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A mass-conserving contact line treatment for second-order conservative phase field methods based on the generalized Navier boundary condition

T0 review · 3 major / 6 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read A no-flux phase-field boundary condition plus a GNBC slip model gives second-order conservative phase-field methods a mass-conserving, physically accurate contact line treatment.

desk verdict Solid, carefully argued methods paper that fills a real gap in CDI contact line treatments; deserves serious review despite a scoped validity claim and some missing reproducibility details. read the letter →

arxiv 2412.16843 v1 pith:5XOMVY7T submitted 2024-12-22 physics.flu-dyn

Reed L. Brown , Shahab Mirjalili , Makrand A. Khanwale , Ali Mani This is my paper · ORCID
classification physics.flu-dyn
keywords contactlinegeneralizedNavierboundaryconditionconservativediffuseinterfacemodelphase-fieldmethodsmassconservationspuriousslipcurvaturecorrectiontwo-phaseCouetteflow
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

The paper proposes a way to simulate contact lines—where a fluid-fluid interface meets a solid wall—with second-order conservative phase field methods that otherwise cannot enforce a separate contact-angle boundary condition. The treatment uses a no-flux boundary condition on the phase field, which conserves mass exactly, and a slip boundary condition for velocity built on the generalized Navier boundary condition, which supplies the contact-line physics. To make the slip model accurate, the authors reformulate derivatives in terms of an approximate signed distance function and add a curvature correction, yielding Model 4A (Equation 30). They validate it on an equilibrium drop that relaxes to the correct contact angle and on two-phase Couette flow that matches molecular dynamics reference data, and they show spurious slip at equilibrium converges away under simultaneous mesh and interface-thickness refinement.

What carries the argument

The load-bearing object is the contact-line slip velocity $u^Y_{\mathrm{slip}}$ in the generalized Navier boundary condition, specifically the version the authors call Model 4A (Equation 30). It is built from three ingredients: (i) the no-flux phase-field boundary condition that conserves mass; (ii) a transformation to the approximate signed distance function $\psi = \epsilon \ln(\phi/(1-\phi))$ to reduce discretization error; and (iii) the curvature correction $(1+\kappa_p\psi)$ derived from the identity $\cos(\theta_{\phi_0}) = (1 + \kappa_p \psi)\cos(\theta_{0.5})$ for concentric circular contours, which eliminates spurious slip at equilibrium. A supporting identity is the alternative chemical potential $\mu = 6\sigma\left[\phi(1-\phi)(1-2\phi)(1-|\nabla\psi|^2)/\epsilon - \phi(1-\phi)\nabla^2\psi\right]$, which improves accuracy near contact lines.

What would settle it

Run a prescribed equilibrium drop with Model 4A but refine the mesh while holding the interface thickness fixed: if $\max|u^Y_{\mathrm{slip}}|$ does not shrink, the claim that the curvature correction eliminates model error fails. Alternatively, compare Model 4A and Model 4B in a high-capillary-number moving contact-line test where the interface visibly departs from phase-field equilibrium; if 4B matches reference data better, the equilibrium assumption that selected version A is violated.

Watch

Extended reading notes

Core claim

The central claim is that contact line dynamics can be incorporated into the second-order conservative diffuse interface (CDI) model without sacrificing mass conservation and without a second phase-field boundary condition: the no-flux boundary condition (Equation 8) enforces mass conservation, while the generalized Navier boundary condition, adapted to a one-sided discrete treatment, models both static and dynamic contact angles through the uncompensated Young's stress. The paper identifies two sources of error in the naive GNBC adaptation—discretization error from differentiating the sharply varying phase field, and model error from assuming every contour meets the wall at the equilibrium angle—and removes them by writing derivatives in terms of the smooth function $\psi = \epsilon \ln(\phi/(1-\phi))$ and multiplying the $\cos(\theta_{\mathrm{eq}})$ term by $(1 + \kappa_p \psi)$, the curvature correction derived from concentric circular contours at phase-field equilibrium. The resulting Model 4A exhibits vanishing spurious slip on a prescribed equilibrium drop, converges under refinement, and reproduces molecular dynamics reference shapes for moving contact lines in symmetric and asymmetric Couette flow.

