Splashing velocity of a viscous liquid squeezed between two parallel disks

David Brutin; Navin Kumar Chandra; Pierre Perrier

REVIEW 4 major objections 5 minor 25 references

Splashing velocity of a viscous liquid squeezed between two parallel disks

T0 review · 4 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read A viscous liquid squeezed between parallel disks ejects a peripheral splash whose bulk speed is set by the momentum-averaged ejection velocity, not the peak velocity, and including both local and convective inertia makes theory match…

desk verdict A clean parameter-free theory for squeeze-film splash velocity, but the mapping from measured front speed to the momentum-average is asserted, not shown, and that is the load-bearing step. read the letter →

arxiv 2608.07960 v1 pith:K63BTYZT submitted 2026-08-08 physics.flu-dyn

Navin Kumar Chandra , Pierre Perrier , David Brutin This is my paper · ORCID

classification physics.flu-dyn

keywords squeezeflowsplashvelocitymomentum-averagedejectionliquidinertiaparalleldisksviscoushigh-speedimagingbloodspatter

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

The paper tries to establish why existing theory overpredicts the speed of the splash thrown out when a disk falls onto a liquid film, and to supply a corrected prediction. It argues that earlier theory left out part of the liquid's inertia and compared its prediction to the wrong observable: the peak ejection velocity. Once both local and convective inertia are kept in the disk's equation of motion, and once the splash speed is represented by a momentum-averaged ejection velocity, the model matches high-speed experiments for water and glycerol-water mixtures and independent data spanning wide parameter ranges.

What carries the argument

The central object is the momentum-averaged ejection velocity (Eq. 5), $v_{e,\mathrm{avg}} = (1/h_0)\int_0^{h_0} v_e\, dh$, derived from the rate of radial momentum ejection $\dot{\psi} = 2\pi R h \rho v_e^2$ divided by the total splash mass $m \approx \pi R^2 h_0 \rho$. It replaces the peak value $\max(v_e)$ as the observable for bulk splash speed. It is fed by the coupled system: kinematic mass conservation $v_r = u r/(2h)$, the top-disk equation of motion $M\,d^2u/dt^2 = Mg - F_v - F_i$, and the corrected inertial resistance $F_i = \rho\pi R^4\left(3u^2/(16h^2) + \dot{u}/(8h)\right)$, with $F_v = 3\pi\mu u R^4/(2h^3)$. The ODE system is solved numerically, and $v_{e,\mathrm{avg}}$ is computed by integrating $v_e$ over $h$ from $0$ to $h_0$.

What would settle it

Measure the radial velocity profile across the thickness of the ejected splash at several instants and compare the fastest front speed to both $\max(v_e)$ and $v_{e,\mathrm{avg}}$; if the front tracks the fastest parcel, or if a measurable liquid film remains trapped between the disks so that $m \approx \pi R^2 h_0 \rho$ is the wrong normalizer, the proposed agreement would fail.

Watch

Extended reading notes

Core claim

The bulk splash velocity of a squeezed viscous film is not the maximum radial velocity reached at the disk edge during squeezing. The relevant speed is the momentum-averaged value $v_{e,\mathrm{avg}} = (1/h_0)\int_0^{h_0} v_e\, dh$, obtained by conserving linear momentum of the coherent splash. The paper shows that correcting the inertial resistance term by including both local and convective acceleration (Eq. 4) and then averaging the ejection velocity over the film thickness brings theory into good agreement with their experiments and with literature data.

Load-bearing premise

The measured fastest-moving bulk splash front moves at the momentum-averaged velocity, which holds only if internal viscous resistance homogenizes velocity across ejected layers and nearly all the initial film mass is pulled into the splash before fragmentation.

Editorial extensions

If this is right

The splash velocity is fixed by the coupled ODEs (Eqs. 1, 2, 4) plus Eq. 5, with no fitted parameters, so any new squeeze experiment can be compared to a unique prediction.
The relevant observable for splash propagation is the momentum-averaged ejection velocity, so future analyses should stop using peak ejection velocity as the proxy for splash speed.
Both local and convective inertia must enter the inertial resistance; dropping either one reintroduces a large overestimate.
The corrected model works across roughly three orders of magnitude in viscosity and one order in disk radius, including independent data, so it can serve as a leading-order baseline for splash morphology and fragmentation.
The result gives a quantitative input for bloodstain-spatter modelling, where bulk splash speed is the leading-order quantity.

