pith. machine review for the scientific record. sign in

arxiv: 2605.13196 · v1 · submitted 2026-05-13 · ⚛️ physics.bio-ph · cond-mat.soft· physics.comp-ph· physics.flu-dyn

Recognition: no theorem link

Onsager-variational formulation of diffuse-domain methods for computational modeling of microscale fluid-structure interactions

Xinpeng Xu

Pith reviewed 2026-05-14 01:30 UTC · model grok-4.3

classification ⚛️ physics.bio-ph cond-mat.softphysics.comp-phphysics.flu-dyn
keywords diffuse-domain methodsOnsager variational principlefluid-structure interactionsdiffuse interfacessurface dynamicsthermodynamic consistencymicroscale flowsactive shells
0
0 comments X

The pith

Onsager variational principle embeds sharp-interface energies into bulk to derive diffuse-domain models for fluid-structure interactions.

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

The paper shows how to build diffuse-domain methods for microscale fluid-structure problems by replacing sharp surfaces with volumetric representations on regular grids. It embeds the free-energy and dissipation functionals from sharp surfaces into the bulk via a diffuse delta density, then obtains all equations from a single Rayleighian constructed with Onsager's principle. This automatically distinguishes conserved fields transported by full material velocity from tangential internal variables that need projected objective rates. The same construction recovers known models for scalar transport on rigid or deformable interfaces and for near-wall hydrodynamics, then produces new coupled models for vesicles with surface viscosity and for active shells. A reader cares because the approach supplies one thermodynamically consistent route instead of case-by-case extensions.

Core claim

Embedding sharp-surface free-energy and dissipation functionals into the bulk through a diffuse surface delta density and deriving the governing equations from the Rayleighian distinguishes balance-law fields, internal nonconserved order parameters, and rate variables; conserved surface densities follow the full material velocity while tangential vector and tensor variables require projected objective or co-rotational rates within their admissible tangential spaces.

What carries the argument

Onsager variational principle applied to a Rayleighian that incorporates sharp-surface functionals via a diffuse delta density.

If this is right

  • The construction recovers established DDM models for scalar transport on rigid and deformable interfaces together with their sharp-interface limits.
  • It produces coupled diffuse-domain models for multicomponent deformable vesicles that include surface viscosity, tangential slip, and finite areal compressibility.
  • Active stresses enter through active work power while the passive part remains thermodynamically consistent.
  • The same variational structure applies to interfacial hydrodynamics near rigid walls and to active shells carrying chemical and tangential vector order.

Where Pith is reading between the lines

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

  • The same embedding technique could be tested on problems where surface tension and bending rigidity compete, such as vesicle fission, to check whether the diffuse formulation preserves energy dissipation rates.
  • Because the framework separates kinematic rates from constitutive choices, it may allow direct substitution of different surface rheologies without re-deriving the bulk equations.
  • Numerical implementations could be checked for conservation of total energy and entropy production when the diffuse width is held fixed but grid resolution is increased.

Load-bearing premise

Embedding sharp-surface free-energy and dissipation functionals into the bulk through a diffuse surface delta density correctly recovers the sharp-interface limits and respects the distinct transport rules for conserved versus tangential internal variables.

What would settle it

Numerical solution of a test problem such as a rigid sphere sedimenting in viscous fluid, with the diffuse-interface width systematically reduced, must converge in drag force and velocity field to the known sharp-interface Stokes solution.

Figures

Figures reproduced from arXiv: 2605.13196 by Xinpeng Xu.

