Optimal rigid brush for fluid capture

Basile Radisson; Emmanuel Si\'efert; Fabian Brau; Hadrien Bense; Hoa-Ai B\'eatrice Hua; Lucie Domino

REVIEW 3 major objections 4 minor 48 references

Optimal rigid brush for fluid capture

T0 review · 3 major / 4 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read This paper claims that liquid capture by a rigid brush withdrawn from a bath obeys a parameter-free Darcy-law equation, and that captured volume peaks at a computable brush porosity.

desk verdict Strong and useful paper; the Heaviside permeability closure above the bath is the main unvalidated assumption, but the central results survive scrutiny. read the letter →

arxiv 2411.15970 v1 pith:MFZG6WRI submitted 2024-11-24 physics.flu-dyn

Basile Radisson , Hadrien Bense , Emmanuel Siéfert , Lucie Domino , Hoa-Ai Béatrice Hua , Fabian Brau This is my paper · ORCID

classification physics.flu-dyn

keywords viscousentrainmentrigidbrushDarcy'slawanisotropicporousmediumcapillaryriseJurinheightoptimalporositydipcoating

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 tries to establish that the liquid captured by a rigid brush pulled from a bath is not governed by vague scaling laws but by a single nonlinear partial differential equation derived from Darcy's law, with every coefficient fixed by brush geometry and fluid properties. The equation tracks the spatio-temporal position of the air-liquid interface inside the brush, so it predicts not only the maximum entrained height but the entire volume history. If correct, it turns brush design into a calculation: for fixed retraction speed and immersion depth there is an optimal porosity, and the model gives its value. The authors validate the prediction with 3D-printed brushes, finding measured optimal pillar radii close to the computed values for two retraction speeds.

What carries the argument

The load-bearing object is the nonlinear diffusion-type PDE (Eq. (9a)) for $H$, obtained by expanding the pressure in powers of the aspect ratio $\delta = 2L_0/D$. The mechanism is the competition between the brush's upward motion and gravity-driven drainage through the longitudinal and transverse permeabilities $k_\parallel$ and $k_\perp$, with $k_\perp$ set to zero above the bath level by the Heaviside closure $k_\perp(z) = k_\perp\theta(-z)$. The equation's coefficients are fixed by three dimensionless numbers: $\mathcal V = V/V_\parallel$ (retraction speed versus Darcy drainage speed), $\bar\delta = \delta(k_\perp/k_\parallel)^{1/2}$ (radial versus vertical flow), and $\bar h_J = h_J/L_0$ (initial capillary rise versus immersion depth). The optimal-porosity result follows by re-expressing these parameters in terms of porosity and maximizing the final captured volume.

What would settle it

Track tracer particles in the liquid column above the bath level during withdrawal: if the radial velocity is measurably nonzero in the region $z>0$, the Heaviside closure $k_\perp(z) = k_\perp\theta(-z)$ is wrong and Eq. (9a) cannot be the full description. A complementary check is to measure $h_0(t)$ for very slow withdrawals and compare the timing of the interface maximum with the model; a systematic shift with $\bar\delta$ would indicate missing transverse drainage.

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Extended reading notes

Core claim

The central claim is that a brush withdrawn from a bath behaves as an anisotropic porous medium whose transverse permeability switches off above the bath level, and that under this closure Darcy's law yields Eq. (9a) for $H(\bar r,\bar t)$, the interface height relative to the Jurin height. The equation involves three dimensionless groups: the retraction speed relative to the gravitational drainage speed, the radial-to-vertical flow ratio, and the Jurin height relative to immersion depth, with no fitting parameters. Its solutions reproduce the observed non-monotonic rise-and-fall of the interface at low speed and monotonic rise at high speed, and collapse the measured maximum heights onto a master curve. The same equation yields the captured volume at the end of retraction, which is non-monotonic in porosity; the predicted optimal pillar radii, $R = 729\,\mu\mathrm{m}$ at $V = 10\,\mathrm{mm/min}$ and $R = 571\,\mu\mathrm{m}$ at $V = 100\,\mathrm{mm/min}$, match the measured maxima near $R = 700$-$750\,\mu\mathrm{m}$ and $R = 550$-$600\,\mu\mathrm{m}$.

