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arxiv: 2605.25167 · v1 · pith:ZLW47UUW · submitted 2026-05-24 · physics.flu-dyn

Energetic variational formulation for electrohydrodynamics of surfactant-laden droplets

Hangjie Ji , Jian-Guo Liu This is my paper

Reviewed by Pith T0 review T1 audit T2 compute T3 formal T4 kernel 2026-06-29 23:32 UTCgrok-4.3pith:ZLW47UUWrecord.jsonopen to challenge →

classification physics.flu-dyn
keywords energetic variational formulationelectrohydrodynamicssurfactant-laden dropletsOnsager principleStokes flowmoving interfaceMaxwell stressesdroplet dynamics
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The pith

Minimizing the Rayleighian derives the full set of equations for surfactant-laden electrohydrodynamic droplets.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops an energetic variational framework for two-phase Stokes flow coupled with surfactant transport on a moving interface and electrostatic effects. It applies Onsager's principle by minimizing the Rayleighian, which sums the rate of change of free energy and a dissipation functional, subject to the incompressibility constraint. This single minimization produces the bulk Stokes equations, the interfacial stress balance that includes both Marangoni and Maxwell stresses, the electrostatic equation, the surface transport equation for insoluble surfactant, and the moving contact-line dynamics. The same structure also yields a reduced mean-curvature model and a one-dimensional graph formulation for sessile droplets, together with a numerical scheme. The approach matters because it builds thermodynamic consistency directly into models used for digital microfluidics and related liquid-handling technologies.

Core claim

By minimizing the Rayleighian, defined as the sum of the rate of change of the free energy and the dissipation functional, subject to the incompressibility constraint, the formulation simultaneously yields the Stokes equations in each bulk phase, the interfacial stress-balance condition incorporating Marangoni and Maxwell stresses, the electrostatic equation, the surface transport equation for insoluble surfactant concentration, and the moving contact-line dynamics.

What carries the argument

The Rayleighian functional, minimized subject to the incompressibility constraint following Onsager's principle.

If this is right

  • Replacing the viscous dissipation functional with Rayleigh dissipation produces a reduced model in which surfactant-laden droplets evolve by motion by mean curvature.
  • Representing sessile droplets as graphs reduces the full system to a one-dimensional coupled model for liquid height, surfactant concentration, and electric potential.
  • A first-order implicit-explicit scheme can be applied to the reduced graph system to compute the coupled effects of surfactant transport and electric fields.
  • The variational structure automatically incorporates energy dissipation into the dynamics of the interface and the surfactant.

Where Pith is reading between the lines

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

  • The same Rayleighian construction may extend to three-dimensional non-graph geometries while preserving the same stress and transport equations.
  • Energy-stable discretizations could be designed by mimicking the continuous variational structure at the discrete level.
  • The framework might accommodate additional physics, such as soluble surfactants, by adjusting only the free-energy functional.

Load-bearing premise

The system dynamics are obtained by minimizing the Rayleighian subject to the incompressibility constraint, following Onsager's principle, with specific choices for the free energy and dissipation functionals.

What would settle it

A direct check showing that the derived interfacial stress balance fails to recover the standard combination of viscous, Marangoni, and Maxwell stresses would falsify the claim.

Figures

Figures reproduced from arXiv: 2605.25167 by Hangjie Ji, Jian-Guo Liu.

Figure 1
Figure 1. Figure 1: (Left) Schematics of a sessile drop on solid substrate. The liquid domain Ω− rests on the substrate z = 0 and is surrounded by the gas domain Ω+. The free surface St meets the solid substrate at the contact line Ct. The unit normal n on St points outward from Ω− into Ω+. The dynamic contact angle θCL is measured inside the drop between the substrate and the tangent to St at Ct. (Right) Contact-line geometr… view at source ↗
Figure 2
Figure 2. Figure 2: Droplet spreading with electrostatics in a homogeneous electric field concentration c(x, t) is assumed to be spatially uniform, h(x, 0) = −2(x − 0.5)(x + 0.5), c(x, 0) = 0.5. (6.22) The numerical simulations are carried out using the IMEX scheme described in Algorithm 1. The electrostatic problem for ϕ is solved in the bulk domain Ω±, where the entire computational domain Ω = [−1, 1] × [0, 1], while the he… view at source ↗
Figure 3
Figure 3. Figure 3: Droplet transport induced by an inhomogeneous electric field (6.23). Next, we consider droplet dynamics induced by a spatially varying substrate potential, ϕ0(x, 0) = ( 0, −1 ⩽ x < 0, 1, 0 ⩽ x < 1 . (6.23) Unlike the homogeneous case, the imposed substrate potential generates an electric field that breaks the left-right symmetry of the system [PITH_FULL_IMAGE:figures/full_fig_p020_3.png] view at source ↗
read the original abstract