Figure 1
Figure 1. Figure 1: FIG. 1. Experimental motivation for modeling complex dynamics on curved surfaces. Left: liquid–liquid phase separation on [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Schematic illustration of the diffuse-domain embedding and its biological motivation from deformable active surfaces. [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Schematic illustration of a moving contact line in immiscible two-phase flow on a rigid solid surface. (left) Diffuse [PITH_FULL_IMAGE:figures/full_fig_p016_3.png] view at source ↗
read the original abstract

Direct numerical simulation of microscale fluid--structure interactions in multicomponent and multiphase flows requires methods that can represent moving boundaries together with fields constrained to evolving interfaces. Diffuse-domain methods (DDMs) address this geometric difficulty by replacing sharp surfaces with diffuse volumetric representations on regular computational domains. Here we formulate DDMs using Onsager's variational principle. Instead of extending sharp-interface equations and boundary conditions term by term, we embed sharp-surface free-energy and dissipation functionals into the bulk through a diffuse surface delta density and derive the governing equations from the Rayleighian. The framework distinguishes balance-law fields, internal nonconserved order parameters, and kinematic or constitutive rate variables. It also clarifies a key moving-surface distinction: conserved surface densities are transported by the full material surface velocity, whereas explicitly tangential vector and tensor internal variables require projected objective or co-rotational rates within their admissible tangential state spaces. For scalar transport on rigid and deformable interfaces, and for interfacial hydrodynamics near rigid walls, the formulation recovers established DDM models and their sharp-interface limits. The same variational construction yields coupled diffuse-domain models for multicomponent deformable vesicles with surface viscosity, tangential slip, and finite areal compressibility, and for active shells carrying chemical and tangential vector order. These results provide a unified route to thermodynamically consistent passive DDMs for interfacial and surface dynamics, while allowing active stresses through active work power. The framework is relevant to soft matter, microfluidic interfaces, biological membranes, and morphogenetic surface 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 paper develops an Onsager-variational formulation of diffuse-domain methods (DDMs) for microscale fluid-structure interactions. Sharp-surface free-energy and dissipation functionals are embedded into the bulk via a diffuse surface delta density; the governing equations then follow from minimization of the Rayleighian. The framework distinguishes balance-law fields from internal nonconserved order parameters and from kinematic rate variables, and enforces distinct transport rules: conserved surface densities advect with the full material velocity while tangential vector/tensor fields evolve under projected objective or co-rotational rates. The construction recovers known DDM models for scalar transport on rigid and deformable interfaces, near-wall hydrodynamics, and their sharp-interface limits; it is further applied to multicomponent deformable vesicles with surface viscosity, tangential slip and finite compressibility, and to active shells carrying chemical and tangential order.

Significance. If the derivations hold, the work supplies a single thermodynamically consistent route to passive and active DDMs for interfacial and surface dynamics. The explicit separation of transport classes and the allowance for active work power are genuine strengths that could streamline construction of new models while preserving consistency with sharp limits. The approach is directly relevant to soft-matter hydrodynamics, microfluidic interfaces, biological membranes and morphogenetic problems.

major comments (2)
  1. [§3] §3 (derivation of the Rayleighian): the embedding of the sharp-surface dissipation functional through the diffuse delta density is asserted to recover the correct tangential viscous stresses, yet the manuscript provides no explicit asymptotic expansion or matched-asymptotics calculation confirming that the bulk dissipation integral converges to the sharp-surface dissipation as the interface width ε→0. This step is load-bearing for the claim of recovering established sharp-interface models.
  2. [§5] §5 (vesicle and active-shell examples): the distinction between full material advection for conserved densities and projected objective rates for tangential vectors is stated formally, but the resulting weak-form equations for the coupled hydrodynamics are not written out; without these explicit expressions it is impossible to verify that the variational construction indeed produces the expected tangential slip and surface-viscosity terms.
minor comments (2)
  1. [§2] Notation for the diffuse delta density and the projection operators onto the tangential plane should be introduced once and used consistently; several passages reuse the same symbol for the diffuse indicator and its gradient, which risks confusion.
  2. [Introduction] The abstract and introduction cite recovery of “established DDM models” but do not list the specific references or equations being recovered; a short table or explicit side-by-side comparison in the text would strengthen the claim.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the positive assessment and constructive comments. We address each major comment below and indicate the revisions we will make to strengthen the manuscript.

read point-by-point responses

  1. Referee: [§3] §3 (derivation of the Rayleighian): the embedding of the sharp-surface dissipation functional through the diffuse delta density is asserted to recover the correct tangential viscous stresses, yet the manuscript provides no explicit asymptotic expansion or matched-asymptotics calculation confirming that the bulk dissipation integral converges to the sharp-surface dissipation as the interface width ε→0. This step is load-bearing for the claim of recovering established sharp-interface models.