Load-bearing premise

The model assumes that liquid above the bath level cannot drain sideways: the transverse permeability is exactly zero there, and this step is imposed rather than measured, so significant sideward drainage above the bath would change the interface equation and the predicted optimal porosity.

Editorial extensions

If this is right

For a given brush and liquid there is no optimal retraction speed or immersion depth: captured volume grows monotonically with both, so the practical design lever is brush geometry rather than operating speed.
At fixed speed and depth, captured volume is non-monotonic in porosity, and the optimal porosity saturates at high speed to approximately $1 - \tilde h_J^2/4$, a value insensitive enough that near-optimal brushes collect almost the same volume.
The maximum interface height follows the quasi-master curve $H_0^m \simeq [\bar h_J + \alpha(\bar\delta)]\mathcal V$ for speeds below a transition value, giving a universal law for predicting maximum capture.
The same model, with the appropriate permeabilities, describes two parallel plates withdrawn from a bath, so the framework extends to other anisotropic porous geometries beyond pillar brushes.

Reading between the lines

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

A testable extension is to replace the Heaviside transverse-permeability step with a smooth decay above the bath; if that decay length is set by the meniscus scale, the optimal porosity could shift at low speeds, which high-speed imaging of the rim could probe.
Because the captured-volume curve is flat near its maximum, small manufacturing errors in pillar radius barely change the collected mass; this tolerance is implicit in the paper's numbers and could be used to set fabrication tolerances for practical brush designs.
For nectar-feeding animals with brush-like tongues, the model suggests an optimal papilla spacing for a given nectar viscosity and lapping speed; the paper notes the biological context but does not map animal tongue geometries onto the predicted optimum.
The optimal porosity is not a material constant: it depends on immersion depth and brush width through $\bar\delta$ and $\bar h_J$, so translating the result into industrial dip-coating requires specifying the full operating conditions.

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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

Radisson et al. have done something genuinely useful: they turn the old scaling-law picture of viscous entrainment from a brush into a working PDE for the interface, identify the three dimensionless groups that control it, and show the predicted master curve really does collapse data from 13 brushes and a plate geometry. The optimal-radius prediction (729 µm at 10 mm/min, 571 µm at 100 mm/min against measured maxima in the same ballpark) is a concrete, falsifiable claim. The permeability closures come from independent COMSOL simulations, not from the capture experiments, so the interface predictions are not curve fits. That is a real achievement.

The main soft spot is the Heaviside transverse permeability, k_perp(z) = k_perp Theta(-z). It is imposed because a constant k_perp underestimates the entrained height, and it is central to the dynamics above the bath. The SI's physical rationale—the air-liquid interface prevents radial expansion—is plausible but not tested. If lateral drainage above the bath is non-negligible, the PDE gains a radial-diffusion term in that region and the optimal-porosity curves shift. I do not think this sinks the paper: the authors are transparent about the assumption, the agreement across a wide parameter range is strong, and the optimal radius is insensitive to the multiplicative D^2 factor. But a direct test, e.g. stopping the withdrawal and watching whether the meniscus relaxes radially, would settle it.

Two smaller caveats. The phrase "without any fitting parameter" is a little generous because the effective brush diameter D is chosen per speed when computing masses (15.8 mm vs 16.9 mm). That does not change the optimal-radius prediction, but it is a chosen value, not a measured one. Also the paper reports no code or raw data, which makes re-implementation slower than it should be.

Overall: the central argument holds up. This deserves a serious referee. I would ask the authors for either a stopped-withdrawal lateral-drainage measurement or a direct estimate of the transverse velocity above the bath, and for a clear statement of how D is defined. With those, the paper is publishable as is.