Reading between the lines

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

If the homogenization argument is right, there should be a dimensionless group comparing the viscous homogenization time of the ejected layer to the squeeze time; the transition from dome-shaped splashes (water) to disk-shaped splashes (viscous liquids) should be encoded in that group, which the paper leaves implicit.
The early spray-like ejecta, excluded here, may set initial conditions for fragmentation; coupling the bulk $v_{e,\mathrm{avg}}$ with an ejecta velocity model could predict droplet size distributions in blood spatter.
Applying the momentum-averaging logic to non-Newtonian liquids would replace $F_v$ with a viscoelastic force, and the resulting prediction should be testable with existing squeeze-flow rheometry data.
The framework's use of the entire film mass $m \approx \pi R^2 h_0 \rho$ could be checked directly by weighing the residual film left between the disks after impact; if the residual is large, $v_{e,\mathrm{avg}}$ should be renormalized by the ejected mass only.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Read it. The new piece is the combination of a corrected inertia term (local plus convective, already in Lang et al.) with the momentum-averaged ejection velocity as the splash observable. That averaging idea is simple and, as far as the citations show, genuinely absent. The ODE system is parameter-free, Eq 5 follows from momentum conservation, and the collated dataset spans a useful range. Credit where due: this is a clean, testable claim.

Soft spots. The biggest is the bridge from theory to experiment. They measure the fastest-moving splash front, then compare to ve_avg, the center-of-mass speed of all ejected liquid. Unless internal stresses wipe out velocity gradients fast, the leading edge moves faster than the average. The paper's homogenization argument is qualitative and, for water, the viscous diffusion time across a millimeter layer is about a second, while the squeeze lasts a millisecond. That order-of-magnitude mismatch is not addressed. It is possible the front is actually slowed by a different mechanism—fast parcels catching up and merging with slow fluid—but the paper doesn't say that. The residual-film assumption (all mass m = πR² h0 ρ goes into the splash) is also asserted, not measured. Eq 2 has a dimensional typo: d²u/dt² for u a velocity should be du/dt. No raw data or error bars are given, just a claim of <2.5% slope error.

None of this kills the paper. The empirical agreement might still be real, but the comparison is testing an unverified map between observable and theory. A referee should ask for direct evidence: e.g., particle tracking or simulation showing the front speed equals ve_avg, and a measurement of the residual film.

Recommendation: send to peer review. It's a serious, parameter-free attempt with a plausible central idea, and the subfield (blood spatter, squeeze-flow) would benefit from an independent check. Reading group: yes, it'll generate good discussion.

Referee Report

4 major / 5 minor

Summary. Chandra, Perrier, and Brutin revisit the squeeze-film splashing of a viscous liquid between two parallel disks. They argue that previous theory over-predicts the measured splash velocity for two reasons: the inertial resistance of the liquid was incompletely treated, and the peak ejection velocity was used as the representative splash velocity. Their framework couples the kinematic relation Eq. (1), the disk equation of motion Eq. (2), and an inertia term Eq. (4) that includes both local and convective radial acceleration. The proposed observable is a momentum-averaged ejection velocity, Eq. (5), obtained by integrating the radial momentum flux over the entire ejection and dividing by the nominal initial film mass. The model has no fitted parameters and is compared with the authors' high-speed measurements on water and two glycerol-water mixtures (Fig. 3) and with literature data over a wider parameter range (Fig. 4b).

Significance. If the central claim holds, the paper provides a parameter-free, physically motivated leading-order prediction for squeeze-driven splash velocity, with the practical virtue of requiring only geometric and material inputs. The authors correctly identify the two most plausible sources of the historical discrepancy and give a reproducible derivation of Eqs. (4) and (5). Validation against independent data from Bazilevsky and Rozhkov over two to three orders of magnitude in mass, radius, and viscosity is a genuine strength. The main weakness is that the measured 'fastest-moving splash front' is identified with the momentum-averaged quantity v_e,avg without direct verification, and this identification is doubtful for the low-viscosity cases in which viscous homogenization is too slow. The paper therefore presents a plausible and useful framework, but its central quantitative claim is not yet established.

major comments (4)