    Authors: We agree that an explicit matched-asymptotics verification would strengthen the load-bearing claim. In the revised manuscript we will add a concise appendix that performs the leading-order asymptotic analysis of the bulk dissipation integral as ε→0, confirming convergence to the sharp-surface tangential viscous dissipation. This addition will be placed after the main derivation in §3 and will not alter the variational construction itself. revision: yes

  2. Referee: [§5] §5 (vesicle and active-shell examples): the distinction between full material advection for conserved densities and projected objective rates for tangential vectors is stated formally, but the resulting weak-form equations for the coupled hydrodynamics are not written out; without these explicit expressions it is impossible to verify that the variational construction indeed produces the expected tangential slip and surface-viscosity terms.

    Authors: We acknowledge that the explicit weak forms would facilitate direct verification. In the revised manuscript we will append the full weak-form statements for the coupled hydrodynamics in §5, obtained directly from the Rayleighian stationarity condition. These expressions will explicitly display the tangential-slip and surface-viscosity contributions arising from the projected objective rates and the diffuse delta embedding, allowing immediate comparison with the expected sharp-interface terms. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained from external variational principle

full rationale

The paper starts from the external Onsager variational principle, embeds previously established sharp-surface free-energy and dissipation functionals into the bulk via a diffuse delta density, and derives the Rayleighian and governing equations directly. The distinctions between balance-law fields, nonconserved order parameters, and kinematic rates, as well as the transport rules for conserved versus tangential variables, follow from the variational structure and standard kinematic projections without re-expressing inputs as outputs. Recovery of known DDM models and sharp-interface limits is presented as a consistency verification rather than a fitted prediction. No load-bearing step reduces to self-citation, ansatz smuggling, or renaming; the central construction remains independent of the target results.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The central claim rests on the validity of Onsager's variational principle as the starting point and on the mathematical correctness of embedding sharp-surface functionals via a diffuse delta density; no free parameters are introduced in the abstract.

axioms (1)
  • standard math Onsager's variational principle (Rayleighian minimization)
    Invoked to derive governing equations from embedded free-energy and dissipation functionals.
invented entities (1)
  • diffuse surface delta density no independent evidence
    purpose: To embed sharp-surface free-energy and dissipation functionals into the bulk domain
    Introduced as the mechanism that replaces sharp interfaces with volumetric representations while preserving variational structure.

pith-pipeline@v0.9.0 · 5575 in / 1406 out tokens · 47967 ms · 2026-05-14T01:30:07.844910+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

93 extracted references · 93 canonical work pages

  1. [1]

    phase separation

    This non-monotonic choice is useful when contact-line slip and wall relaxation are important [41, 44, 70]. Rayleighian and dynamic equations— Using the constrained balance (61) and minimizing the Rayleighian R[v,J ϕ] = ˙F+ Φ− R Ω P∇·[(1−ψ)v]dVwith respect tovandJ ϕ gives ρ(1−ψ) (∂ tv+v· ∇v) =−(1−ψ)∇P+∇· ηb(ϕ, ψ) ∇v+∇v T + (1−ψ)ˆµϕ∇ϕ,(67a) ∂t[(1−ψ)ϕ] +∇·[(...

    work page

  2. [2]

    II, IV B, and V, we collect here only the sharp-surface identities that are used later

