From wicking to anti-wicking: A universal framework for capillary dynamics

Aniruddha Saha; Sadaf Sobhani

REVIEW 4 major objections 4 minor 17 references

From wicking to anti-wicking: A universal framework for capillary dynamics

T0 review · 4 major / 4 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 A single parameter, the product of damping and forcing, orders capillary rise dynamics across channel geometries.

desk verdict Zeta = <D>·<F> is not a sufficient statistic, so the paper's central universal-parameter claim and optimization objective are unsupported; desk-reject. read the letter →

arxiv 2411.15440 v1 pith:B7D2BGXO submitted 2024-11-23 physics.flu-dyn

Aniruddha Saha , Sadaf Sobhani This is my paper · ORCID

classification physics.flu-dyn

keywords capillaryrisewickinganti-wickingdampedsystemgeneticalgorithmaxisymmetricchannelslubricationapproximationmicrofluidics

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

Capillary rise in a channel is usually thought to depend separately on geometry, wettability, and applied pressure. This paper argues that for axisymmetric channels with gradually varying radius, the front height follows a linear damped-system equation, and that a single number—the height-averaged product of damping and forcing, $\zeta = \langle D\rangle \langle F\rangle$—ranks the whole spectrum from fast wicking to strongly slowed (anti-wicking) rise. Because higher $\zeta$ consistently predicts faster initial rise and earlier equilibration in the solutions the authors compute, they use $\zeta$ as the objective in a genetic algorithm that searches over channel shapes. The optimized shapes are reported to rise faster and to enter the viscous regime earlier than the two previous best-known channel geometries, with $\zeta$ about 30% higher under the same normalization.

What carries the argument

The load-bearing identity is the damped-system form of the governing equation, $\frac{d\bar{h}}{d\bar{t}} = -D(\bar{h})\bar{h} + F(\bar{h})$, where the damping coefficient $D$ gathers viscous resistance, gravitational effects, and geometric slope, and the forcing $F$ gathers capillary driving plus any external pressure ramp. The new object is $\zeta = \langle D\rangle \cdot \langle F\rangle$, the product of the height-averaged coefficients, introduced by analogy with $\eta = D\cdot F$ for the constant-coefficient equation. The genetic algorithm in Eq. (6) maximizes $J(x) = \zeta$ over the shape parameters of the radius profile $R(x) = \hat{R} + \tilde{R}\cos(n\pi/2 - 2\pi x/\lambda)$, and the monotonic $\zeta$–height relationship is what allows the optimizer to sidestep solving the full integro-differential system at every step.

What would settle it

Simulate or measure, for a pair of channels deliberately constructed to have equal $\zeta$ but strongly different $D(h)$ and $F(h)$ profiles, the full rise curves from Eq. (4) or from experiment; if the curves differ in ordering or equilibration, the averaging collapse fails and the genetic-algorithm objective in Eq. (6) is not a faithful proxy for rise speed.

Watch

Extended reading notes

Core claim

The paper claims that the dimensionless front dynamics reduce to $\frac{d\bar{h}}{d\bar{t}} = -D(\bar{h})\bar{h} + F(\bar{h})$, with $D$ and $F$ expressed in terms of channel radius, slope, contact angle, density contrast, viscosity, and external pressure. The central discovery is that the height-averaged product $\zeta = \langle D\rangle \langle F\rangle$ behaves like the constant-coefficient product $\eta = D\cdot F$ in a linear damped system: the authors find a monotonic relation between $\zeta$ and rise height across hydrophilic and hydrophobic channels, with and without external pressure. Using $\zeta$ as the fitness function $J(x) = \langle D(x)\rangle \cdot \langle F(x)\rangle$ in a genetic algorithm, the paper reports new channel geometries whose modeled rise exceeds that of the power-law optimal channel of Gorce et al. and the polynomial optimal channel of Figliuzzi and Buie. The conclusion is that a single scalar can characterize both wicking enhancement and inhibition in the same framework.

Load-bearing premise

The load-bearing premise is that replacing the height-dependent damping and forcing with their averaged values keeps the ranking of rise speeds intact, so that the single product $\zeta = \langle D\rangle \langle F\rangle$ is enough to compare channels; if two channels with the same $\zeta$ but very different $D$ and $F$ profiles rise at different rates, the framework loses its predictive power.