[Eq. (2)] Equation (2) is dimensionally inconsistent as written: u is defined as the vertical velocity, so M d^2u/dt^2 has units of kg m/s^3 rather than force. The intended first-order equation is M du/dt = Mg - F_v - F_i. Because Eq. (2) is one of the coupled equations solved numerically to produce all theoretical predictions, the manuscript must state the correct form and confirm that the numerics used the first-order version.
[Paragraph defining psi and v_e,avg, and Eq. (5)] The load-bearing identification of the experimentally tracked 'fastest-moving splash front' with the momentum-averaged v_e,avg is not justified. For water (nu ~ 1e-6 m^2/s), the viscous diffusion time across even a 1 mm ejected layer is about 1 s, far longer than the squeeze/ejection time h0/u0 ~ 1 ms; thus internal viscous resistance cannot homogenize the layers of the water splash before the front speed is measured. If the front instead tracks the fastest parcel, the comparison in Figs. 3 and 4b tests a different quantity from Eq. (5). I request a direct test, such as particle tracking or PIV inside the splash, or a quantitative homogenization-time estimate with measured velocity profiles.
[Numerical-solution paragraph before Fig. 3] The manuscript acknowledges that the film thickness never reaches zero and that 'momentum averaging should be done considering a mass ... smaller than the mass of the initial liquid film,' but then adopts m ~ pi R^2 h0 rho on the hypothesis that residual liquid is pulled into the splash. No evidence is provided for this hypothesis, and it is least plausible for the low-viscosity cases in which the claimed agreement is best. This normalizer directly enters Eq. (5); the authors should quantify the ejected mass, for example from the images or from a mass balance, or bound the resulting error.
[Figs. 3 and 4b] The validation figures contain no error bars and no point-by-point residuals or quantitative agreement metric. The statement that slope standard errors are below 2.5% addresses only the linear fits used to estimate u0 and v_e,avg, not the spread between theory and experiment, the uncertainty in h0, or the pixel-resolution limit (0.11 mm/pixel at 5000 fps, corresponding to roughly 0.55 m/s per frame in front speed). Without raw data or a table of values, the 'good agreement' claimed in Figs. 3 and 4b cannot be assessed quantitatively.

minor comments (5)

[Numerical solution paragraph] The sentence 'We numerically solve equation 5 to compute the momentum-averaged splash velocity' is confusing: Eq. (5) is a quadrature, while the coupled system (1), (2), and (4) is what is actually solved. Please describe the algorithm as evolving the disk motion and accumulating the integral over dh.
[Experimental methods] Please clarify how the effective mass M is determined from the lever-arm geometry; it is a key input to Eq. (2), and no uncertainty is reported for it.
[Figure 1 reference] The text refers to 'Figure 1a' for the water splash, but Figure 1 as printed has no visible panel labels; please correct the citation or add the labels.
[Typographical issues] There are several typographical and spacing errors, including 'Acknowledgment-This' and missing spaces such as 'ofthesplashvelocity'; these should be corrected throughout.
[Data availability] The data availability statement says no data or software support the manuscript; for a validation-focused paper, providing the raw measurement tables and the MATLAB solver would materially improve reproducibility and allow the referee and readers to check the agreement claimed in Figs. 3 and 4b.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the derivation is self-contained, parameter-free, and validated against independent data.

full rationale

The paper's derivation chain is not circular. Eq 1 is a kinematic constraint from mass conservation; Eq 2 is the top-disk equation of motion; Eqs 3-4 derive the inertial resistance including local and convective inertia; Eq 5 defines a momentum-averaged ejection velocity from the radial momentum flux, with no free parameters fitted to the splash-velocity measurements. The comparison in Figs 3 and 4 uses only geometric, material, and measured inputs (R, h0, rho, mu, u0, effective M), and the theory is also checked against independent literature data from Bazilevsky and Rozhkov, which are external works rather than self-citations. The main physical assumptions—depth-averaged plug flow, homogenization of ejected-layer velocities, and the hypothesis that most of the residual film is pulled into the splash—are stated openly and are approximations that could affect accuracy, but they do not define the predicted quantity in terms of the experimental outcome. Choosing ve_avg rather than max(ve) is a physical modeling choice motivated by momentum conservation, not a parameter fitted to the observed splash front. No load-bearing self-citation chain or imported uniqueness theorem appears. Thus no circular step can be quoted and reduced to the paper's own inputs; the appropriate verdict is no significant circularity.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

No fitted constants. All inputs are geometry, material properties, and measured initial conditions. The theoretical content rests on the plug-flow kinematic model, the inertial pressure closure, and the momentum-averaging hypothesis that ties theory to the measured observable.