    Differential geometry and kinematics of a moving vesicle surface To connect this appendix section directly to the diffuse-domain formulation of a moving vesicle surface in Secs. II, IV B, and V, we collect here only the sharp-surface identities that are used later. The diffuse-domain embedding has already been formulated in Sec. II; in this appendix we wo...

    work page

  3. [3]

    V In Sec

    Sharp-interface model: OVP formulation consistent with Sec. V In Sec. IV B, we provide a passiveextensible-surface model with the areal-compression fieldc aΓ, the chemical concentrationc mΓ, and the tangential velocityV Γ∥, including the scalar membrane free energies, passive viscous surface dynamics, and the bulk hydrodynamics inside and outside the clos...

    work page

  4. [4]

    The local embedded material balance is ∂t(δϵci) +∇·(δ ϵciV) =−∇·(δ ϵJi) +δ ϵki,(B1) whereJ i is the nonconvective surface flux andk i is a source

    Scalar balance laws and shifted geometry chemical potential Scalar balance laws— Letc i denote a scalar surface balance-law field, withi= a,m. The local embedded material balance is ∂t(δϵci) +∇·(δ ϵciV) =−∇·(δ ϵJi) +δ ϵki,(B1) whereJ i is the nonconvective surface flux andk i is a source. In Sec. V,J a =0,k a = 0, whileJ m andk m are generally nonzero. We...

    work page

  5. [5]

    Polar contribution to the free-energy rate: diffuse and sharp forms We first record the polar part of the free-energy rate in the diffuse-domain model and then give its intrinsic sharp-surface counterpart. Only the terms involving the polar rate and the tangential velocity are displayed in this subsection; the scalar-balance and shape-variation terms are ...

    work page

  6. [6]

    Variations of surface viscous and polar dissipation functionals and active work power We next evaluate the variations of the surface viscous dissipation functional and the surface polar dissipation functional, as well as of the active power input. Diffuse-Domain Model— The contributions from surface viscosity and polar flow alignment to the dissipation fu...

    work page

  7. [7]

    Kirby,Micro- and Nanoscale Fluid Mechanics: Transport in Microfluidic Devices(Cambridge University Press, 2010)

    B. Kirby,Micro- and Nanoscale Fluid Mechanics: Transport in Microfluidic Devices(Cambridge University Press, 2010)

    work page 2010

  8. [8]

    Helfrich, Elastic properties of lipid bilayers: Theory and possible experiments, Zeitschrift f¨ ur Naturforschung C28, 693 (1973)

    W. Helfrich, Elastic properties of lipid bilayers: Theory and possible experiments, Zeitschrift f¨ ur Naturforschung C28, 693 (1973)

    work page 1973

  9. [9]

    Arroyo and A

    M. Arroyo and A. DeSimone, Relaxation dynamics of fluid membranes, Physical Review E79, 031915 (2009)

    work page 2009

  10. [10]

    Dziuk and C

    G. Dziuk and C. M. Elliott, Finite element methods for surface pdes, Acta Numerica22, 289 (2013)

    work page 2013

  11. [11]

    R¨ atz and A

    A. R¨ atz and A. Voigt, PDEs on Surfaces: A Diffuse Interface Approach, Communications in Mathematical Sciences4, 575 (2006)

    work page 2006

  12. [12]

    C. M. Elliott and B. Stinner, Analysis of a diffuse interface approach to an advection diffusion equation on a moving surface, Mathematical Models and Methods in Applied Sciences19, 787 (2009)

    work page 2009

  13. [13]

    X. Li, J. Lowengrub, A. R¨ atz, and A. Voigt, Solving pdes in complex geometries: A diffuse domain approach, Communi- cations in Mathematical Sciences7, 81 (2009)

    work page 2009

  14. [14]

    F. C. Keber, E. Loiseau, T. Sanchez, S. J. DeCamp, L. Giomi, M. J. Bowick, M. C. Marchetti, Z. Dogic, and A. R. Bausch, Topology and dynamics of active nematic vesicles, Science345, 1135 (2014)

    work page 2014

  15. [15]

