform into oblate shape. At sufficiently high electric field strengths, the droplet undergoes an Electrohydrodynamic instability, aligning their axis at an angle to the direction of electric field due to the imbalance in induce electric torque. Upon further increasing the field strength, the droplets display oscillatory deformation before settling into a transiently stable configuration. The current study identifies and characterizes the distinct regimes of droplet behaviour, from initial deformation at low electric fields to oscillatory behaviour at high field strengths depending upon the relative dielectric and fluid properties of the liquid phases.

Figures

Figures reproduced from arXiv: 2607.21638 by Bertrand Lecordier (CORIA), Franck Lefebvre (CORIA), IUF), Mostafa S Shadloo (CORIA, Saurabh Sharma (CORIA).

**Figure 1.** Figure 1: Experimental setup showing the image acquisition system for (a) Shadow imaging technique (b) 3D Lagrangian Particle Tracking technique. High resolution shadow imaging [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗

**Figure 2.** Figure 2: Experimentally captured images of a deformed droplet under different electric field strengths (E₀): (a) 4.50 kV/cm, (b) 4.75 kV/cm, (c) 5.25 kV/cm, and (d) 5.75 kV/cm However, at higher electric fields, E₀>4.5 kV/cm the droplet response departs from this simple monotonic trend and displays an oscillatory (Fig. 2c and 2d) evolution of the aspect ratio before reaching equilibrium. The origin of this behavior… view at source ↗

**Figure 3.** Figure 3: Variation of droplet orientation angle and aspect ratio at different electric field strengths (E₀): (a) 4.50 kV/cm, (b) 4.75 kV/cm, (c) 5.25 kV/cm, and (d) 5.75 kV/cm. 3D PTV (Shake the box Technique) [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗

**Figure 4.** Figure 4: Sequence of droplet deformation dynamics captured in raw images from three synchronized cameras, showing seeded glass tracer particles under an applied electric field strength of 3.75 kV for a droplet with a diameter of 4.5 mm. It is also important to note that the Shake-the-Box (STB) Lagrangian Particle Tracking technique has been employed here primarily as a preliminary demonstration to assess its suitab… view at source ↗

**Figure 5.** Figure 5: Temporal evolution of the flow vector field in the vicinity of the droplet [PITH_FULL_IMAGE:figures/full_fig_p011_5.png] view at source ↗

discussion (0)

Reference graph

Works this paper leans on

2 extracted references · 1 canonical work pages

[1]

& Vlahovska, P

Brosseau Q. & Vlahovska, P. M. (2017). Streaming from the Equator of a Drop in an External Electric Field. Physical review letter, 119, 034501. doi: 10.1103/PhysRevLett.119.034501. Garton, C. G., & Krasucki, Z. (1964). Bubbles in insulating liquids: stability in an electric field. Proc. R. Soc. London, A 1964, 280,

work page doi:10.1103/physrevlett.119.034501 2017
[429]

Studies in electrohydrodynamics. I. Circulation produced in a drop by an electric field

Karp, J. R., Lecordier, B. & Shadloo, M. S. (2024). Electrohydrodynamic flows inside a neut rally buoyant leaky dielectric drop. Physics of Fluids, 36 (5): 053323, doi: https://doi.org/10.1063/5.0204569. 22nd LISBON Laser Symposium 2026 13 M. S. Abbasi, R. Song, S. Cho, & J. Lee (2020). Electro-hydrodynamics of emulsion drop lets: Physical insights to app...

arXiv 2024

[1] [1]

& Vlahovska, P

Brosseau Q. & Vlahovska, P. M. (2017). Streaming from the Equator of a Drop in an External Electric Field. Physical review letter, 119, 034501. doi: 10.1103/PhysRevLett.119.034501. Garton, C. G., & Krasucki, Z. (1964). Bubbles in insulating liquids: stability in an electric field. Proc. R. Soc. London, A 1964, 280,

work page doi:10.1103/physrevlett.119.034501 2017

[2] [429]

Studies in electrohydrodynamics. I. Circulation produced in a drop by an electric field

Karp, J. R., Lecordier, B. & Shadloo, M. S. (2024). Electrohydrodynamic flows inside a neut rally buoyant leaky dielectric drop. Physics of Fluids, 36 (5): 053323, doi: https://doi.org/10.1063/5.0204569. 22nd LISBON Laser Symposium 2026 13 M. S. Abbasi, R. Song, S. Cho, & J. Lee (2020). Electro-hydrodynamics of emulsion drop lets: Physical insights to app...

arXiv 2024