assumptions (6)

domain assumption The radial flow is depth-averaged and independent of z, so vr = ur/(2h) holds throughout the film and at the rim.
Invoked to derive Eq 1 and used in Eq 3 for the pressure field; ignores vertical velocity structure and possible rim effects.
standard math Liquid is incompressible and mass conservation is the standard kinematic constraint.
Used to write Eq 1; standard continuity for axisymmetric incompressible flow.
domain assumption The inertial pressure field obeys Eq 3 with boundary pressure zero at r=R, and the viscous pressure is not included in Fi.
Used to derive Eq 4; neglects coupling between viscous and inertial pressure and assumes p=0 at the rim.
domain assumption The top disk deceleration is governed by Eq 2 with both Fv and Fi acting simultaneously, and the gravity term is negligible.
Used to solve for u(t) and h(t); the stated equation has a typo (d²u/dt² for velocity).
ad hoc to paper The measured bulk splash front speed equals the momentum-averaged velocity of Eq 5, which requires internal resistance to homogenize velocities and assumes the entire initial film mass, including residual liquid, joins the splash.
This is the key mapping from theory to experiment; the paper states it as a hypothesis about internal homogenization and residual mass pull-in.
domain assumption Air drag and capillary forces on the splash are negligible during the experimental timescale.
Invoked to justify the momentum balance for the splash; valid for high-speed splashes but not at low speeds.

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Cite this review

Pith. "Pith review of Splashing velocity of a viscous liquid squeezed between two parallel disks." pith.science (2026). https://pith.science/paper/K63BTYZT

@misc{pith2026260807960,
  author       = {Pith},
  title        = {Pith review of: Splashing velocity of a viscous liquid squeezed between two parallel disks},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/K63BTYZT}},
  note         = {Machine review of arXiv:2608.07960}
}

read the original abstract

Squeezing of a liquid film between two approaching solid surfaces can generate a high-speed peripheral splash. Despite extensive studies on squeeze-film hydrodynamics, quantitative prediction of the splash velocity remains unresolved, with existing theory significantly overestimating experiments. Here, we present a theoretical framework to predict the ejection velocity of viscous liquid squeezed between two parallel circular disks. We show that discrepancy of theory from experiments arise due to incomplete treatment of liquid inertia and from using peak ejection velocity to represent splash velocity. By accounting for both local and convective inertia, and introducing a momentum-averaged ejection velocity, we obtain good agreement with experiments over a wide range of parameters.

Figures

Figures reproduced from arXiv: 2608.07960 by the authors.

**Figure 1.** FIG. 1. Experimental images showing the left-half of side view from high-speed visualization of splashing phenomenon. [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗

**Figure 2.** FIG. 2. (a) Schematic representation of a viscous liquid [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗

**Figure 3.** FIG. 3. Experimental values and their corresponding theo [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

Figures from the paper (1 more)

**Figure 4.** Figure 4: FIG. 4. Comparison of theory and experimental values of [PITH_FULL_IMAGE:figures/full_fig_p004_4.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

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Pith tools

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

[1] [1]

S. Gart, B. Chang, B. Slama, R. Goodnight, S. H. Um, and S. Jung, Dynamics of squeezing fluids: Clapping wet hands, Phy. Rev. E (2013)

work page 2013

[2] [2]

C. Ge, P. Liu, Q. Qu, T. Hu, and J. Zhang, Effect of aircraft tire wear on water spray and water displacement drag, Aerospace Science and Technology147, 109006 (2024)

work page 2024

[3] [3]

E. Moss, A. Krassnokutski, B. Skews, and R. Paton, Highly transient squeeze-film flows, Journal of fluid me- chanics671, 384 (2011)

work page 2011

[4] [4]

Krassnokutski, E

A. Krassnokutski, E. Moss, and B. Skews, An experimen- tal study of highly transient squeeze-film flows, Physics of Fluids25(2013)

work page 2013

[5] [5]

J. Lang, R. Nathan, and Q. Wu, Experimental study of transient squeezing film flow, Journal of Fluids Engineer- ing141, 081110 (2019)

work page 2019

[6] [6]

R. M. Kenedi,Perspectives in Biomedical Engineering: Proceedings of a Symposium organised in association with the Biological Engineering Society and held in the Univer- sity of Strathclyde, Glasgow, June 1972(Springer, 1973)

work page 1972

[7] [7]

Lang and Q

J. Lang and Q. Wu, Modeling of the transient cere- brospinal fluid flow under external impacts, European Journal of Mechanics-B/Fluids87, 171 (2021)

work page 2021

[8] [8]

Shuler and S

S. Shuler and S. Advani, Transverse squeeze flow of con- centrated aligned fibers in viscous fluids, Journal of Non- Newtonian Fluid Mechanics65, 47 (1996)

work page 1996