    L. A. Hoffmann, L. N. Carenza, J. Eckert, and L. Giomi, Theory of defect-mediated morphogenesis, Science Advances8, eabk2712 (2022)

    work page 2022

  16. [16]

    Z. Wang, M. C. Marchetti, and F. Brauns, Patterning of morphogenetic anisotropy fields, Proceedings of the National Academy of Sciences120, e2220167120 (2023)

    work page 2023

  17. [17]

    Salbreux and F

    G. Salbreux and F. J¨ ulicher, Mechanics of active surfaces, Physical Review E96, 032404 (2017)

    work page 2017

  18. [18]

    Khoromskaia and G

    D. Khoromskaia and G. Salbreux, Active morphogenesis of patterned epithelial shells, eLife12, e75878 (2023)

    work page 2023

  19. [19]

    G. Hou, J. Wang, and A. Layton, Numerical methods for fluid-structure interaction—a review, Commun. Comput. Phys. 12, 337 (2012)

    work page 2012

  20. [20]

    B. E. Griffith and N. A. Patankar, Immersed methods for fluid–structure interaction, Annual review of fluid mechanics52, 421 (2020)

    work page 2020

  21. [21]

    Feng and E

    Z.-G. Feng and E. E. Michaelides, Heat transfer in particulate flows with direct numerical simulation (dns), International Journal of Heat and Mass Transfer52, 777 (2009)

    work page 2009

  22. [22]

    M. S. Shadloo, G. Oger, and D. Le Touz´ e, Smoothed particle hydrodynamics method for fluid flows, towards industrial applications: Motivations, current state, and challenges, Computers & Fluids136, 11 (2016)

    work page 2016

  23. [23]

    Hoogerbrugge and J

    P. Hoogerbrugge and J. Koelman, Simulating microscopic hydrodynamic phenomena with dissipative particle dynamics, Europhysics Letters19, 155 (1992)

    work page 1992

  24. [24]

    Jacqmin, Contact-line dynamics of a diffuse fluid interface, Journal of Fluid Mechanics402, 57 (2000)

    D. Jacqmin, Contact-line dynamics of a diffuse fluid interface, Journal of Fluid Mechanics402, 57 (2000)

    work page 2000

  25. [25]

    Mokbel, H

    D. Mokbel, H. Abels, and S. Aland, A phase-field model for fluid–structure interaction, Journal of Computational Physics 372, 823 (2018)

    work page 2018

  26. [26]

    Hong and Q

    Q. Hong and Q. Wang, Thermodynamically consistent hybrid computational models for fluid-particle interactions, Journal of Computational Physics513, 113147 (2024)

    work page 2024

  27. [27]

    Glowinski, T.-W

    R. Glowinski, T.-W. Pan, T. I. Hesla, D. D. Joseph, and J. Periaux, A fictitious domain approach to the direct numerical simulation of incompressible viscous flow past moving rigid bodies: application to particulate flow, Journal of computational physics169, 363 (2001)

    work page 2001

  28. [28]

    Nakayama and R

    Y. Nakayama and R. Yamamoto, Simulation method to resolve hydrodynamic interactions in colloidal dispersions, Physical Review E71, 036707 (2005)

    work page 2005

  29. [29]

    Yamamoto, J

    R. Yamamoto, J. J. Molina, and Y. Nakayama, Smoothed profile method for direct numerical simulations of hydrodynam- ically interacting particles, Soft Matter17, 4226 (2021)

    work page 2021

  30. [30]

    Tanaka and T

    H. Tanaka and T. Araki, Simulation method of colloidal suspensions with hydrodynamic interactions: Fluid particle dynamics, Phys. Rev. Lett.85, 1338 (2000)

    work page 2000

  31. [31]

    Kockelkoren, H

    J. Kockelkoren, H. Levine, and W.-J. Rappel, Computational approach for modeling intra-and extracellular dynamics, Physical Review E68, 037702 (2003)

    work page 2003

  32. [32]