Editorial extensions

If this is right

If $\zeta$ is truly universal, then channel design for either fast wicking or deliberate flow inhibition reduces to maximizing or minimizing a single scalar, rather than solving the full integro-differential equation.
The optimized geometries produced by the genetic algorithm raise the modeled liquid height beyond the previously reported power-law and polynomial optimal channels, implying the prior shapes were not global optima in this design space.
The damped-system form suggests a direct analogy with electrical RC circuits, where the product of damping and forcing plays the role of a dynamic response parameter, potentially enabling circuit-inspired intuition for microfluidic transport.
Because $\zeta$ also tracks the onset of the viscous regime, it can be used to predict when the $h \propto \sqrt{t}$ behavior begins, which matters for timing-sensitive applications such as paper-based diagnostics.

Reading between the lines

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

The paper's constant-coefficient argument shows that the product $\eta = D\cdot F$ matters for the analogue, but it does not prove that the height-averaged product $\zeta$ preserves the ordering for the full variable-coefficient problem; that is the assumption a reader should test.
Since $\zeta$ combines two quantities that enter the solution separately (equilibrium height is $F/D$ and relaxation rate is $D$), designs with identical $\zeta$ could have different balances of equilibrium and speed; the optimization would then be selecting a proxy without separating these effects.
One could test the framework further by fabricating two channels with equal $\zeta$ but different $D$ and $F$ profiles and measuring whether the rise curves collapse, which would confirm averaging is legitimate beyond the simulation cases shown.
The radius parametrization is like a truncated Fourier series; extending it with more terms may let the same optimizer find even faster shapes, but the gain would be bounded by the validity of the lubrication approximation.

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.

Desk Editor's Note

read the letter

Quick take: the paper's central parameter zeta = · does not do what it claims. The constant-coefficient equation used as motivation gives h(t) = F/D (1 - e^{-Dt}), which depends on D and F separately, not only on their product. Two cases with identical zeta can have equilibrium heights a factor of 100 apart, and a lower-zeta system can have a larger initial slope. So the monotonic ranking and the GA objective in Eq. (6) are unsupported. This is a load-bearing flaw, not a nit.

What's genuinely useful: the paper sets up a compact first-order framework dh/dt = -D(h)h + F(h) for axisymmetric channels, and it covers a wide design space including hydrophilic and hydrophobic cases with external pressure. The idea of using a scalar objective for genetic optimization of channel shape is attractive, and benchmarking against the geometries of [7] and [15] is the right way to calibrate. The optimized shapes do produce faster modeled rise in the full equation, which suggests the search itself might be fine; the problem is the claimed justification via zeta.

Where it falls apart: the leap from constant coefficients to height-averaged coefficients is never justified. The paper asserts monotonicity from Fig. 2 without a proof, and the stress-test counterexample shows that even for the simplest case, higher product does not imply faster rise. The averaging step could easily reorder cases. Also, the derivation from Eqs. (1)-(2) to (5) is too compressed; there are apparent dimensional and sign issues (the OCR doesn't help, but the published text reads garbled in places). No experimental validation is given, and no code or data are released, which makes the optimization hard to reproduce.

Bottom line: the framework is standard Lucas-Washburn with a new averaging gimmick, and the gimmick fails on inspection. The paper is not ready for peer review as written. A serious revision would need to either prove a ranking theorem for zeta under the actual h-dependent coefficients, or replace the objective with something concretely tied to rise time (e.g., time to reach 90% of equilibrium). If the authors did that, the optimization study might be worth revisiting. As is, I'd desk-reject.

Referee Report

4 major / 4 minor

Summary. The manuscript proposes a one-dimensional energy-balance model for capillary rise in axisymmetric channels with slowly varying radius, written in compact form as dh/dt = -D(h)h + F(h). It introduces a height-averaged product zeta = <D><F> as a universal parameter that is claimed to order rise dynamics from wicking to anti-wicking, and it uses a genetic algorithm with objective J = zeta to optimize channel geometries. The optimized geometries are compared with the prior designs of Gorce et al. [7] and Figliuzzi and Buie [15], and the paper reports faster rise and a roughly 30% increase in zeta. The central claim is that this single parameter characterizes rise dynamics across geometric, wetting, and pressure conditions.