The coupling of surfactant-laden droplet dynamics and electric fields plays an important role in liquid-handling technologies such as digital microfluidics. We develop an energetic variational framework for the coupled dynamics of two-phase Stokes flow with surfactant transport on a moving interface and electrostatic effects. Based on Onsager's principle, the governing equations are derived by minimizing the Rayleighian, defined as the sum of the rate of change of the free energy and the dissipation functional, subject to the incompressibility constraint. This formulation simultaneously yields the Stokes equations in each bulk phase, the interfacial stress-balance condition incorporating Marangoni and Maxwell stresses, the electrostatic equation, the surface transport equation for insoluble surfactant concentration, and the moving contact-line dynamics. By replacing the viscous dissipation functional with Rayleigh dissipation, we also derive a reduced model for surfactant-laden droplets evolving by motion by mean curvature. Representing sessile droplets as graphs further reduces the system to a one-dimensional coupled electrohydrodynamic model for the liquid height, surfactant concentration, and electric potential. A first-order implicit-explicit scheme is proposed for the graph system, and numerical results illustrate the coupled effects of surfactant transport and electric fields on droplet dynamics.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript develops an energetic variational formulation for the electrohydrodynamics of surfactant-laden droplets based on Onsager's principle. The governing equations are obtained by minimizing the Rayleighian (rate of free-energy change plus dissipation) subject to incompressibility; this is claimed to simultaneously produce the bulk Stokes equations in each phase, the interfacial stress balance incorporating Marangoni and Maxwell stresses, the electrostatic equation, the surface transport equation for insoluble surfactant, and moving contact-line dynamics. The paper also derives a reduced mean-curvature-flow model by replacing viscous dissipation with Rayleigh dissipation, reduces the sessile-droplet problem to a 1D graph formulation, and presents a first-order IMEX scheme with illustrative numerical results.

Significance. If the central variational derivation is free of algebraic error, the work supplies a thermodynamically consistent framework that unifies bulk flow, interface conditions, surfactant transport, and electrostatics. Such energetic formulations are valuable for constructing structure-preserving discretizations and for extending to more complex multiphysics droplet problems. The reduction steps to mean-curvature flow and the 1D graph model, together with the numerical illustration, add practical utility.

major comments (2)
  1. [Variational derivation (Rayleighian minimization)] The load-bearing step is the functional derivative of the Rayleighian with respect to the normal interface velocity (the step that produces the interfacial stress balance). The manuscript must explicitly display this calculation, including the contributions from the surfactant-dependent surface tension in the free energy and from the electrostatic energy, to confirm that the Marangoni term (surface gradient of tension) and the Maxwell-stress jump [[T^M]]·n appear with correct signs and without omitted terms. Any hidden assumption about how the electric potential couples to interface motion would undermine the strongest claim.
  2. [Definition of dissipation functional] The dissipation functional must be stated with sufficient precision (bulk viscous dissipation plus any interfacial or contact-line contributions) so that the reader can verify that the resulting bulk Stokes equations and the normal-stress jump are recovered exactly when the variation is performed subject to the incompressibility constraint.
minor comments (2)
  1. [Introduction] The abstract and introduction should cite the specific prior energetic-variational works on two-phase flow or electrohydrodynamics that the present construction extends.
  2. [Graph formulation] In the 1D graph reduction, the notation for the reduced variables (height h, surfactant concentration, potential) should be introduced with a clear diagram or coordinate definition to avoid ambiguity when the curvature and normal vectors are written in 1D form.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. The suggestions will improve the clarity of the variational derivation in the manuscript. We address each major comment below.

read point-by-point responses

  1. Referee: [Variational derivation (Rayleighian minimization)] The load-bearing step is the functional derivative of the Rayleighian with respect to the normal interface velocity (the step that produces the interfacial stress balance). The manuscript must explicitly display this calculation, including the contributions from the surfactant-dependent surface tension in the free energy and from the electrostatic energy, to confirm that the Marangoni term (surface gradient of tension) and the Maxwell-stress jump [[T^M]]·n appear with correct signs and without omitted terms. Any hidden assumption about how the electric potential couples to interface motion would undermine the strongest claim.