    F. H. Fenton, E. M. Cherry, A. Karma, and W.-J. Rappel, Modeling wave propagation in realistic heart geometries using the phase-field method, Chaos15(2005)

    work page 2005

  33. [33]

    Levine and W.-J

    H. Levine and W.-J. Rappel, Membrane-bound turing patterns, Physical Review E72, 061912 (2005). 36

    work page 2005

  34. [34]

    Shao, W.-J

    D. Shao, W.-J. Rappel, and H. Levine, Computational model for cell morphodynamics, Physical review letters105, 108104 (2010)

    work page 2010

  35. [35]

    B. A. Camley, Y. Zhao, B. Li, H. Levine, and W.-J. Rappel, Periodic migration in a physical model of cells on micropatterns, Physical review letters111, 158102 (2013)

    work page 2013

  36. [36]

    Deckelnick and V

    K. Deckelnick and V. Styles, Stability and error analysis for a diffuse interface approach to an advection–diffusion equation on a moving surface, Numerische Mathematik139, 709 (2018)

    work page 2018

  37. [37]

    Lerv˚ ag and J

    K. Lerv˚ ag and J. Lowengrub, Analysis of the diffuse-domain method for solving pdes in complex geometries, Communica- tions in Mathematical Sciences13, 1473 (2015)

    work page 2015

  38. [38]

    Wang and Q

    X. Wang and Q. Du, Modelling and simulations of multi-component lipid membranes and open membranes via diffuse interface approaches, Journal of Mathematical Biology56, 347 (2008)

    work page 2008

  39. [39]

    J. S. Lowengrub, A. R¨ atz, and A. Voigt, Phase-field modeling of the dynamics of multicomponent vesicles: Spinodal decomposition, coarsening, budding, and fission, Physical Review E79, 031926 (2009)

    work page 2009

  40. [40]

    Aland, S

    S. Aland, S. Egerer, J. Lowengrub, and A. Voigt, Diffuse interface models of locally inextensible vesicles in a viscous fluid, Journal of Computational Physics277, 32 (2014)

    work page 2014

  41. [41]

    Z. Wen, N. Valizadeh, T. Rabczuk, and X. Zhuang, Hydrodynamics of multicomponent vesicles: A phase-field approach, Computer Methods in Applied Mechanics and Engineering432, 117390 (2024)

    work page 2024

  42. [42]

    K. E. Teigen, X. Li, J. Lowengrub, F. Wang, and A. Voigt, A diffuse-interface approach for modelling transport, diffusion and adsorption/desorption of material quantities on a deformable interface, Communications in Mathematical Sciences7, 1009 (2009)

    work page 2009

  43. [43]

    Aland, J

    S. Aland, J. Lowengrub, and A. Voigt, Two-phase flow in complex geometries: A diffuse domain approach, Computer Modeling in Engineering & Sciences: CMES57, 77 (2010)

    work page 2010

  44. [44]

    Nestler and A

    M. Nestler and A. Voigt, A diffuse interface approach for vector-valued PDEs on surfaces, Communications in Mathematical Sciences22, 1749 (2024)

    work page 2024

  45. [45]

    Kloppe and S

    M. Kloppe and S. Aland, A phase-field model of elastic and viscoelastic surfaces in fluids, Computer Methods in Applied Mechanics and Engineering428, 117090 (2024)

    work page 2024

  46. [46]

    Chen and J

    Y. Chen and J. Lowengrub, Tumor growth in complex, evolving microenvironmental geometries: a diffuse domain approach, Journal of Theoretical Biology361, 14 (2015)

    work page 2015

  47. [47]

    M. Gao, Z. Li, and X. Xu, Simulation method of microscale fluid-structure interactions: Diffuse-resistance-domain ap- proach, Physical Review Fluids10, 094001 (2025)

    work page 2025

  48. [48]