Significance. If valid, a single parameter that ranks capillary-rise performance across geometry, wettability, and applied pressure would be practically useful and would unify disparate design rules. The paper has real strengths: Eq. (4) is derived from an energy-balance formulation rather than fitted ad hoc, the model is benchmarked against external geometries from [7] and [15], and the optimized shapes are concrete falsifiable predictions. However, the central universal-parameter claim is not supported by the evidence in the manuscript. The motivating constant-coefficient system itself shows that the product D*F is not a sufficient statistic for the rise curve, and no proof or controlled numerical experiment is given that replacing D(h) and F(h) by height averages preserves the ordering of rise dynamics. Because the optimization objective in Eq. (6) is exactly this product, the reported design improvements are not tied to a well-defined rise-height or rise-time objective. The core contribution therefore needs substantial reformulation, not just local revision.

major comments (4)

[After Eq. (5), constant-coefficient motivation] The claim that eta = D*F controls rise behavior is false for the very equation used to motivate it. For dh/dt = -D h + F with h(0) = 0, the solution is h(t) = (F/D)(1 - e^{-Dt}), so the equilibrium height is F/D and the relaxation time is 1/D; these are independent pieces of information, not functions of eta only. Equal-eta pairs such as (D,F) = (10,10) and (1,100) have the same eta = 100 but equilibrium heights 1 and 100 and relaxation times 0.1 and 1. Lower-eta cases can also rise faster initially: (D,F) = (1,1000) has eta = 10^3 and initial slope 1000, while (D,F) = (1000,100) has eta = 10^5 and initial slope 100. The text's statements that higher eta gives faster initial rise and earlier equilibration are therefore contradicted by the manuscript's own model, and this undermines the analogy used to justify zeta.
[Paragraph defining zeta after Eq. (5)] The step from the constant-coefficient discussion to the height-averaged product zeta = <D><F> is not justified. Equation (4) has h-dependent D and F; replacing them by their averages changes the solution, and no argument is given to show that geometries with larger zeta rise faster at every time or even at a fixed target height. Fig. 2(a) is a colored ensemble of trajectories, not a controlled test: the curves vary simultaneously in D(h), F(h), and zeta, so the visual correlation does not establish the claimed monotonic relation. The statement that 'the monotonic relationship between rise height and zeta across the entire design space' is observed would need at minimum a quantitative measure, such as time to a fixed height as a function of zeta with independent variation of D and F, before it can support optimization.
[Eq. (6), optimization objective] Because the cost function J = <D><F> is the very quantity whose validity as a ranking scalar is unestablished, the optimization results do not show that the resulting geometries achieve faster rise for any concrete performance measure. The text reports that the optimized curves 'surpass the heights predicted by [7,15]' and a '~30% increase', but the 30% figure refers to zeta, not to a rise-height or rise-time improvement. In the constant-coefficient case, maximizing zeta can select a large-F/large-D geometry that equilibrates quickly at a low height, so the design recommended by Eq. (6) is not necessarily the one that reaches a prescribed height first. The claim that these geometries are optimal is therefore not supported by the stated objective.
[Eq. (2)] Equation (2) as printed contains delta-dot-E on both sides and appears dimensionally inconsistent; as the stated basis for Eq. (4), it must be rewritten with all terms explicitly defined. The gravitational potential derivative should presumably involve dh/dt, but none appears in the first term as written. Without a corrected statement of this balance, the reduction to Eq. (4) cannot be checked.

minor comments (4)

[Pressure ramp definition, after Eq. (4)] The coefficients P_A and P_B in the pressure ramp P = P_A + P_B t are introduced but never given values or units, so the hydrophobic-pressure cases in Fig. 2 are not reproducible.
[Fig. 2 caption] The phrase 'reduced effective viscosity' should be defined quantitatively; the non-dimensional mu-bar is an integral measure of channel resistance, not a physical viscosity, and the text should state exactly what is plotted.
[Calibration paragraph, near R(h) = (1-h/l)^(5/6)] The calibration of the geometric parameters R_hat, R_tilde, and lambda to the power-law profile is not described quantitatively; please report the fitted values and the fitting procedure so the benchmark comparison is reproducible.
[Reference [16]] The manuscript refers to Supplementary Material for numerical validation, but the validation details are not included in the text; either include the supplementary material in the submission or summarize the validation in the paper.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the governing equation is derived from an energy balance and the design comparisons use external prior geometries; the ζ-based optimization is self-consistent but not circular.

