Expansion of a hole in a viscoelastic liquid sheet

Howard A. Stone; Rodolfo Brand\~ao; Tachin Ruangkriengsin

REVIEW 3 major objections 3 minor 300 references

Expansion of a hole in a viscoelastic liquid sheet

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

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Pith's one-line read Viscoelastic stress accelerates hole expansion in viscous polymer films.

desk verdict Solid asymptotic extension of Newtonian hole-retraction theory to Oldroyd-B, but a concrete typo in the first-order mass equation must be fixed before the derivation is reproducible. read the letter →

arxiv 2608.04345 v1 pith:NEO3NBNN submitted 2026-08-05 physics.flu-dyn

Tachin Ruangkriengsin , Rodolfo Brandão , Howard A. Stone This is my paper · ORCID

classification physics.flu-dyn

keywords viscoelasticliquidsheetholeexpansionOldroyd-Bfluidthin-filmasymptoticsTaylor-Culickretractionexponentialgrowthpolymericnormalstressescapillary

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 asks whether viscoelastic stresses change the way a hole expands in a thin liquid sheet, and it gives a specific answer: in a highly viscous polymer solution modeled by the Oldroyd-B equations, polymer stretching makes the hole grow exponentially faster than in a Newtonian film. The paper derives thin-film equations on the scale of the hole and matches them to a tip-region force balance, then solves the reduced model in two parameter regimes. In the weakly viscoelastic limit the dimensionless radius obeys $R_e(T)\approx e^{(1/2+\alpha Wi\beta_p)T}$ with $\alpha=(12-6\log 2-\pi)/21\approx 0.224$, and in the ultra-dilute limit it obeys $R_e(T)\approx e^{(1/2+\alpha^*\beta_p)T}$ with $\alpha^*(Wi)>0$. The same stress-redistribution mechanism produces a film that thickens near the retracting edge. These results give a quantitative target for experiments on polymeric films, which had left the role of viscoelasticity unresolved.

What carries the argument

The central machinery is an asymptotic decomposition of the film into a hole-scale region of length $r_0$ and a tip-scale region of length $h_0\ll r_0$, joined by a matching condition derived from a global force balance over the tip. On the hole scale the leading-order flow is extensional and plug-like, giving thin-film equations for thickness $H(R,T)$, radial velocity $U_R(R,T)$, and polymeric stresses $\Sigma_{RR},\Sigma_{ZZ},\Sigma_{\theta\theta}$; the tip enters only through the effective boundary condition $H(2\beta_s R^{-1}\partial_R(RU_R)+2\beta_s\partial_R U_R+\Sigma_{RR}-\Sigma_{ZZ})=-1$ at the edge. A change of variables $\xi=R/R_e(T)$ immobilizes the moving boundary, and the hole radius is recovered from $\mathrm{d}\log R_e/\mathrm{d}T=\tilde U(1,T)$, so a steady state in the moving frame implies exponential growth. The first-order correction to the edge velocity, $U_{\mathrm{st}}(1)=\alpha=(12-6\log 2-\pi)/21$, is obtained analytically in the small-$Wi$ limit and as $\alpha^*(Wi)$ by quadrature in the ultra-dilute limit.

What would settle it

Direct numerical simulation of the full axisymmetric Stokes–Oldroyd-B equations with initially relaxed polymers at $Wi=0.5$, $\beta_p=0.1$ should give a long-time edge-velocity correction near $\alpha^*(0.5)\approx 0.077$; a correction that is absent or negative would falsify the thin-film reduction.

Watch

Extended reading notes

Core claim

Working in the large-Ohnesorge-number regime, the paper claims that the exponential hole growth known for viscous Newtonian sheets persists for Oldroyd-B films, and that viscoelasticity systematically accelerates it. The acceleration is captured by a first-order correction to the edge velocity: at small Weissenberg number the long-time growth exponent becomes $1/2+\alpha Wi\beta_p$, while in the ultra-dilute limit it becomes $1/2+\alpha^*(Wi)\beta_p$, with the two formulas agreeing in their overlap regime since $\alpha^*(Wi)=\alpha Wi+O(Wi^2)$. The mechanism is that the retracting flow stretches polymers azimuthally and compresses them radially, generating normal-stress differences that modify the stress balance at the tip and drive a stronger outward extensional flow; the accompanying rim thickening only partially offsets this effect. The ultra-dilute analysis also identifies $Wi=1$ (equivalently the extensional-flow threshold $Wi_{\mathrm{ext}}=1/2$) as the point where the Oldroyd-B azimuthal stress growth becomes unbounded, so the predictions are stated for $Wi<1$.

Load-bearing premise

The analysis assumes the polymer chains are completely relaxed at the instant the hole is punctured; the paper itself notes that a film carrying prestress from earlier deformation changes the leading-order solution qualitatively, so the predicted growth rates would not apply to such a film.

Editorial extensions

If this is right

Even a small polymer viscosity fraction $\beta_p$ changes the exponent of exponential hole growth by $\alpha Wi\beta_p$ at small $Wi$, so the Newtonian rate $1/2$ is a lower bound in this regime.
In the ultra-dilute regime the acceleration persists for every $Wi<1$, with $\alpha^*(Wi)$ rising linearly at small $Wi$ and saturating near $\alpha^*\approx 0.117$ as $Wi\to 1^-$.
The film thickness develops a localized thickening near the retracting edge while staying nearly uniform away from it, consistent with experiments that see no sustained rim.
The two asymptotic predictions coincide in the overlap $Wi\ll 1$, $\beta_p\ll 1$, giving a single formula that interpolates through the common limit.
At $Wi=1$ the Oldroyd-B description of the azimuthal stress at the edge breaks down, marking the range of validity of the predicted growth law.

Reading between the lines

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

(Extension) The computed corrections could account for the reported shortfall of the Newtonian viscous theory against polymer-film experiments, making the acceleration experimentally accessible.
(Extension) The same tip-matching and moving-frame strategy should transfer to planar sheets and viscoelastic threads, where inertia may need to be retained; testing the predicted sign and magnitude there would separate the mechanism from geometry-specific details.
(Extension) Because the leading-order solution changes qualitatively when the initial polymeric stress is nonzero, experiments must control or characterize the deformation history of the film before comparing with the predicted exponents.
(Extension) Full numerical solutions of the reduced thin-film model at order-one values of both $Wi$ and $\beta_p$ would fill the gap between the two asymptotic regimes; the formulas here provide natural benchmarks for such a calculation.

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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

You should know two things about this paper. First, the science is good: it is the first systematic thin-film asymptotic framework for a freely suspended Oldroyd-B sheet, and it produces explicit formulas for the enhancement of the exponential hole-growth rate in two limits (Wi→0 and β_p→0), with the Newtonian result recovered exactly. Second, the printed derivation of the headline constant α contains a concrete error. Equation (6.10)/(6.16) writes the advection term as -1/(2ξ)∂H/∂ξ; it should be -ξ/2 ∂H/∂ξ. With the printed term, the divergence term cancels it and the equation reduces to stationary-frame mass conservation, which cannot produce the steady-state equation (6.27) used to compute α. The corrected coefficient gives (6.27) exactly. This looks like a transcription slip—the subsequent algebra and the numerics use the corrected form—but as printed the manuscript is not reproducible. A referee should ask for this to be fixed.

Now the credit. The paper properly extends the Munro–Lister and Savva–Bush results to viscoelasticity, and the tip-matched boundary condition survives the generalization. The two asymptotic routes agree in the overlap regime, and the numerical solutions converge to the analytically derived steady states. The mechanism—azimuthal stretching and radial compression of polymers modifying the tip stress balance—is clearly explained. I checked parts of the algebra; beyond the mass-equation typo, it is consistent.

Soft spots, in order. The initially relaxed polymer state (2.5) is a genuine limitation for the melt experiments that motivate the work; the paper says so in §7, but it means the formulas will not directly apply to those experiments unless a prestress is imposed. The long-time exponent (6.38) uses integrability of U(1,T)-α, supported only by numerics; that is a minor gap. The Oldroyd-B model diverges at Wi≥1, and the paper is honest about it. None of these undercuts the asymptotic result as derived.

Recommendation: send it to a serious referee. The typo is easy to fix, and the central contribution is original and carefully executed. I would cite it.