Referee Report

3 major / 4 minor

Summary. This paper analyzes the liquid entrained by a rigid brush withdrawn from a liquid bath. The brush is modeled as an anisotropic Darcy porous medium with longitudinal permeability k∥ and a transverse permeability that is switched off above the bath level, k⊥(z)=k⊥θ(−z). A δ-expansion in the aspect ratio leads to a nonlinear PDE (Eq. (9a)) for the interface height H(r,t) in terms of three dimensionless parameters (V, D7FF̄δ, D7FF̄hJ). The authors validate the PDE against side-view measurements of the interface for 13 printed brushes and for a parallel-plate geometry, propose a master curve for the maximum interface rise, and derive an optimal porosity (equivalently pillar radius) that maximizes captured mass at a given withdrawal speed. Experiments confirm the predicted optimal radii R≈729 µm and R≈571 µm for V=10 and 100 mm/min.

Significance. If correct, the paper provides a useful quantitative framework for liquid capture by anisotropic porous media, capturing non-monotonic interface dynamics and yielding a design rule for optimal brushes. The main PDE is not fitted to the capture experiments; the permeability functions are obtained independently from COMSOL simulations and prior asymptotic results. The experimental comparison is broad (13 brushes, two oils, plate geometry), and the optimal-radius predictions are explicit and falsifiable. The derivation is presented in detail in the SI, including approximate analytical solutions and error estimates. The main caveats are that the Heaviside transverse-permeability closure is imposed rather than measured, and the perturbation expansion is used at aspect ratios where it is formally uncontrolled; both concerns can in principle be addressed with targeted checks.

major comments (3)

[Model, Eq. (2); SI Eq. (14)] The closure k⊥(z)=k⊥θ(−z) is load-bearing: it removes radial Darcy flow above the bath and is therefore responsible for the form of Eq. (9a) and for the predicted optimal radius. The only justification offered is that the air-liquid interface 'prevent[s] any radial expansion,' but a free interface can deform and permit lateral drainage. The agreement with experiments is suggestive but does not independently verify the step, because the parallel-plate validation uses the same closure and the permeability functions are obtained from independent numerical simulations rather than from a measurement of the radial flow. I ask for a direct test: for example, stop the withdrawal before the brush clears the bath and record whether the entrained column above the bath changes shape or drains sideways, or measure the radial component of the flow/meniscus in the region z>0. Without such a test, the central predictive claim rests on an unverified assumption; if lateral drainage above the bath is non-negligible, Eq. (9a) would need an additional radial-diffusion term and the optimum would shift.
[Model, Eqs. (6)-(9); SI Eq. (27)] The PDE is derived by expanding the pressure in δ² and neglecting a nonlinear term, assuming δ ≪ 1, yet Fig. 3A reports comparisons up to δ≈1.27 and Fig. 3B up to δ≈1.30. The SI gives a 3.5% relative-error bound for the neglected term only in a restricted parameter range (0≤D7FF̄δ≤2, 0≤V≤5, 0.1≤D7FF̄hJ≤0.5), which does not cover all experiments. Please quantify the truncation error over the full experimental parameter space, or compare Eq. (9a) with a direct numerical solution of the unsimplified system at the largest δ used. This is a correctness-risk concern rather than an observed failure, but it is load-bearing because the 'good agreement up to δ of order 1' is asserted rather than demonstrated.
[Fig. 4C; SI 'Discussion about the value of D'] The mass comparison uses D=15.8 mm at V=10 mm/min and D=16.9 mm at V=100 mm/min, with the SI noting that D is not well defined for the printed brushes. Since the dimensional mass scales with D², the choice of D does not move the predicted optimal radius, but it weakens the 'no fitting parameter' claim for the dimensional mass curves. I ask the authors to report the geometric range of D for the brushes used, or to compare with a measurement-based D (e.g., the rim-pillar diameter) rather than selecting D per speed.