full rationale

The central derivation chain is first-principles: Eqs. (1)-(3) are written from free-surface energy, gravitational potential, viscous dissipation, and an external pressure ramp, and Eq. (4) is the resulting compact integro-differential equation. No fitted constant or pre-supposed rise law enters the derivation of D(h) and F(h). The parameter ζ = ⟨D⟩·⟨F⟩ is defined after Eq. (5) as a product of height-averaged coefficients of Eq. (4), rather than being fitted to any subset of the target rise curves. The comparisons against Gorce et al. [7] and Figliuzzi and Buie [15] are external benchmarks, and the paper's own geometries are explicitly calibrated to the power-law shape from [7] before optimization, so the validation is not self-referential. The genetic algorithm in Eq. (6) maximizes ζ and the resulting rise is then computed from the same Eq. (4); this makes the optimization self-consistent, but it does not make the claim 'higher ζ implies faster rise' an identity or a fitted tautology. Whether height-averaging D and F preserves the ordering of solutions is a mathematical validity question, not a circularity: the constant-coefficient counterexamples noted by the reader show that η = D·F is not a sufficient statistic, which would undermine the optimization's guarantee, but that is a correctness risk rather than a derivation loop. No load-bearing self-citations or author-imported uniqueness theorems appear in the paper. Therefore no circular step can be exhibited from the text, and the appropriate circularity score is 0.

Assumptions & free parameters 2 free parameters · 4 assumptions · 1 invented entities

The model leans on standard low-Reynolds-number capillary assumptions and a specific geometric parameterization. The main load-bearing addition is the zeta parameter, which is not independently measured. The free parameters listed are the calibration to a prior theoretical optimum and the unspecified pressure ramp coefficients; the channel shape variables in the genetic algorithm are design variables rather than fitted constants.

free parameters (2)

power-law calibration shape coefficients = R(h) = (1 - h/l)^(5/6)
Chosen to reproduce the optimal geometry of Gorce et al. [7] for benchmarking; this is a fit to a prior theoretical result, not an independent measurement.
external pressure ramp coefficients P_A, P_B = not specified in text
Introduced for hydrophobic anti-wicking cases as P = P_A + P_B t; numerical values are not reported, so the anti-wicking solutions cannot be reproduced from the paper alone.

assumptions (4)

domain assumption Lubrication approximation with parabolic velocity profile and negligible inertia is valid for all studied geometries, including optimized and hydrophobic forced-flow cases.
Invoked before Eq. (2) and after Eq. (5); the model is first order in h, yet the paper discusses an inertial h proportional to t regime, so the assumption is load-bearing and not checked for the optimized shapes.
domain assumption The radius parameterization R = R_hat + R_tilde cos(n pi/2 - 2 pi x/lambda) covers the complete axisymmetric design space with dR/dx much less than 1.
Introduced after Eq. (5); steep-walled or non-sinusoidal shapes may violate the lubrication condition, so the complete design space claim rests on this representation.
ad hoc to paper The product zeta = <D> times <F> is a sufficient and monotonic descriptor of rise dynamics.
Central assertion in the paragraph defining zeta and in Eq. (6); no proof is given, and the constant-coefficient analogue shows the product does not determine the full solution.
domain assumption The surface free-energy balance in Eq. (1) is the correct rate expression for a moving meniscus in a varying-radius channel.
The terms and signs in Eq. (1) are nonstandard and not derived from a free-energy functional in the text; the subsequent compact form Eq. (4) depends on it.

invented entities (1)

none
purpose: No new physical object is introduced.
The parameter zeta is a composite scalar derived from model coefficients, not a new entity with an independent falsifiable handle.

how reviews work

0 comments

Cite this review

Pith. "Pith review of From wicking to anti-wicking: A universal framework for capillary dynamics." pith.science (2026). https://pith.science/paper/B7D2BGXO

@misc{pith2026241115440,
  author       = {Pith},
  title        = {Pith review of: From wicking to anti-wicking: A universal framework for capillary dynamics},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/B7D2BGXO}},
  note         = {Machine review of arXiv:2411.15440}
}

read the original abstract

The dynamics of capillary rise under different geometric and fluid conditions have the common signatures of rapid rise followed by an equilibrium state that describe the underlying competing forces. We present a new interpretation of capillary dynamics using a linear damped system where modulation of damping and forcing characteristics are achieved using axisymmetric channels with sinusoidal variation in radius. The complete axisymmetric design space ranging from hydrophilic channels that enable spontaneous imbibition to hydrophobic channels, that required external pressure mechanisms is modeled and the force dynamics is split into simultaneous damping and forcing characteristics. We introduce the product of damping and forcing terms as the new parameter that effectively characterizes rise dynamics across various geometric and flow conditions, encompassing both flow-enhancing and flow-inhibiting scenarios. The monotonic nature of this parameter enables the development of a stochastic optimization method that can determine optimal channel geometries for controlled capillary rise.