Referee Report

3 major / 3 minor

Summary. The paper studies the axisymmetric expansion of a hole in a highly viscous liquid sheet described by the Oldroyd-B model, in the large-Ohnesorge-number limit. The authors derive extensional thin-film equations on the hole scale and an effective boundary condition at the retracting tip from an asymptotic force balance in the tip region, following the approach of Ahsan et al. (2026). The reduced model is then analyzed in two complementary limits. In the weakly viscoelastic limit Wi<<1, a regular perturbation expansion yields the long-time growth rate (1/2 + alpha Wi beta_p)T with alpha=(12-6 log 2 - pi)/21, and numerical solutions of the reduced equations are shown to converge to the analytically derived steady state. In the ultra-dilute limit beta_p<<1 with Wi fixed, the first-order correction to the edge velocity is expressed as a quadrature involving hypergeometric functions, giving growth rate (1/2 + alpha*(Wi) beta_p)T; the two regimes are shown to overlap as Wi->0. The central claim is that viscoelastic stresses accelerate hole expansion relative to the Newtonian result Re(T)=e^{T/2} and induce thickness variations with thickening near the edge.

Significance. If the results hold, the paper provides the first systematic asymptotic framework for viscoelastic hole expansion in freely suspended sheets and gives a concrete, parameter-free prediction for the leading viscoelastic correction to the exponential growth rate. The strengths of the work include the exact recovery of the Newtonian limit, the absence of any fitting parameters, the analytical derivation of alpha, the consistency of the two asymptotic expansions in their overlap region, and the explicit numerical verification of convergence to the steady states. The main limitations are the initial condition of relaxed polymers (2.5), the Oldroyd-B constitutive restriction to dilute solutions, and the reliance on numerical observation for one integrability assumption; the authors acknowledge several of these in Sections 7 and 8. These limitations do not undermine the internal consistency of the asymptotic construction, but they should be stated precisely in the final version.

major comments (3)

[Eqs. (6.10), (6.16), (6.27)] The mass-conservation equation is misprinted. From the moving-frame transformation (5.12) and the mass equation (5.13), with U[0]=1/(2xi) and U[0](1)=1/2, the first-order advection term is -(xi/2) partial H[1]/partial xi, not -(1/(2xi)) partial H[1]/partial xi. With the printed coefficient, the divergence term (1/(2xi)) partial H[1]/partial xi cancels the advection term, so the printed steady-state version of (6.16) would imply d(xi U)/d xi = 0. That contradicts the steady-state equation (6.27), which is the equation actually used to derive the exponent alpha in (6.34). Since this equation is load-bearing for the headline result (6.38), the manuscript is not reproducible as written. Please correct (6.10) and (6.16) and confirm that the numerical scheme and the steady-state algebra use the corrected form.
[After Eq. (6.36)] The replacement of the integral of U(1,T) by alpha T + o(T) requires that U(1,T)-alpha be integrable. The text states that this is supported only by numerical observation. This assumption is load-bearing for the exponential-rate prediction (6.38): if the approach to the steady value were only algebraically slow, a logarithmic correction to the growth rate could appear. Please either supply a proof or a quantitative numerical demonstration (e.g., a measured decay rate with an error bound, or a fitted asymptotic decay), or state explicitly that (6.38) is conditional on this integrability assumption.
[Eqs. (7.2) and Section 7] The ultra-dilute expansion relies critically on the initially relaxed polymeric stress condition (5.17). The authors correctly note in Section 7 that a nonzero initial polymeric stress changes the leading-order solution qualitatively. Because this condition is also the assumption under which the small-Wi analysis is applied to experiments (Section 8), the paper should state more prominently that the prediction applies only to films that are freshly punctured from a relaxed state, and should indicate what prestress levels would invalidate the exponential-rate formula.

minor comments (3)

[Figures 2, 3, 5, 6] Several axis labels are incomplete: for example, 'Film thickness ( , T)' and 'Velocity ( , T)' are missing the variables H and U. Please complete the labels.
[Section 6.2.1] The statement that no initial condition is imposed on the velocity, followed by the initial distribution (6.26), should clarify that T=0 in the small-Wi outer problem is the beginning of the O(1) evolution after the initial stress boundary layer, not the physical instant of puncture.
[References] The citation 'Munro (2018)' in Section 3 and 'Munro & Lister (2018)' in Section 6.1 should be checked for consistency, since both are used to attribute the same asymptotic decomposition.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the growth-rate corrections are derived from the stated equations, not fitted; the printed advection typo in §6 is a reproducibility issue, not circularity.

full rationale

The central quantitative claims are self-contained. The parameters Wi, beta_p, beta_s, lambda, mu_s and mu_p are prescribed physical inputs; the constants alpha=(12-6 log 2-pi)/21 and alpha*(Wi) are obtained by solving the reduced boundary-value problems (6.16)-(6.18) and (7.3)-(7.5)/(7.15), not by fitting to the predicted growth rate. The Newtonian limit is recovered independently (R_e=e^{T/2}, Eq. (6.9)) and agrees with published results, which confirms the framework rather than renaming it. Self-citations appear—Ahsan et al. (2026) for the domain-decomposition method and Ruangkriengsin et al. (2025) for the O(Wi) initial stress boundary layer—but the load-bearing tip boundary condition is re-derived in §4 and the exponent alpha follows from the paper's own mass and momentum equations; the boundary-layer citation is auxiliary and does not supply the numerical value of alpha. In the ultra-dilute limit, alpha*(Wi) is evaluated by quadrature of an analytically derived integral, not by fitting. The only notable defect is a printed algebraic inconsistency: the advection term -1/(2xi) dH/dxi in (6.10)/(6.16) cancels the mass-flux term, whereas the steady-state equation (6.27) used to compute alpha corresponds to the moving-frame term -(xi/2) dH/dxi. This is a reproducibility/correctness error, not a circular reduction. The unproved integrability of U(1,T)-alpha is a stated numerical observation and a rigor limitation, not circularity.

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

The central claim rests on standard rheological and thin-film assumptions, plus two specific technical premises: the initially relaxed polymer state and the unproven integrability of the edge-velocity transient. No new particles, forces, or entities are introduced, and no free parameters are fitted to data: beta_p and Wi are inputs, while alpha and alpha*(Wi) are derived constants.

assumptions (5)

domain assumption Oldroyd-B constitutive model with a single relaxation time lambda, equations (2.3)-(2.4)
The model is appropriate for dilute polymer solutions, not the polymer melts in the motivating experiments; the authors acknowledge this limitation in the conclusions.
domain assumption Initially relaxed polymers, Sigma_p = 0 at t = 0, equation (2.5)
The derived acceleration relies on this condition; section 7 states that nonzero initial polymeric stress changes the leading-order solution qualitatively.
domain assumption Separation of scales epsilon = h0/r0 much less than 1 and inertia neglected with Oh much greater than 1/epsilon, equations (2.1) and (2.17)
These are the standard thin-film and high-Ohnesorge assumptions that justify the two-region decomposition and the Stokes-flow tip balance.
ad hoc to paper U(1,T) - alpha is integrable at long times, asserted after equation (6.36)
The final formula (6.38) takes the integral of U(1,T) - alpha to be O(1) without a proof; numerical results suggest rapid approach to the steady state, but this remains an unproven premise.
domain assumption The tip-scale problem reduces to a two-dimensional Cartesian Stokes problem, section 4.1
The azimuthal curvature and azimuthal stress terms are O(epsilon) smaller at the tip, which is valid for hole radii much larger than the film thickness, but it excludes strongly three-dimensional tip shapes.