minor comments (4)

[Eq. (12)] The notation is confusing because V denotes both the dimensional captured volume and its dimensionless rescaled value V=V/VI; please use distinct symbols such as Ṽ for the rescaled quantity.
[SI caption of Fig. 6] The caption says 'for three value of R' and should read 'for three values of R'; similar small grammar issues appear elsewhere in the SI.
[Fig. 3B] The grey area representing the region spanned by theory is not defined quantitatively; please state the precise range of D7FF̄δ and D7FF̄hJ used to generate it.
[Eq. (14)] The fitted expressions for A(D7FF̄hJ) and B(D7FF̄hJ) are presented without error estimates or fitting range; adding these would help readers assess the accuracy of Eq. (14).

Circularity Check

1 steps flagged · score 2.0 of 10

Central derivation is not circular: Eq. (9a) is a parameter-free Darcy-law prediction tested against independent experiments; only D is tuned in one mass comparison and the Heaviside k⊥ closure is an openly stated modeling assumption, not a fitted parameter.

fitted input called prediction [Main text Fig. 4C caption; SI "Discussion about the value of D used to compute the mass of liquid captured".]
"D = 15.8 mm and D = 16.9 mm were used in the theory for V = 10 mm/min and V = 100 mm/min, respectively. ... We found a good agreement with the data in Fig. 4C of the main text using D = 15.8 mm and D = 16.9 mm for V = 10 mm/min and V = 100 mm/min, respectively."

The theoretical mass curves in Fig. 4C are presented as predictions, but the brush diameter D, which enters the dimensional mass through a D^2 prefactor, is not independently measured or fixed for each speed: different values are chosen for the two retraction speeds to make the computed curves agree with the measured masses. This makes D^2 a two-point fitted scale factor in the mass comparison. The central optimal-radius claim is not vitiated because the paper explicitly notes that the predicted optimal radius is unaffected by a multiplicative constant (D^2 only scales the mass), so R_opt remains an unfitted output of Eq. (9a). This is a minor fitted input in a secondary comparison, not a load-bearing circularity.

full rationale

The derivation chain is essentially self-contained. Eq. (9a) follows from Darcy's law (Eq. 2), a two-term pressure expansion in the aspect ratio δ (Eqs. 6-8), and the boundary conditions p(r,-L)=ρgL and p(r,h)=-ρghJ. The permeabilities k∥ and k⊥ are obtained from independent COMSOL Stokes-flow simulations (SI Eqs. 8-11), not fitted to the capture experiments, and they extend published asymptotic expressions [43,44]. The Jurin height hJ is independently derived from surface-energy minimization and checked against experiments (SI Eq. 5, Fig. 5D). The three dimensionless groups V, δ‾, and h‾J emerge naturally from the derivation, and Eq. (9a) is a genuine nonlinear PDE whose numerical solution is compared with experiments with no fitted parameters. The non-monotonic H0(t) curves, the master-curve behavior H0^m ≈ [h‾J+α(δ‾)]V, and the optimal-porosity curves in Fig. 4 are outputs of this PDE, not inputs; the auxiliary fits for α(δ‾), V_T, and S(h‾J) are fits to the model's own numerics, so they are not circular with respect to the experimental data. Two caveats keep the score above zero. First, the closure k⊥(z)=k⊥θ(−z) (Model Eq. 2; SI Eq. 14) is a structural assumption explicitly motivated by the observations that a constant k⊥ "underestimates the height of the liquid column entrained by the brush compared to experiments" and that the data require a third dimensionless group. This is an empirically constrained ansatz, and no independent measurement of radial flow above the bath is presented; the parallel-plate validation uses the same Heaviside assumption, so it is not an independent check. This is a limitation or correctness risk, but not a circular reduction: no continuous parameter is fitted to the capture data, and the PDE retains substantial predictive content, such as the optimal radius. Second, in the mass comparison of Fig. 4C, the brush diameter D is chosen per speed (D=15.8 mm at V=10 mm/min, D=16.9 mm at V=100 mm/min), so D^2 acts as a two-point scale factor for the absolute mass. This is a minor fitted input, but it does not affect the optimal-radius prediction, which is the central design claim. No load-bearing self-citations or imported uniqueness theorems appear in the argument; the cited prior permeability asymptotics are standard and independently reproduced. The central result is therefore not equivalent to its inputs by construction.