Figures

Figures reproduced from arXiv: 2411.15440 by the authors.

**Figure 2.** FIG. 2. Non-dimensional height ((a) and (b)) and viscosity ((c) and (d)) as function of time for capillaries with varying shapes [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 3.** FIG. 3. Evolution of the channel surface through Genetic [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

17 extracted references · 17 canonical work pages

[7]

Gorce, I

J.-B. Gorce, I. J. Hewitt, and D. Vella, Langmuir 32, 1560 (2016)

work page 2016
[15]

Figliuzzi and C

B. Figliuzzi and C. Buie, Journal of Fluid Mechanics731, 142 (2013)

work page 2013
[1]

Lucas, Kolloid-Zeitschrift 23, 15 (1918)

R. Lucas, Kolloid-Zeitschrift 23, 15 (1918)

work page 1918
[2]

E. W. Washburn, Physical review 17, 273 (1921)

work page 1921
[3]

M. Tani, D. Ishii, S. Ito, T. Hariyama, M. Shimomura, and K. Okumura, Plos One 9, e96813 (2014)

work page 2014
[4]

Czachor, Journal of Hydrology 328, 604 (2006)

H. Czachor, Journal of Hydrology 328, 604 (2006)

work page 2006
[5]

Tsori, Langmuir 22, 8860 (2006)

Y. Tsori, Langmuir 22, 8860 (2006)

work page 2006
[6]

B. Ma, D. Li, and H. Yang, Physics of Fluids 34 (2022)

work page 2022

Show all 17 references

[8]

N. A. Dudukovic, E. J. Fong, H. B. Gemeda, J. R. DeOtte, M. R. Cer´ on, B. D. Moran, J. T. Davis, S. E. Baker, and E. B. Duoss, Nature 595, 58 (2021)

2021
[9]

M. E. A. B. Amara, P. Perre, and S. B. Nasrallah, Jour- nal of Porous Media 19 (2016)

2016
[10]

Erickson, D

D. Erickson, D. Li, and C. Park, Journal of colloid and interface science 250, 422 (2002)

2002
[11]

Berthier, D

J. Berthier, D. Gosselin, A. Pham, G. Delapierre, N. Bel- gacem, and D. Chaussy, Langmuir 32, 915 (2016)

2016
[12]

Anoop and A

R. Anoop and A. Sen, Physical Review E 92, 013024 (2015)

2015
[13]

Anfodillo, V

T. Anfodillo, V. Carraro, M. Carrer, C. Fior, and S. Rossi, New Phytologist 169, 279 (2006)

2006
[14]

Reyssat, L

M. Reyssat, L. Courbin, E. Reyssat, and H. A. Stone, Journal of Fluid Mechanics 615, 335 (2008)

2008
[16]

See Supplemental Material for explanation of numerical method and validation
[17]

J. H. Holland, Scientific american 267, 66 (1992)

1992

Pith tools

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

[1] [7]

Gorce, I

J.-B. Gorce, I. J. Hewitt, and D. Vella, Langmuir 32, 1560 (2016)

work page 2016

[2] [15]

Figliuzzi and C

B. Figliuzzi and C. Buie, Journal of Fluid Mechanics731, 142 (2013)

work page 2013

[3] [1]

Lucas, Kolloid-Zeitschrift 23, 15 (1918)

R. Lucas, Kolloid-Zeitschrift 23, 15 (1918)

work page 1918

[4] [2]

E. W. Washburn, Physical review 17, 273 (1921)

work page 1921

[5] [3]

M. Tani, D. Ishii, S. Ito, T. Hariyama, M. Shimomura, and K. Okumura, Plos One 9, e96813 (2014)

work page 2014

[6] [4]

Czachor, Journal of Hydrology 328, 604 (2006)

H. Czachor, Journal of Hydrology 328, 604 (2006)

work page 2006

[7] [5]

Tsori, Langmuir 22, 8860 (2006)

Y. Tsori, Langmuir 22, 8860 (2006)

work page 2006

[8] [6]

B. Ma, D. Li, and H. Yang, Physics of Fluids 34 (2022)

work page 2022