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Cite this review

Pith. "Pith review of Expansion of a hole in a viscoelastic liquid sheet." pith.science (2026). https://pith.science/paper/NEO3NBNN

@misc{pith2026260804345,
  author       = {Pith},
  title        = {Pith review of: Expansion of a hole in a viscoelastic liquid sheet},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/NEO3NBNN}},
  note         = {Machine review of arXiv:2608.04345}
}

abstract

Experiments on highly viscous polymeric films show that punctured holes expand exponentially in time, without sustained accumulation of liquid near the rim. This response departs from the Taylor--Culick description, in which displaced liquid accumulates in a growing rim that moves at constant speed. Although these differences were initially attributed to viscoelasticity, they were later rationalized using a purely viscous theory, leaving the role of viscoelastic stresses unresolved. We analyze the expansion of an axisymmetric hole in a freely suspended viscoelastic liquid sheet described by the Oldroyd-B model. Exploiting the separation of length scales between hole radius and film thickness, we derive extensional thin-film equations on the scale of the hole and an effective boundary condition from an asymptotic force balance in the tip region. Analytical solutions are obtained for weak viscoelasticity, $Wi\ll 1$, and the ultra-dilute limit, $\mu_p\ll\mu_s$, where $Wi$ is the Weissenberg number, while $\mu_s$ and $\mu_p$ are solvent and polymeric viscosities, respectively. For weak viscoelasticity, the dimensionless hole radius grows approximately as $e^{(0.5+\alpha Wi \beta_p)T}$, where $\alpha=(12 - 6\log 2-\pi)/21\approx 0.224$ and $\beta_p=\mu_p/(\mu_s+\mu_p)$. In the ultra-dilute limit, the radius grows approximately as $e^{(0.5+\alpha^{*}\beta_p )T}$, where $\alpha^{*}(Wi)>0$ is evaluated numerically. In both regimes, viscoelastic stresses increase the exponential growth rate relative to the Newtonian limit and induce film-thickness variations, with thickening near the retracting edge. This acceleration arises from azimuthal stretching and radial compression of the polymers, which redistribute stresses in the film and modify the stress balance at the tip, leading to a stronger outward radial extensional flow.

Figures

Figures reproduced from arXiv: 2608.04345 by the authors.

**Figure 1.** Schematic illustration of an expanding hole in a viscoelastic liquid sheet. In this regime, the initial film thickness can be expressed as ℎ(𝑟, 0) = ℎ0 for 𝑟 𝑟0 − 1 ≫ 𝜖. (2.2) Together, (2.1) and (2.2) motivate a decomposition of the problem into two regions: a holescale region, in which the radial length scale is set by the initial hole radius and the vertical length scale by the film thickness, and a tip-scale re… view at source ↗

**Figure 2.** Time evolution of the normalized first-order correction to the film thickness, H (𝜉, 𝑇) = 𝐻˜ [1] (𝜉, 𝑇)/𝛽𝑝. (a) Film thickness H (𝜉, 𝑇) at 𝑇 = 0, 1, 2 and 5, compared with the analytically derived steady-state solution. (b) Film thickness at the edge, H (1, 𝑇), as a function of time. 1.0 1.5 2.0 2.5 3.0 3.5 0.050 0.075 0.100 0.125 0.150 0.175 0.200 0.225 0.250 V e l o c i t y ( , T) (a) T=0 T=1 T=2 T=5 Steady state … view at source ↗

**Figure 3.** Time evolution of the normalized first-order correction to the velocity, U (𝜉, 𝑇) = 𝑈˜ [1] (𝜉, 𝑇)/𝛽𝑝. (a) Velocity U (𝜉, 𝑇) at 𝑇 = 0, 1, 2 and 5, compared with the analytically derived steady-state solution. (b) Velocity at the edge, U (1, 𝑇), as a function of time. 𝑊𝑖. Expanding the velocity 𝑈˜ (1, 𝑇) in (5.18), we find 𝑅𝑒 (𝑇) = exp ∫ 𝑇 0 𝑈˜ (1, 𝑇˜) 𝑑𝑇˜ ≈ exp 𝑇 2 + 𝛽𝑝𝑊𝑖 ∫ 𝑇 0 U (1, 𝑇˜) 𝑑𝑇˜ , (6.36) where the… view at source ↗

Figures from the paper (4 more)

**Figure 4.** Figure 4: First-order correction to the steady-state edge velocity, 𝑈˜ {1} st (1) = 𝛼 ∗ , computed numerically from (7.15) for 0 ⩽ 𝑊𝑖 < 1. The dashed line shows the small-𝑊𝑖 asymptotic approximation 𝛼 ∗ (𝑊𝑖) ∼ 𝛼𝑊𝑖, where 𝛼 = (12 − 6 log 2 − 𝜋)/21, as obtained in (6.34). the long…

**Figure 5.** Figure 5: Time evolution of the first-order corrections to the film thickness, 𝐻˜ {1} (𝜉, 𝑇), and velocity, 𝑈˜ {1} (𝜉, 𝑇). (a) Film thickness 𝐻˜ {1} (𝜉, 𝑇) at 𝑇 = 0, 1, 2, and 10, compared with the analytically derived steady-state solution. (b) Velocity 𝑈˜ {1} (𝜉, 𝑇) at 𝑇 = 0, …

**Figure 6.** Figure 6: Time evolution of the first-order corrections to the film thickness at the edge, 𝐻˜ {1} (1, 𝑇), and velocity at the edge, 𝑈˜ {1} (1, 𝑇), for different values of 𝑊𝑖. (a) Film thickness at the edge 𝐻˜ {1} (1, 𝑇) for 𝑊𝑖 = 0.1, 0.3, 0.5, and 0.7. (b) Velocity at the edge 𝑈…

**Figure 7.** Figure 7: Summary of the two asymptotic regimes and their predictions for hole expansion. (a) Schematic parameter plane (𝑊𝑖, 𝛽𝑝) showing the validity regions of the small-𝑊𝑖 expansion, 𝑊𝑖 ≪ 1, and the ultra-dilute expansion, 𝛽𝑝 ≪ 1, together with their overlap. Markers indicate …

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Reference graph

Works this paper leans on

300 extracted references · 80 canonical work pages

[1]

Theorie mecanique de la chaleur , author=. Ann. Chim. Phys , volume=
[2]

Journal of fluid mechanics , volume=

Lubrication of soft viscoelastic solids , author=. Journal of fluid mechanics , volume=. 2016 , publisher=

2016
[3]

Nature , volume=

Some applications of photography , author=. Nature , volume=
[4]

Some experiments on the dynamics of liquid films , author=. J. Appl. Phys. , volume=. 1959 , publisher=

1959
[5]

Atomization of undulating liquid sheets , author=. J. Fluid Mech. , volume=. 2007 , publisher=

2007
[6]

Interfacial Flows---The Power and Beauty of Asymptotic Methods , pages=

Viscoelastic lubrication using the second-order fluid , author=. Interfacial Flows---The Power and Beauty of Asymptotic Methods , pages=. 2025 , publisher=

2025
[7]

On the coefficient of viscous traction and its relation to that of viscosity , author=. Proc. R. Soc. London Ser. A , volume=. 1906 , publisher=

1906
[8]

Fragmentation versus cohesion , author=. J. Fluid Mech. , volume=. 2020 , publisher=

2020

Show all 300 references

[9]

The effect of applied pressure on the shape of a two-dimensional liquid curtain falling under the influence of gravity , author=. J. Fluid Mech. , volume=. 1993 , publisher=

1993
[10]

Hole formation and growth in freely standing polystyrene films , author=. Phys. Rev. E , volume=. 1999 , publisher=

1999
[11]

Evidence of convective constraint release during hole growth in freely standing polystyrene films at low temperatures , author=. Phys. Rev. E , volume=. 2005 , publisher=

2005
[12]

G. I. Taylor , journal=. The dynamics of thin sheets of fluid. 1959 , publisher=

1959
[13]

Comments on a ruptured soap film , author=. J. Appl. Phys. , volume=. 1960 , publisher=

1960
[14]

W. R. McEntee and K. J. Mysels , journal=. Bursting of soap films. 1969 , publisher=

1969
[15]

Frankel and K

S. Frankel and K. J. Mysels , journal=. Bursting of soap films. 1969 , publisher=

1969
[16]

Breaking of liquid films and threads , author=. Phys. Fluids , volume=. 1983 , publisher=

1983
[17]

Nonlinear rupture of free films , author=. Phys. Fluids A , volume=. 1993 , publisher=

1993
[18]

Models for thin viscous sheets , author=. Eur. J. Appl. Math. , volume=. 1996 , publisher=

1996
[19]

Stefan , journal=

J. Stefan , journal=
[20]

Surface tension driven flows , author=. SIAM J. Appl. Math. , volume=. 1983 , publisher=

1983
[21]

Viscous bursting of suspended films , author=. Phys. Rev. Lett. , volume=. 1995 , publisher=

1995
[22]

Science , volume=

The life and death of ``bare'' viscous bubbles , author=. Science , volume=. 1998 , publisher=

1998
[23]

The Initial Surface Tension--Driven Flow of a Wedge of Viscous Fluid , author=. SIAM J. Appl. Math. , volume=. 2005 , publisher=

2005
[24]

, journal=

Eggers, J. , journal=. Post-breakup solutions of. 2014 , publisher=

2014
[25]