Assumptions & free parameters 5 free parameters · 6 assumptions · 0 invented entities

The ledger shows that the central claim rests on an experimentally independent permeability closure (COMSOL fits plus an asymptotic benchmark) and on several modeling simplifications, the most fragile being the Heaviside transverse permeability. No new physical entities are introduced. The only parameter tuned to the capture experiments is the effective brush diameter D used in one mass-comparison figure; this does not move the predicted optimal radius. Auxiliary fits to the model's own numerics (V_T, alpha) affect regime labels and master-curve presentation, not the derivation of the PDE.

free parameters (5)

Longitudinal permeability fit F_parallel(phi) = k_parallel/d^2 = (1/4)[1 - sqrt(3) R/d]^4.6 [1 - (sqrt(3)/(8 pi))(ln(1-phi) + K - 35(1-phi))], K = 1.498
Coefficients (4.6, 35, K) fit COMSOL permeability simulations; they are inputs from a numerical closure, not fitted to brush-capture experiments. Central predictions depend on this closure.
Transverse permeability fit F_perp(phi) = k_perp/d^2 = (1/12.4)[1 - 2R/d]^2.5 [1 - 0.21(ln(1-phi) + K)]
Fit to COMSOL simulations of transverse flow; independent of the capture experiments but part of the model's input.
Effective brush diameter D used in mass curves = D = 15.8 mm at V = 10 mm/min; D = 16.9 mm at V = 100 mm/min
D is not uniquely defined for printed brushes; values are chosen within the plausible rim-definition range to match measured absolute mass in Fig. 4C. The predicted optimal radius is unaffected by this multiplicative scaling, but the phrase 'without any fitting parameter' is overstated.
Simplified k_perp/k_parallel fit coefficients = a = 3.4e-4, b = 8.86 in k_perp/k_parallel = (1/2)(1 - a exp[b(1-phi)])^2
Fit to the ratio of the two permeability fits, used in Eq. (9b) and in plotting; not fitted to experiments.
Numerical-solution response fits V_T and alpha = V_T = 1/(1 + 2.63 hJ^(6/5)); alpha^2(x) = tanh[1/(2x)^2]
These coefficients are fits to the model's own numerically computed H0, used for regime boundaries and the master-curve collapse; they do not use experimental capture data to set constants.

assumptions (6)

domain assumption Darcy's law with anisotropic permeabilities k_parallel and k_perp represents the brush as a continuum porous medium.
Invoked in Model, Eq. (2). The brush has O(100) pillars with spacing comparable to pillar radius, so continuum treatment smooths pore-level flow details.
ad hoc to paper Transverse permeability vanishes above the bath: k_perp(z) = k_perp Theta(-z).
Introduced after Eq. (2) and SI Eq. (14) to match the observed three-parameter dynamics. It is physically motivated by the air-liquid interface but not independently validated, and it introduces a discontinuity in the radial velocity.
domain assumption Pressure boundary conditions: hydrostatic pressure at the immersed bottom, capillary pressure at the interface, and hydrostatic pressure in the bath at the rim.
Used to solve Eq. (8); assumes no dynamic pressure coupling at the meniscus and no film or curvature corrections.
standard math Small aspect ratio delta = 2L0/D, with a perturbation expansion in delta^2 truncated at order delta^2.
Expansion Eq. (7). The authors acknowledge the formal condition delta << 1 but report agreement up to delta about 1.27.
standard math The nonlinear (d_r H)^2 term in the PDE can be neglected.
SI after Eq. (26) bounds the induced error at 3.5% for the stated parameter ranges; this is a numerical check, not a proof for all regimes.
domain assumption Initial state is an equilibrated Jurin height with no residual flow.
Eq. (10) and Methods; experiments wait until the capillary rise reaches steady state before retraction.