    S. L. Veatch and S. L. Keller, Separation of liquid phases in giant vesicles of ternary mixtures of phospholipids and cholesterol, Biophysical Journal85, 3074 (2003)

    work page 2003

  49. [49]

    Nitschke and A

    I. Nitschke and A. Voigt, Active nematodynamics on deformable surfaces, Proceedings of the Royal Society A481, 20240380 (2025)

    work page 2025

  50. [50]

    Qian, X.-P

    T. Qian, X.-P. Wang, and P. Sheng, A variational approach to moving contact line hydrodynamics, J. Fluid Mech.564, 333 (2006)

    work page 2006

  51. [51]

    Doi and A

    M. Doi and A. Onuki, Dynamic coupling between stress and composition in polymer solutions and blends, Journal de Physique II2, 1631 (1992)

    work page 1992

  52. [52]

    Doi, Onsager’s Variational Principle in Soft Matter, Journal of Physics: Condensed Matter23, 284118 (2011)

    M. Doi, Onsager’s Variational Principle in Soft Matter, Journal of Physics: Condensed Matter23, 284118 (2011)

    work page 2011

  53. [53]

    Doi,Soft Matter Physics(Oxford University Press, 2013)

    M. Doi,Soft Matter Physics(Oxford University Press, 2013)

    work page 2013

  54. [54]

    Doi, Onsager principle as a tool for approximation, Chinese Physics B24, 020505 (2015)

    M. Doi, Onsager principle as a tool for approximation, Chinese Physics B24, 020505 (2015)

    work page 2015

  55. [55]

    Doi, Onsager principle in polymer dynamics, Prog

    M. Doi, Onsager principle in polymer dynamics, Prog. Polym. Sci.112, 101339 (2021)

    work page 2021

  56. [56]

    Xu and T

    X. Xu and T. Qian, Hydrodynamic boundary conditions derived from onsager’s variational principle, Procedia IUTAM 20, 144 (2017)

    work page 2017

  57. [57]

    H. Wang, T. Qian, and X. Xu, Onsager’s variational principle in active soft matter, Soft Matter17, 3634 (2021)

    work page 2021

  58. [58]

    H. Wang, B. Zou, J. Su, D. Wang, and X. Xu, Variational methods and deep ritz method for active elastic solids, Soft Matter18, 6015 (2022)

    work page 2022

  59. [59]

    W. Yu, T. Qian, and Q. Wang, Onsager principle-based domain embedding and numerical approximations for Allen–Cahn- type models, Journal of Computational Physics543, 114403 (2025)

    work page 2025

  60. [60]

    W. Yu, Q. Wang, Z. Zhang, and T. Qian, Onsager principle-based domain embedding for thermodynamically consistent Cahn–Hilliard model in arbitrary domain, arXiv preprint arXiv:2508.06830 (2025)

    work page arXiv 2025

  61. [61]

    Aland and C

    S. Aland and C. Wohlgemuth, A phase-field model for active contractile surfaces, arXiv preprint arXiv:2306.16796 (2023)

    work page arXiv 2023

  62. [62]

    Aris,Vectors, Tensors and the Basic Equations of Fluid Mechanics(Dover Publications, New York, 1989) unabridged, corrected republication of the 1962 original

    R. Aris,Vectors, Tensors and the Basic Equations of Fluid Mechanics(Dover Publications, New York, 1989) unabridged, corrected republication of the 1962 original

    work page 1989

  63. [63]

    Federer,Geometric Measure Theory, Die Grundlehren der mathematischen Wissenschaften, Vol

    H. Federer,Geometric Measure Theory, Die Grundlehren der mathematischen Wissenschaften, Vol. 153 (Springer, New York, 1969)

    work page 1969

  64. [64]

    S. M. Allen and J. W. Cahn, A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening, Acta Metallurgica27, 1085 (1979)

    work page 1979

  65. [65]