Capillary retraction of the edge of a stretched viscous sheet , author=. J. Fluid Mech. , volume=. 2018 , publisher=

2018
[26]

Bursting dynamics of thin free liquid films from Newtonian and viscoelastic solutions , author=. Phys. Rev Lett. , volume=. 1997 , publisher=

1997
[27]

The formation of thick borders on an initially stationary fluid sheet , author=. Phys. Fluids , volume=. 1999 , publisher=

1999
[28]

On the bursting of viscous films , author=. Phys. Fluids , volume=. 1999 , publisher=

1999
[29]

The retraction of the edge of a planar liquid sheet , author=. Phys. Fluids , volume=. 2002 , publisher=

2002
[30]

Viscous sheet retraction , author=. J. Fluid Mech. , volume=. 2009 , publisher=

2009
[31]

Asymptotic behavior of a retracting two-dimensional fluid sheet , author=. Phys. Fluids , volume=. 2011 , publisher=

2011
[32]

Deka and J.-L

H. Deka and J.-L. Pierson , journal=. Revisiting the. 2020 , publisher=

2020
[33]

Sanjay and U

V. Sanjay and U. Sen and P. Kant and D. Lohse , journal=. Taylor--. 2022 , publisher=

2022
[34]

Journal of Fluid Mechanics , volume=

Thin-sheet flow between coalescing bubbles , author=. Journal of Fluid Mechanics , volume=. 2015 , publisher=

2015
[35]

Scaling laws and dynamics of bubble coalescence , author=. Phys. Rev. Fluids , volume=. 2017 , publisher=

2017
[36]

Electrohydrodynamic rotation of drops at large electric Reynolds numbers , author=. J. Fluid Mech. , volume=. 2016 , publisher=

2016
[37]

A 2D electrohydrodynamic model for electrorotation of fluid drops , author=. J. Colloid Interface Sci. , volume=. 2002 , publisher=

2002
[38]

2015 , publisher=

Applied Partial Differential Equations , author=. 2015 , publisher=

2015
[39]

2002 , publisher=

Spectral methods for incompressible viscous flow , author=. 2002 , publisher=

2002
[40]

Nanofluidic devices and their applications , author=. Anal. Chem. , volume=. 2008 , publisher=

2008
[41]

Abramowitz and I

M. Abramowitz and I. A. Stegun , Edition =. Handbook of Mathematical Functions , Year =
[42]

and Yariv, E

Abu Hamed, M. and Yariv, E. , Journal =. Boundary-induced electrophoresis of uncharged conducting particles: near-contact approximation , Volume =
[43]

and Yariv, E

Abu Hamed, M. and Yariv, E. , Journal =. Induced-charge electrokinetic flows about polarizable nano-particles: the thick-Debye-layer limit , Volume =
[44]

Laminar boundary layer flows with surface reactions , author=. Ind. Eng. Chem. , volume=. 1957 , publisher=

1957
[45]

and Taylor, T

Acrivos, A. and Taylor, T. D. , Journal =. Heat and mass transfer from single spheres in Stokes flow , Volume =
[46]

and Goddard, J

Acrivos, A. and Goddard, J. D. , Journal =
[47]

, Journal =

Acrivos, A. , Journal =. Heat transfer at high
[48]

and Lo, T

Acrivos, A. and Lo, T. S. , Journal =. Deformation and breakup of a single slender drop in an extensional flow , Volume =
[49]

Ajayi, O. O. , Journal =. A note on Taylor's electrohydrodynamic theory , Volume =
[50]

, Journal =

Ajdari, A. , Journal =. Pumping liquids using asymmetric electrode arrays , Volume =
[51]

and Bocquet, L

Ajdari, A. and Bocquet, L. , Journal =
[52]

M. A. Aggour and H. A. Tchelepi and H. Y. Al-Youssef , title =. Journal of Petroleum Science and Engineering , year =
[53]

Rectification and diffusion of self-propelled particles in a two-dimensional corrugated channel , author=. Phys. Rev. E , volume=. 2013 , publisher=

2013
[54]

Alexander, B. M. and Prieve, D. C. , Journal =. A hydrodynamic technique for measurement of colloidal forces , Volume =
[55]

Allan, R. S. and Mason, S. G. , Journal =
[56]

and Chen, C

Allison, S. and Chen, C. and Stigter, D. , Journal =. The length dependence of translational diffusion, free solution electrophoretic mobility, and electrophoretic tether force of rigid rod-like model duplex
[57]

The cataphoresis of gas bubbles in water , author=. Proc. Roy. Soc. London A , volume=. 1924 , publisher=

1924
[58]

S. A. Anbah and G. V. Chilingar and C. M. Beeson , title =. Annals of the New York academy of Sciences , year =
[59]

and Long, D

Andre, P. and Long, D. and Ajdari, A. , Journal =
[60]

Andersen, M. B. and Frey, J. and Pennathur, S. and Bruus, H. , Journal =. Surface-dependent chemical equilibrium constants and capacitances for bare and 3-cyanopropyldimethylchlorosilane coated silica nanochannels , Volume =
[61]

Anderson, J. L. and Lowell, M. E. and Prieve, D. C. , Journal =
[62]

Anderson, J. L. and Idol, W. K. , Journal =. Electroosmosis through pores with nonuniformly charged walls , Volume =
[63]

Anderson, J. L. , Journal =. Effect of nonuniform zeta potential on particle movement in electric fields , Volume =
[64]

Anderson, J. L. , Journal =. Colloid transport by interfacial forces , Volume =
[65]

Self-starting micromotors in a bacterial bath , author=. Phys. Rev. Lett. , volume=. 2009 , publisher=

2009
[66]

Europhys

Active ratchets , author=. Europhys. Lett. , volume=. 2011 , publisher=

2011
[67]

Archer and T

S. Archer and T. T. Li and A. T. Evans and S. T. Britland and H. Morgan , title =. Biochem Biophys Res Commun , year =
[68]

and Rangel, R.H

Ardekani, A.M. and Rangel, R.H. and Joseph, D.D. , Journal =. Two spheres in a free stream of a second-order fluid , Volume =
[69]

Emulsions stabilised solely by colloidal particles , author=. Adv. Colloid Interface Sci. , volume=. 2003 , publisher=

2003
[70]

Bach, G. A. and Hollingsworth, A. D. and Koch, D. L. , Journal =. Electrophoretic mobility of rigid rodlike particles in dilute aqueous and glycerol suspensions:
[71]

Baker, D. R. and Verbrugge, M. W. , Journal =. The role of charge separation in the response of electrochemical systems , Volume =
[72]

and Babich, Y.A

Baran, A.A. and Babich, Y.A. and Tarovsky, A.A. and Mischuk, N.A. , Journal =. Superfast electrophoresis of ion-exchanger particles , Volume =
[73]

and Mishchuk, N

Barany, S. and Mishchuk, N. A. and Prieve, D. C. , Journal =. Superfast electrophoresis of conducting dispersed particles , Volume =
[74]

Electrophoresis in strong electric fields , author=. Adv. Colloid Interface Sci. , volume=. 2009 , publisher=

2009
[75]

and Chen, D

Barcilon, V. and Chen, D. P. and Eisenberg, R. S. , Issn =. SIAM J. Appl. Math. , Number =
[76]

Baret and O

J.-C. Baret and O. J. Miller and V. Taly and M. Ryckelynck and A. El-Harrak and L. Frenz and C. Rick and M. L. Samuels and J. B. Hutchison and J. J. Agresti and D. R. Link and D. A. Weitz and A. D. Griffiths , title =. Lab Chip , year =
[77]

Baret and F

J.-C. Baret and F. Kleinschmidt and A. El Harrak and A. D. Griffiths , title =. Langmuir , year =
[78]

Baret , title =

J.-C. Baret , title =. Lab Chip , year =
[79]

Biomicrofluidics , volume=

Dynamic double layer effects on ac-induced dipoles of dielectric nanocolloids , author=. Biomicrofluidics , volume=. 2010 , publisher=

2010
[80]

Batchelor,G. K. , Publisher =
[81]

, Journal =

Batchelor, G.K. , Journal =. On steady laminar flow with closed streamlines at large Reynolds number , Volume =
[82]

Batchelor, G. K. , Journal =. Slender-Body theory for particles of arbitrary cross-section in
[83]

Batchelor, G. K. , Journal =. The stress system in a suspension of force-free particles , Volume =
[84]