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Cite this review

Pith. "Pith review of Optimal rigid brush for fluid capture." pith.science (2026). https://pith.science/paper/MFZG6WRI

@misc{pith2026241115970,
  author       = {Pith},
  title        = {Pith review of: Optimal rigid brush for fluid capture},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/MFZG6WRI}},
  note         = {Machine review of arXiv:2411.15970}
}

read the original abstract

Parallel assemblies of slender structures forming brushes are common in our daily life from sweepers to pastry brushes and paintbrushes. This type of porous objects can easily trap liquid in their interstices when removed from a liquid bath. This property is exploited to transport liquids in many applications ranging from painting, dip-coating, brush-coating to the capture of nectar by bees, bats and honeyeaters. Rationalizing the viscous entrainment flow beyond simple scaling laws is complex due to its multiscale structure and the multidirectional flow. Here, we provide an analytical model, together with precision experiments with ideal rigid brushes, to fully characterize the flow through this anisotropic porous medium as it is withdrawn from a liquid bath. We show that the amount of liquid entrained by a brush varies non-monotonically during the withdrawal at low speed, is highly sensitive to the different parameters at play and is very well described by the model without any fitting parameter. Finally, an optimal brush geometry maximizing the amount of liquid captured at a given retraction speed is derived from the model and experimentally validated. These optimal designs open routes towards efficient liquid manipulating devices.

Figures

Figures reproduced from arXiv: 2411.15970 by the authors.

**Figure 1.** D shows some typical temporal evolution of the interface height measured at the center of the brush, h0(t) = h(0, t), when the retraction speed V is varied while keeping the immersion depth L0 constant (Movies S1 and S2). The interface first moves up at the same speed as the pillars (dashed curve in Fig. 1D), before slowing down to reach a maximum value h m 0 where it stops and starts moving down. The higher the ret… view at source ↗

**Figure 2.** FIG. 2. (A) Evolution of the longitudinal and transverse permeabilities rescaled by [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 3.** FIG. 3. (A) Comparison between some temporal variations [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗

Figures from the paper (7 more)

**Figure 4.** Figure 4: FIG. 4. (A) Evolution of the rescaled volume captured by a brush, [PITH_FULL_IMAGE:figures/full_fig_p006_4.png]

**Figure 5.** Figure 5: FIG. 5 [PITH_FULL_IMAGE:figures/full_fig_p011_5.png]

**Figure 6.** Figure 6: F shows the evolution of k⊥/k∥ as a function of ϕ together with the ratio of the fits (11) and (9). The good agreement between the data and this ratio highlights the good quality of the fits. Nevertheless, the ratio of these fits yields a rather cumbersome expression. …

**Figure 7.** Figure 7: FIG. 7. (A) Evolution of [PITH_FULL_IMAGE:figures/full_fig_p015_7.png]

**Figure 8.** Figure 8: FIG. 8. (A) Evolution of the transition speed [PITH_FULL_IMAGE:figures/full_fig_p016_8.png]

**Figure 9.** Figure 9: A shows the evolution of ¯δ0 as a function of V and h¯ J . The three cross indicates three illustrative cases shown in Fig. 9B. For example, when V = 0.1 and h¯ J = 0.1, ¯δ0 ≃ 0.45. Therefore, if ¯δ ≤ 0.45, the evolution of H0 is well captured by H1D 0 obtained from th…

**Figure 10.** Figure 10: FIG. 10. (A) Comparison between some temporal variations of [PITH_FULL_IMAGE:figures/full_fig_p022_10.png]

Discussion (0). Continue with ORCID to comment.

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