    Authors: We agree that explicitly displaying the functional derivative calculation is essential for rigor. In the revised manuscript, we will add a dedicated subsection or appendix detailing the variation of the Rayleighian with respect to the normal interface velocity. This will include the explicit contributions from the surfactant-dependent surface tension term in the free energy, which produces the Marangoni stress as the surface gradient of tension, and from the electrostatic energy, which produces the Maxwell stress jump [[T^M]]·n with the correct signs. The electric potential is determined by solving the electrostatic equations in the bulk domains with appropriate interface conditions, and its coupling to the interface motion enters solely through the energy variation without additional assumptions. revision: yes

  2. Referee: [Definition of dissipation functional] The dissipation functional must be stated with sufficient precision (bulk viscous dissipation plus any interfacial or contact-line contributions) so that the reader can verify that the resulting bulk Stokes equations and the normal-stress jump are recovered exactly when the variation is performed subject to the incompressibility constraint.

    Authors: We will revise the manuscript to state the dissipation functional with greater precision, explicitly writing the bulk viscous dissipation as the integral over each phase of 2μ|D(u)|^2 and specifying any interfacial dissipation or contact-line dissipation terms. This precise definition will enable readers to directly verify the recovery of the Stokes equations in the bulk and the normal stress jump at the interface upon performing the variation subject to incompressibility. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation applies Onsager principle to independent functionals

full rationale

The paper constructs free-energy and dissipation functionals explicitly from physical ingredients (surfactant-dependent tension, electrostatic energy, viscous dissipation) and applies the standard Onsager variational principle subject to incompressibility. The resulting Euler-Lagrange equations are the claimed PDE system; nothing in the provided abstract or description shows the target interfacial stress balance or transport laws being inserted into the functionals or used to define the Rayleighian. No self-citation chain, fitted-parameter renaming, or ansatz smuggling is indicated. The central claim therefore remains an independent derivation rather than a tautology.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the application of Onsager's principle to a Rayleighian constructed from free energy and dissipation functionals for the two-phase electrohydrodynamic system with surfactant.

axioms (1)
  • domain assumption Onsager's principle: governing equations obtained by minimizing the Rayleighian (free-energy rate of change plus dissipation) subject to incompressibility
    Stated directly in the abstract as the basis for deriving all the listed equations.

pith-pipeline@v0.9.1-grok · 5731 in / 1249 out tokens · 40377 ms · 2026-06-29T23:32:31.442980+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

4 extracted references

  1. [1]

    Choi, Kihwan, Ng, Alphonsus HC, Fobel, Ryan & Wheeler, Aaron R2012 Digital microfluidics.Annual review of analytical chemistry5(1), 413–440

    Abdelgawad, Mohamed & Wheeler, Aaron R2009 The digital revolution: a new paradigm for microfluidics.Advanced Materials21(8), 920–925. Choi, Kihwan, Ng, Alphonsus HC, Fobel, Ryan & Wheeler, Aaron R2012 Digital microfluidics.Annual review of analytical chemistry5(1), 413–440. Energetic variational formulation for droplet dynamics27 Chu, Weiqi, Ji, Hangjie, ...

    2023

  2. [2]

    Doi, Masao2013Soft matter physics

    Doi, Masao2011 Onsager’s variational principle in soft matter.Journal of Physics: Condensed Matter23(28), 284118. Doi, Masao2013Soft matter physics. oxford university press. Gao, Yuan & Liu, Jian-Guo2021aGradient flow formulation and second order numerical method for motion by mean curvature and contact line dynamics on rough surface. Interfaces and Free ...

    1985

  3. [3]

    Addison-wesley Reading, MA. Karimi-Sibaki, E, V akhrushev, Alexander, Kadylnykova, Anastasiia, Wu, Menghuai, Ludwig, Andreas, Bohacek, Jan & Kharicha, Abdellah2023 A multiphase model for exploring electrochemical marangoni flow.Electrochemistry Communications155, 107567. Kim, Chang-Jin “CJ”2001 Micropumping by electrowetting. InASME international mechanic...

    2001

  4. [4]

    coffee stain effect

    Annual Reviews. Nganguia, Herve, Pak, On Shun & Young, Y-N2019 Effects of surfactant transport on electrodeformation of a viscous drop.Physical Review E99(6), 063104. Oron, Alexander, Davis, Stephen H. & Bankoff, S. George1997 Long-scale evolution of thin liquid films.Rev. Mod. Phys.69, 931–980. Osher, Stanley & Sethian, James A1988 Fronts propagating wit...

    1997