Batchelor, G. K. , Journal =. Sedimentation in a dilute dispersion of spheres , Volume =
[85]

Sedimentation in a dilute polydisperse system of interacting spheres

Batchelor, GK , Journal =. Sedimentation in a dilute polydisperse system of interacting spheres. Part 1. General theory , Volume =
[86]

Baygents, J. C. and Saville, D. A. , Booktitle =. The circulation produced in a drop by an electric field: a high field strength electrokinetic model , Volume = 7, Year =
[87]

Baygents, J. C. and Saville, D. A. , Journal =. Electrophoresis of drops and bubbles , Volume =
[88]

Baygents, J. C. and Saville, D. A. , Journal =. Electrophoresis of small particles and fluid globules in weak electrolytes , Volume =
[89]

Bazant, M. Z. and Thornton, K. and Ajdari, A. , Journal =. Diffuse-charge dynamics in electrochemical systems , Volume =
[90]

Bazant, M. Z. and Squires, T. M. , Journal =. Induced-charge electrokinetic phenomena: Theory and microfluidic applications , Volume =
[91]

Bazant, M. Z. and Chu, K. T. and Bayly, B. J. , Journal =. Current-Voltage Relations for Electrochemical Thin Films , Volume =
[92]

Bazant, M. Z. and Ben, Y. , Journal =
[93]

Bazant, M. Z. and Kilic, M. S. and Storey, B. D. and Ajdari, A. , Journal =. Nonlinear electrokinetics at large voltages , Volume =
[94]

Bazant, M. Z. and Kilic, M. S. and Storey, B. D. and Ajdari, A. , Journal =. Towards an understanding of induced-charge electrokinetics at large applied voltages in concentrated solutions , Volume =
[95]

Bazant, M. Z. and Squires, T. M. , Journal =. Induced-charge electrokinetic phenomena , Volume =
[96]

Bazant, M. Z. and Storey, B. D. and Kornyshev, A. A. , Journal =. Double layer in ionic liquids: Overscreening versus crowding , Volume =
[97]

Faraday Discuss

The surface of neat water is basic , author=. Faraday Discuss. , volume=. 2009 , publisher=

2009
[98]

Exact Poisson-Boltzmann solution for the interaction of dissimilar charge-regulating surfaces , author=. Phys. Rev. E , volume=. 1999 , publisher=

1999
[99]

Effect of charge convection on gravitational settling of drop in uniform electric field , author=. Phys. Fluids , volume=. 2020 , publisher=

2020
[100]

and Chang, H.-C

Ben, Y. and Chang, H.-C. , Journal =. Nonlinear
[101]

and Demekhin, E

Ben, Y. and Demekhin, E. A. and Chang, H.-C. , Journal =. Nonlinear electrokinetics and ``superfast'' electrophoresis , Volume =
[102]

Bender, C. M. and Orszag, S. A. , Publisher =. Advanced Mathematical Methods for Scientists and Engineers , Year =
[103]

The rotary motor of bacterial flagella , author=. Annu. Rev. Biochem. , volume=
[104]

Biesheuvel, P. M. and van Soestbergen, M. and Bazant, M. Z. , Date-Modified =. Imposed currents in galvanic cells , Volume =. Electrochimica Acta , Number =
[105]

and Vandepeere, L

van der Biest, O. and Vandepeere, L. G. , Journal =. Electrophoretic deposition of materials , Volume =
[106]

Biesheuvel and L

A. Biesheuvel and L. van Wijngaarden , Journal =
[107]

Biesheuvel, P. M. and van Soestbergen, M. , Journal =. Counterion volume effects in mixed electrical double layers , Volume =
[108]

Bike, S. G. and Prieve, D. C. , Journal =. Electrohydrodynamic lubrication with thin double layers , Volume =
[109]

Bike, S. G. and Prieve, D. C. , Journal =. Measurements of double-layer repulsion for slightly overlapping counterion clouds , Volume =
[110]

, howpublished =

Ahsan, T. , howpublished =
[111]

, Title =

Thomas, L.H. , Title =. Watson Sc. Comp. Lab. Rep. , year =
[112]

Bike, S. G. and Prieve, D. C. , Journal =. Electrohydrodynamics of thin double layers: a model for the streaming potential profile , Volume =
[113]

Bike, S. G. and Lazarro, L. and Prieve, D. C. , Journal =. Electrokinetic Lift of a Sphere Moving in Slow Shear Flow Parallel to a Wall
[114]

Breakup of asymmetric liquid ligaments , author=. Phys. Rev. Fluids , volume=. 2019 , publisher=

2019
[115]

and Magnaudet, J

Pierson, J.-L. and Magnaudet, J. and Soares, E. J. and Popinet, S. , journal=. Revisiting the. 2020 , publisher=

2020
[116]

Bike, S. G. and Prieve, D. C. , Journal =. Electrokinetic Lift of a Sphere Moving in Slow Shear Flow Parallel to a Wall
[117]

, Journal =

Bikerman, J.J. , Journal =
[118]

Particles as surfactants -- similarities and differences , author=. Curr. Opin. Colloid Interface Sci. , volume=. 2002 , publisher=

2002
[119]

A spherical envelope approach to ciliary propulsion , author=. J. Fluid Mech. , volume=. 1971 , publisher=

1971
[120]

and Charlaix, E

Bocquet, L. and Charlaix, E. , Journal =
[121]

Langmuir , Pages =

B. Langmuir , Pages =
[122]

Streaming potentials of granular media:

Bol. Streaming potentials of granular media:. J. Geophys. Res. , Pages =
[123]

and Kappl, M

Bonaccurso, E. and Kappl, M. and Butt, H. J. , Journal =. Hydrodynamic force measurements: boundary slip of water on hydrophilic surfaces and electrokinetic effects , Volume =
[124]

Liquid marbles: properties and applications , author=. Curr. Opin. Colloid Interface Sci. , volume=. 2011 , publisher=

2011
[125]

and Andelman, D

Borukhov, I. and Andelman, D. and Orland, H. , Journal =. Steric effects in electrolytes:
[126]

, Journal =

Booth, F. , Journal =
[127]

, Journal =

Booth, F. , Journal =. The cataphoresis of spherical fluid droplets in electrolytes , Volume =
[128]

, Journal =

Booth, F. , Journal =. Sedimentation Potential and Velocity of Solid Spherical Particles , Volume =
[129]

Boymelgreen, A. M. and Miloh, T. , Journal =. A theoretical study of induced-charge dipolophoresis of ideally polarizable asymmetrically slipping Janus particles , Volume =
[130]

Boymelgreen, A. M. and Miloh, T. , Journal =. Induced-charge electrophoresis of uncharged dielectric spherical Janus particles , Volume =
[131]

Electroviscous effects in charged capillaries , author=. J. Colloid Interface Sci. , volume=. 1995 , publisher=

1995
[132]

Interfacial electrical properties of electrogenerated bubbles , author=. J. Appl. Electrochem. , volume=. 1985 , publisher=

1985
[133]

Particle motion driven by solute gradients with application to autonomous motion: continuum and colloidal perspectives , author=. J. Fluid Mech. , volume=. 2011 , publisher=

2011
[134]

Fluid mechanics of propulsion by cilia and flagella , author=. Ann. Rev. Fluid Mech. , volume=. 1977 , publisher=

1977
[135]

The constrained Brownian movement of spherical particles in cylindrical pores of comparable radius: Models of the diffusive and convective transport of solute molecules in membranes and porous media , author=. J. Colloid Interface Sci. , volume=. 1977 , publisher=

1977
[136]

, Journal =

Brenner, H. , Journal =. The
[137]

1993 , Publisher =

Macrotransport Processes , author=. 1993 , Publisher =

1993
[138]

and De Gennes, P

Brochard, F. and De Gennes, P. G. , Journal =. Dynamics of confined polymer chains , Volume =
[139]

Streaming from the equator of a drop in an external electric field , author=. Phys. Rev. Lett. , volume=. 2017 , publisher=

2017
[140]

2003 , publisher=

Complex variables and applications , author=. 2003 , publisher=

2003
[141]

Brown, A. B. D. and Smith, C. G. and Rennie, A. R. , Journal =. Pumping of water with ac electric fields applied to asymmetric pairs of microelectrodes , Volume =
[142]

and Ajdari, A

Brunet, E. and Ajdari, A. , Journal =. Generalized Onsager relations for electrokinetic effects in anisotropic and heterogeneous geometries , Volume =
[143]

and Ajdari, A

Brunet, E. and Ajdari, A. , Journal =. Thin double layer approximation to describe streaming current fields in complex geometries: Analytical framework and applications to microfluidics , Volume =
[144]