    J. W. Cahn and J. E. Hilliard, Free energy of a nonuniform system. I. interfacial free energy, The Journal of Chemical Physics28, 258 (1958)

    work page 1958

  66. [66]

    Seifert, Curvature-induced lateral phase segregation in two-component vesicles, Physical Review Letters70, 1335 (1993)

    U. Seifert, Curvature-induced lateral phase segregation in two-component vesicles, Physical Review Letters70, 1335 (1993)

    work page 1993

  67. [67]

    J¨ ulicher and R

    F. J¨ ulicher and R. Lipowsky, Domain-induced budding of vesicles, Physical Review Letters70, 2964 (1993)

    work page 1993

  68. [68]

    J¨ ulicher and R

    F. J¨ ulicher and R. Lipowsky, Shape transformations of vesicles with intramembrane domains, Physical Review E53, 2670 (1996)

    work page 1996

  69. [69]

    Taniguchi, Shape deformation and phase separation dynamics of two-component vesicles, Physical Review Letters76, 37 4444 (1996)

    T. Taniguchi, Shape deformation and phase separation dynamics of two-component vesicles, Physical Review Letters76, 37 4444 (1996)

    work page 1996

  70. [70]

    Q. Du, C. Liu, R. Ryham, and X. Wang, Modeling the spontaneous curvature effects in static cell membrane deformations by a phase field formulation, COMMUNICATIONS ON PURE AND APPLIED ANALYSIS4, 537 (2005)

    work page 2005

  71. [71]

    Campelo and A

    F. Campelo and A. Hernandez-Machado, Dynamic model and stationary shapes of fluid vesicles, The European Physical Journal E20, 37 (2006)

    work page 2006

  72. [72]

    M. D. Rueda-Contreras, A. F. Gallen, J. R. Romero-Arias, A. Hernandez-Machado, and R. A. Barrio, On gaussian curvature and membrane fission, Scientific Reports11, 9562 (2021)

    work page 2021

  73. [73]

    Caetano, C

    D. Caetano, C. M. Elliott, M. Grasselli, and A. Poiatti, Multi-component phase separation and small deformations of a spherical biomembrane, Calculus of Variations and Partial Differential Equations65, 72 (2026)

    work page 2026

  74. [74]

    J. W. Barrett, H. Garcke, and R. N¨ urnberg, Gradient flow dynamics of two-phase biomembranes: Sharp interface variational formulation and finite element approximation, SMAI Journal of Computational Mathematics4, 151 (2018)

    work page 2018

  75. [75]

    Winter, Y

    A. Winter, Y. Liu, A. Ziepke, G. Dadunashvili, and E. Frey, Phase separation on deformable membranes: Interplay of mechanical coupling and dynamic surface geometry, Physical Review E111, 044405 (2025)

    work page 2025

  76. [76]

    T. Qian, C. Qiu, and P. Sheng, A scaling approach to the derivation of hydrodynamic boundary conditions, J. Fluid Mech. 611, 333–364 (2008)

    work page 2008

  77. [77]

    Seifert, Configurations of fluid membranes and vesicles, Advances in Physics46, 13 (1997)

    U. Seifert, Configurations of fluid membranes and vesicles, Advances in Physics46, 13 (1997)

    work page 1997

  78. [78]

    L. E. Scriven, Dynamics of a fluid interface: Equation of motion for newtonian surface fluids, Chemical Engineering Science 12, 98 (1960)

    work page 1960

  79. [79]

    Evans and W

    E. Evans and W. Rawicz, Entropy-driven tension and bending elasticity in condensed-fluid membranes, Physical Review Letters64, 2094 (1990)

    work page 2094

  80. [80]

    Rawicz, K

    W. Rawicz, K. C. Olbrich, T. McIntosh, D. Needham, and E. Evans, Effect of chain length and unsaturation on elasticity of lipid bilayers, Biophysical Journal79, 328 (2000)

    work page 2000

Showing first 80 references.