, Journal =

Brunner, E. , Journal =. Theorie der Reaktionsgeschwindigkeit in heterogenen Systemen , Volume =
[145]

, Journal =

Brunner, E. , Journal =. Die kathodische und anodische Stromspannungskurve bei der Elektrolyse von Jod-Jodkaliuml\;osungen , Volume =
[146]

Water surface is acidic , author=. Proc. Nat. Acad. Sci. USA , volume=. 2007 , publisher=

2007
[147]

Buckmaster, J. D. , Journal =. Pointed bubbles in slow viscous flow , Volume =
[148]

, Journal =

Buitenhuis, J. , Journal =. Electrophoresis of fd-virus particles: Experiments and an analysis of the effect of finite rod lengths , Volume =
[149]

and Laocharoensuk, R

Burdick, J. and Laocharoensuk, R. and Wheat, P. M. and Posner, J. D. and Wang, J. , Journal =. Synthetic nanomotors in microchannel networks: directional microchip motion and controlled manipulation of cargo , Volume =
[150]

Carslaw, H. S. and Jaeger, J. C. , Publisher =. Conduction of heat in solids , Year =
[151]

Catchmark, J. M. and Subramanian, S. and Sen, A. , Journal =
[152]

and Schiedt, B

Cervera, J. and Schiedt, B. and Ramirez, P. , journal=. 2007 , publisher=

2007
[153]

and Dorfman, K.D

Chabert, M. and Dorfman, K.D. and de Cremoux, P. and Roeraade, J. and Viovy, J.L. , Journal =. Automated microdroplet platform for sample manipulation and polymerase chain reaction , Volume =
[154]

Chaffey, C. E. and Brenner, H. and Mason, S. G. , Journal =. Particle motions in sheared suspensions , Volume =
[155]

and Yeo, L

Chang, H.-C. and Yeo, L. Y. , Publisher =
[156]

, Journal =

Chang, H.-C. , Journal =
[157]

Chang, Y. C. and Keh, H. J. , Journal =. Diffusiophoresis and electrophoresis of a charged sphere perpendicular to two plane walls , Volume =
[158]

, Journal =

Chapman, D. , Journal =. A contribution to the theory of electrocapillarity , Volume =
[159]

Chen, S. B. and Keh, H. J. , Journal =. Axisymmetric electrophoresis of multiple colloidal spheres , Volume =
[160]

Chen, S. B. and Koch, D. L. , Journal =. Electrophoresis and sedimentation of charged fibers , Volume =
[161]

Chen, S. B. and Koch, D. L. , Journal =. Rheology of dilute suspensions of charged fibers , Volume =
[162]

Chen, P. Y. and Keh, H. J. , Journal =. Diffusiophoresis and electrophoresis of a charged sphere parallel to one or two plane walls , Volume =
[163]

Coalescence of Pickering Emulsion Droplets Induced by an Electric Field , author=. Phys. Rev. Lett. , volume=. 2013 , publisher=

2013
[164]

Chew, W. C. , Journal =. Dielectric enhancement and electrophoresis due to an electrochemical double layer: a uniform approximation , Volume =
[165]

Droplets for ultrasmall-volume analysis , author=. Anal. Chem. , volume=. 2009 , publisher=

2009
[166]

Single drop microextraction using commercial capillary electrophoresis instruments , author=. Anal. Chem. , volume=. 2008 , publisher=

2008
[167]

Chu, K. T. and Bazant, M. Z. , Journal =. Electrochemical Thin Films at and above the Classical Limiting Current , Volume =
[168]

Chu, K. T. and Bazant, M. Z. , Journal =. Nonlinear electrochemical relaxation around conductors , Volume =
[169]

Langmuir , volume=

Evidence of surface charge at the air/water interface from thin-film studies on polyelectrolyte-coated substrates , author=. Langmuir , volume=. 2005 , publisher=

2005
[170]

Cole, J. D. , Publisher =
[171]

D. M. Colegate and C. D. Bain , title =. Phys. Rev. Lett. , year =
[172]

Cooley, M. D. A. and O'Neill, M. E. , Journal =. On the slow rotation of a sphere about a diameter parallel to a nearby plane wall , Volume =
[173]

Cooley, M. D. A. and O'Neill, M. E. , Journal =. On the slow motion generated in a viscous fluid by the approach of a sphere to a plane wall or stationary sphere , Volume =
[174]

Osmotic Propulsion:

C. Osmotic Propulsion:. Phys. Rev. Lett. , Number =
[175]

Soft Matter , volume=

Osmotic propulsion of colloidal particles via constant surface flux , author=. Soft Matter , volume=. 2013 , publisher=

2013
[176]

Journal of Physics: Condensed Matter , volume=

Transport of self-propelling bacteria in micro-channel flow , author=. Journal of Physics: Condensed Matter , volume=. 2012 , publisher=

2012
[177]

Cox, R. G. and Brenner, H. , Journal =
[178]

Cox, R. G. and Brenner, H. , Journal =. The lateral migration of solid particles in
[179]

Cox, R. G. and Zia, I. Y. Z. and Mason, S. G. , Journal =. Particle motions in sheared suspensions XXV. Streamlines around cylinders and spheres , Volume =
[180]

Cox, R. G. , Journal =. The motion of long slender bodies in a viscous fluid
[181]

Cox, R. G. , Journal =
[182]

Cox, R. G. , Journal =. Electroviscous forces on a charged particle suspended in a flowing liquid , Volume =
[183]

Electrically induced pattern formation in thin leaky dielectric films , author=. Phys. Fluids , volume=. 2005 , publisher=

2005
[184]

Stokes flows past gaps in a wall , author=. Proc. Roy. Soc. London A , volume=
[185]

An infinite-series solution for the creeping motion through an orifice of finite length , author=. J. Fluid Mech. , volume=. 1982 , publisher=

1982
[186]

Nano Lett

Nanofluidic diode and bipolar transistor , author=. Nano Lett. , volume=. 2005 , publisher=

2005
[187]

Chaotic induced-charge electro-osmosis , author=. Phys. Rev. Lett. , note=
[188]

Computation of electrical double layer properties in simple electrolytes , author=

Surface ionization and complexation at the oxide/water interface: I. Computation of electrical double layer properties in simple electrolytes , author=. J. Colloid Interface Sci. , volume=. 1978 , publisher=

1978
[189]

Davis, R. H. and Acrivos, A. , Journal =. Sedimentation of noncolloidal particles at low Reynolds numbers , Volume =
[190]

Stresslet asymptotics for a treadmilling swimmer near a two-dimensional corner: hydrodynamic bound states , author=. Proc. Roy. Soc. London. A , volume=. 2012 , publisher=

2012
[191]

Matched asymptotics for a treadmilling low-Reynolds-number swimmer near a wall , author=. Q. J. Mech. Appl. Math. , volume=. 2013 , publisher=

2013
[192]

A nonlinear small-deformation theory for transient droplet electrohydrodynamics , author=. J. Fluid Mech. , volume=. 2017 , publisher=

2017
[193]

Electrohydrodynamics of viscous drops in strong electric fields: numerical simulations , author=. J. Fluid Mech. , volume=. 2017 , publisher=

2017
[194]

A three-dimensional small-deformation theory for electrohydrodynamics of dielectric drops , author=. J. Fluid Mech. , volume=. 2021 , publisher=

2021
[195]

Nature , volume=

Controlled multiple reversals of a ratchet effect , author=. Nature , volume=. 2006 , publisher=

2006
[196]

Dielectric response and conductivity of dilute suspensions of colloidal particles , author=. J. Chem. Soc., Faraday Trans. 2 , volume=. 1981 , publisher=

1981
[197]

Delgado, A. V. and Gonzalez-Caballero, F. and Hunter, R. J. and Koopal, L. K. and Lyklema, J. , Journal =. Measurement and interpretation of electrokinetic phenomena , Volume =
[198]

and Landau, L

Derjaguin, B. and Landau, L. , Journal =. A theory of the stability of strongly charged lyophobic sols and the coalescence of strongly charged particles in electrolytic solution , Volume =
[199]

Derjaguin, B. V. and Dukhin, S. S. and Korotkova, A. A. , Journal =
[200]

Derjaguin, B. V. and Dukhin, S. S. , Booktitle =. Nonequilibrium double layer and electrokinetic phenomena , Volume = 7, Year =
[201]

The Journal of Physical Chemistry , volume=

Kinetics of surfactant adsorption at fluid-fluid interfaces , author=. The Journal of Physical Chemistry , volume=. 1996 , publisher=

1996
[202]

Europhys

Kinetics of surfactant adsorption at fluid/fluid interfaces: non-ionic surfactants , author=. Europhys. Lett.) , volume=. 1996 , publisher=

1996
[203]

and Makino, M

Doi, M. and Makino, M. , Journal =. Electrokinetic boundary condition compatible with the
[204]

Active structuring of colloidal armour on liquid drops , author=. Nat. Commun. , volume=. 2013 , publisher=

2013
[205]

arXiv preprint arXiv:1411.1183 , year=

Driven particles at fluid interfaces acting as capillary dipoles , author=. arXiv preprint arXiv:1411.1183 , year=

arXiv
[206]

arXiv preprint arXiv:1502.05488 , year=

Drag and diffusion coefficients of a spherical particle attached to a fluid interface , author=. arXiv preprint arXiv:1502.05488 , year=

arXiv
[207]

Dorfman, K. D. , Booktitle =. Electrophoresis , Year =
[208]

, Journal =

Dorfman, K.D. , Journal =. DNA electrophoresis in microfabricated devices , Volume =
[209]

, Journal =

Dorm, E. , Journal =
[210]

and Mestel, A

Dubash, N. and Mestel, A. J. , Journal =. Behaviour of a conducting drop in a highly viscous fluid subject to an electric field , Volume =
[211]

and Mestel, A

Dubash, N. and Mestel, A. J. , Journal =. Breakup behavior of a conducting drop suspended in a viscous fluid subject to an electric field , Volume =
[212]

and Prat, M

Dufr\^eche, J. and Prat, M. and Schmitz, P. and Sherwood, J. D. , Journal =. On the apparent permeability of a porous layer backed by a perforated plate , Volume =
[213]

Dukhin number ---

Wikipedia , Note =. Dukhin number ---. 2011 , Bdsk-Url-1 =

2011
[214]

Dukhin, S. S. , Booktitle =. Diffusion-electrical theory of electrophoresis , Volume =
[215]

Dukhin, S. S. and Shilov, V. N. , Journal =. Theory of static polarization of diffuse part of thin electric double layer of spherical particles , Volume =
[216]

Dukhin, S. S. and Semenikhin, V. N. , Journal =
[217]

Dukhin, S. S. and Derjaguin, B. V. , Booktitle =. Equilibrium double layer and electrokinetic phenomena , Volume = 7, Year =
[218]

Dukhin, S. S. and Shilov, V. N. , Publisher =. Dielectric Phenomena and the Double Layer in Disperse Systems and Polyelectrolytes , Year =
[219]

Dukhin, A. S. and Murtsovkin, V. A. , Journal =. Pair Interaction of Particles in Electric-Field .2
[220]

Dukhin, S. S. and Mishchuk, N. A. , Journal =. Unlimited increase in the current through an ionite granule , Volume =
[221]

Dukhin, S. S. and Mishchuk, N. A. and Tarovskii, A. A. and Baran, A. A. , Journal =. Electrophoresis of the 2nd Kind , Volume =
[222]

Dukhin, S. S. and Mishchuk, N. A. and Takhistov, P. B. , Journal =. Electro-osmosis of the second kind and unrestricted current increase in a mixed ionite monolayer , Volume =
[223]

and Mishchuk, N.A

Dukhin, S.S. and Mishchuk, N.A. , Journal =. Concentration polarization of a conducting particle in strong fields , Volume =
[224]

Dukhin, S. S. , Journal =. Electrokinetic phenomena of the 2nd kind and their applications , Volume =
[225]

Dukhin, S. S. , Date-Modified =. Electrophoresis at large. Adv. Colloid Interface Sci. , Pages =
[226]

Dukhin, S. S. , Journal =. Non-equilibrium electric surface phenomena , Volume =
[227]

Langmuir , volume=

Direct observation of the direction of motion for spherical catalytic swimmers , author=. Langmuir , volume=. 2011 , publisher=

2011
[228]

Size dependence of the propulsion velocity for catalytic Janus-sphere swimmers , author=. Phys. Rev. E , volume=. 2012 , publisher=

2012
[229]

Nonlinear dynamics and breakup of free-surface flows , author=. Rev. Mod. Phys. , volume=. 1997 , publisher=

1997
[230]

and Hlushkou, D

Ehlert, S. and Hlushkou, D. and Tallarek, U. , Journal =. Electrohydrodynamics around single ion-permselective glass beads fixed in a microfluidic device , Volume =
[231]

Microfluid Nanofluid , volume=

Nanofluidics: what is it and what can we expect from it? , author=. Microfluid Nanofluid , volume=. 2005 , publisher=

2005
[232]

Europhys

Wall accumulation of self-propelled spheres , author=. Europhys. Lett. , volume=. 2013 , publisher=

2013
[233]

Elton, G. A. H. , Journal =
[234]

Fair, M. C. and Anderson, J. L. , Journal =. Electrophoresis of nonuniformly charged ellipsoidal particles , Volume =
[235]

Fagan, J. A. and Sides, P. J. and Prieve, D. C. , Journal =. Vertical oscillatory motion of a single colloidal particle adjacent to an electrode in an ac electric field , Volume =
[236]

Fagan, J. A. and Sides, P. J. and Prieve, D. C. , Journal =. Vertical motion of a charged colloidal particle near an
[237]

Fagan, J. A. and Paul, J. and Prieve, D. C. , Journal =. Evidence of multiple electrohydrodynamic forces acting on a colloidal particle near an electrode due to an alternating current electric field , Volume =
[238]

Fagan, J. A. and Sides, P. J. and Prieve, D. C. , Journal =. Mechanism of Rectified Lateral Motion of Particles near Electrodes in Alternating Electric Fields below 1
[239]

Feng, J. Q. and Scott, T. C. , Journal =. A computational analysis of electrohydrodynamics of a leaky dielectric drop in an electric field , Volume =
[240]

Nonlinear bubble dynamics , author=. Ann. Rev. Fluid Mech. , volume=. 1997 , publisher=

1997
[241]

Feng, J. Q. , Journal =. Electrohydrodynamic behaviour of a drop subjected to a steady uniform electric field at finite electric Reynolds number , Volume =
[242]

Electrohydrodynamic instabilities in freely suspended viscous films under normal electric fields , author=. Phys. Rev. Fluids , volume=. 2021 , publisher=

2021
[243]

Instability of a planar fluid interface under a tangential electric field in a stagnation point flow , author=. J. Fluid Mech. , volume=. 2022 , publisher=

2022
[244]

arXiv preprint arXiv:2210.04957 , year=

A spectral boundary integral method for simulating electrohydrodynamic flows in viscous drops , author=. arXiv preprint arXiv:2210.04957 , year=

arXiv
[245]

Fischer, T. M. and Dhar, P. , Journal =
[246]

, Journal =

Fixman, M. , Journal =. Thin double layer approximation for electrophoresis and dielectric response , Volume =
[247]

and Uplinger, J

Fologea, D. and Uplinger, J. and Thomas, B. and McNabb, D. S. and Li, J. , journal=. Slowing. 2005 , publisher=

2005
[248]

and Arsenault, A

Fournier-Bidoz, S. and Arsenault, A. C. and Manners, I. and Ozin, G. A. , Journal =
[249]

Frankel, N. A. and Acrivos, A. , Journal =. Heat and mass transfer from small spheres and cylinders freely suspended in shear flow , Volume =
[250]

and Yossifon, G

Frankel, I. and Yossifon, G. and Miloh, T. , Journal =. Dipolophoresis of dielectric spheroids under asymmetric fields , Volume =
[251]

, Journal =

Frumkin, A. , Journal =. New electrocapillary phenomena , Volume =
[252]

Gamayunov, N. I. and Murtsovkin, V. A. and Dukhin, A. S. , Journal =. Pair Interaction of Particles in Electric-Field. 1
[253]

Electrokinetic model for electric-field-induced interfacial instabilities , author=. Phys. Rev. E , volume=. 2014 , publisher=

2014
[254]

and Cayre, O

Gangwal, S. and Cayre, O. J. and Bazant, M. Z. and Velev, O. D. , Journal =
[255]

Bubbles in insulating liquids: stability in an electric field , author=. Proc. Roy. Soc. London. A , volume=. 1964 , publisher=

1964
[256]

Gas flows through constricted shallow micro-channels , author=. J. Fluid Mech. , volume=. 2008 , publisher=

2008
[257]

Jet and progeny formation in the Rayleigh breakup of a charged viscous drop , author=. J. Fluid Mech. , volume=. 2020 , publisher=

2020
[258]

, Journal =

Ghosal, S. , Journal =. Lubrication theory for electro-osmotic flow in a microfluidic channel of slowly varying cross-section and wall charge , Volume =
[259]

Electrokinetic flow and dispersion in capillary electrophoresis , author=. Annu. Rev. Fluid Mech. , volume=. 2006 , publisher=

2006
[260]

and Chen, Z

Ghosal, S. and Chen, Z. , Journal =. A nonlinear equation for ionic diffusion in a strong binary electrolyte , Volume =
[261]

Self-propelled Janus particles in a ratchet: Numerical simulations , author=. Phys. Rev. Lett. , volume=. 2013 , publisher=

2013
[262]

Goldman, A. J. and Cox, R. G. and Brenner, H. , Journal =. The slow motion of two identical arbitrarily oriented spheres through a viscous fluid , Volume =
[263]

Goldman, A. J. and Cox, R.G. and Brenner, H. , Journal =. Slow viscous motion of a sphere parallel to a plane wall --
[264]

Goldman, A. J. and Cox, R.G. and Brenner, H. , Journal =
[265]

and Liverpool, T

Golestanian, R. and Liverpool, T. B. and Ajdari, A. , Journal =. Propulsion of a molecular machine by asymmetric distribution of reaction products , Volume =
[266]

and Liverpool, T

Golestanian, R. and Liverpool, T. B. and Ajdari, A. , Journal =. Designing phoretic micro-and nano-swimmers , Volume =
[267]

, Journal =

Golestanian, R. , Journal =. Anomalous Diffusion of Symmetric and Asymmetric Active Colloids , Volume =
[268]

and Marr, D

Gong, T. and Marr, D. W. M. , Journal =. Electrically switchable colloidal ordering in confined geometries , Volume =
[269]

and Ramos, A

Gonzalez, A. and Ramos, A. and Green, N. G. and Castellanos, A. and Morgan, H. , Journal =. Fluid flow induced by nonuniform ac electric fields in electrolytes on microelectrodes. II. A linear double-layer analysis , Volume =
[270]

, Journal =

Gouy, G. , Journal =. Sur la constitution de la charge electrique a la surface d
[271]

, Journal =

Gouy, G. , Journal =. Electric charge on the surface of an electrolyte , Volume =
[272]

The -potential of gas bubbles , author=. J. Colloid Interface Sci. , volume=. 1995 , publisher=

1995
[273]

I. S. Gradshteyn and I. M. Rhyzhik , title =
[274]

Grafov, B. M. and Chernenko, A. A. , Journal =. Theory of the passage of a constant current through a solution of a binary electrolyte , Volume =
[275]

Green, N. G. and Morgan, H. , Journal =. Dielectrophoretic investigations of sub-micrometre latex spheres , Volume =
[276]

Green, N. G. and Ramos, A. and Gonzalez, A. and Morgan, H. and Castellanos, A. , Journal =
[277]

Gregersen, M. M. and Andersen, M. B. and Soni, G. and Meinhart, C. and Bruus, H. , Journal =
[278]

Gregersen, M. M. and Okkels, F. and Bazant, M. Z. and Bruus, H. , Journal =. Topology and shape optimization of induced-charge electro-osmotic micropumps , Volume =
[279]

Ha, J. W. and Yang, S. M. , Journal =. Deformation and breakup of
[280]

Entropic trapping and escape of long DNA molecules at submicron size constriction , author=. Phys. Rev. Lett. , volume=. 1999 , publisher=

1999
[281]

Lab Chip , volume=

Molecular sieving using nanofilters: past, present and future , author=. Lab Chip , volume=. 2007 , publisher=

2007
[282]

Electrophoresis of surfactant-free bubbles , author=. J. Colloid Interface Sci. , volume=. 2010 , publisher=

2010
[283]

and Suganuma, M

Hasegawa, T. and Suganuma, M. and Watanabe, H. , Journal =. Anomaly of excess pressure drops of the flow through very small orifices , Volume =
[284]

and Brenner, H

Happel, J. and Brenner, H. , Publisher =. Low Reynolds Number Hydrodynamics , Year =
[285]

Harnett, C. K. and Templeton, J. and Dunphy-Guzman, K. A. and Senousy, Y. M. and Kanouff, M. P. , Journal =. Model based design of a microfluidic mixer driven by induced charge electroosmosis , Volume =
[286]

New development in freefem++ , author=. J. Numer. Math. , volume=
[287]

Transport and collective dynamics in suspensions of confined swimming particles , author=. Phys. Rev. Lett. , volume=. 2005 , publisher=

2005
[288]

Streaming currents in a single nanofluidic channel , author=. Phys. Rev. Lett. , volume=. 2005 , publisher=

2005
[289]

Hickey, O. A. and Holm, C. and Harden, J. L. and Slater, G. W. , Journal =. Implicit method for simulating electrohydrodynamics of polyelectrolytes , Volume =
[290]

Hickey, O. A. and Shendruk, T. N. and Harden, J. L. and Slater, G. W. , Journal =. Simulations of Free-Solution Electrophoresis of Polyelectrolytes with a Finite Debye Length Using the
[291]

Higuera, F. J. , Journal =. Breakup of a supported drop of a viscous conducting liquid in a uniform electric field , Volume =
[292]

Electrophoresis of spherical polymer-coated colloidal particles , author=. J. Colloid Interface Sci. , volume=. 2003 , publisher=

2003
[293]

Physical review letters , volume=

Hydrodynamic surface interactions enable Escherichia coli to seek efficient routes to swim upstream , author=. Physical review letters , volume=. 2007 , publisher=

2007
[294]

Hinch, E. J. , Publisher =
[295]

Hinch, E. J. and Acrivos, A. , Journal =. Steady long slender droplets in two-dimensional straining motion , Volume =
[296]

Hinch, E. J. and Acrivos, A. , Journal =. Long slender drops in a simple shear flow , Volume =
[297]

Hinch, E. J. and Sherwood, J. D. , Journal =. Primary electroviscous effect in a suspension of spheres with thin double layers , Volume =
[298]

Hinch, E. J. and Sherwood, J. D. and Chew, W. C. and Sen, P. N. , Journal =. Dielectric response of a dilute suspension of spheres with thin double layers in an asymmetric electrolyte , Volume =
[299]

S. V. Ho and P. W. Sheridan and C. J. Athmer and M. A. Heitkamp and J. M. Brackin and D. Weber and P. H. Brodsky , title =. Environ Sci Technol , year =
[300]

Hoggard, J. D. and Paul, J. and Prieve, D. C. , Journal =. Measurement of the streaming potential and streaming current near a rotating disk to determine its zeta potential , Volume =

Pith tools

Reviewed August 8, 2026 · model on record in the stance chip above.

[1] [1]

Theorie mecanique de la chaleur , author=. Ann. Chim. Phys , volume=

[2] [2]

Journal of fluid mechanics , volume=

Lubrication of soft viscoelastic solids , author=. Journal of fluid mechanics , volume=. 2016 , publisher=

2016

[3] [3]

Nature , volume=

Some applications of photography , author=. Nature , volume=

[4] [4]

Some experiments on the dynamics of liquid films , author=. J. Appl. Phys. , volume=. 1959 , publisher=

1959

[5] [5]

Atomization of undulating liquid sheets , author=. J. Fluid Mech. , volume=. 2007 , publisher=

2007

[6] [6]

Interfacial Flows---The Power and Beauty of Asymptotic Methods , pages=

Viscoelastic lubrication using the second-order fluid , author=. Interfacial Flows---The Power and Beauty of Asymptotic Methods , pages=. 2025 , publisher=

2025

[7] [7]

On the coefficient of viscous traction and its relation to that of viscosity , author=. Proc. R. Soc. London Ser. A , volume=. 1906 , publisher=

1906

[8] [8]

Fragmentation versus cohesion , author=. J. Fluid Mech. , volume=. 2020 , publisher=

2020