Load-bearing premise

The slip model's accuracy rests on the assumption that the diffuse interface stays close to phase-field equilibrium during dynamic contact-line motion, so that the relation $\epsilon|\nabla\phi| = \phi(1-\phi)$ and the curvature-corrected contour identity remain valid; the authors explicitly choose version A over version B on this basis.

Editorial extensions

If this is right

  • Second-order conservative phase field solvers gain a contact-line treatment that conserves mass locally and exactly, without Lagrange-multiplier compensation over the whole domain.
  • The treatment resolves the shear-stress singularity at the contact line through a slip velocity proportional to the uncompensated Young's stress, so the dynamic contact angle can deviate from equilibrium as physics requires.
  • Because the slip model is designed to vanish when the contact angle equals its equilibrium value, static contact-line problems reach steady states with correct contact-angle geometry.
  • If the convergence results hold, practitioners can choose mesh and interface-thickness parameters with the knowledge that spurious slip and wetted-length errors diminish under simultaneous refinement.
  • The same $\psi$-transformation and curvature-correction ideas apply to other second-order phase-field models and could be adapted to the Cahn-Hilliard GNBC to reduce spurious slip at equilibrium.

Reading between the lines

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

  • Because the paper restricts demonstrations to two dimensions, a natural next step is a three-dimensional extension in which the planar curvature $\kappa_p$ generalizes to the in-plane principal curvature; the contour-relation argument suggests the same correction form should hold along each wall-tangent direction.
  • Model 4A and Model 4B may separate in regimes where the interface deviates from phase-field equilibrium; a head-to-head test at high capillary numbers would show whether the equilibrium assumption or the $|\nabla\psi|$-based discretization error dominates.
  • The local, no-flux mass conservation may be a practical advantage for simulations with phase change or scalar transport, where a global Lagrange-multiplier correction could interfere with source terms; this is an untested consequence of the paper's design.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 6 minor

Summary. This paper proposes a boundary treatment for contact lines in the conservative diffuse interface (CDI) model, a second-order conservative Allen-Cahn-type phase field method. The treatment combines a no-flux wall boundary condition for the phase field (Eq. 8), which conserves the total amount of φ and hence mass, with a slip boundary condition for the velocity based on the generalized Navier boundary condition (GNBC). Because the CDI operator is second order, only one phase-field boundary condition is available, so mass conservation is enforced at the wall and contact line physics is encoded in the slip law. The slip is decomposed into a viscous Navier-slip part and a Young's-slip part whose wall integral equals (σ/β)(cos θ − cos θ_eq) (Eq. 19). The authors present four generations of Young's-slip models (1A/B through 4A/B), introducing (i) a transformation to an approximate signed-distance variable ψ = ε ln(φ/(1−φ)) that reduces discretization error, and (ii) a curvature correction (1 + κ_p ψ) that eliminates model error for a prescribed equilibrium circular drop (Eq. 27). Using exact symbolic evaluation of the slip on a prescribed equilibrium phase field, they show that only Models 4A/B eliminate spurious slip and converge under joint mesh/interface-thickness refinement (Fig. 7). Model 4A (Eq.

Significance. If the results hold, this is a solid, well-scoped contribution that fills a genuine gap: second-order conservative phase field methods have lacked a mass-conserving contact line treatment that allows dynamic contact angles, because only one phase-field boundary condition is available. The analysis in Section 3.1 is the paper's strongest feature: the decomposition of spurious slip into discretization error and model error, evaluated exactly with symbolic math for a prescribed equilibrium phase field, provides a clean and checkable way to compare slip models, and the transformation to the signed-distance variable ψ (Eq. 24) is practically valuable and transferable. The curvature correction (Eq. 27) follows from explicit contour geometry, and the convergence evidence in Figures 7, 9, and 10 supports the claim that Models 4A/B eliminate spurious slip under joint mesh/interface refinement. The moving-contact-line validation is anchored by independent molecular-dynamics reference data [42], which is not self-referential.

major comments (3)
  1. [Section 5 (both test cases); Section 2, Eqs. (8)–(9)] The title-level claim of mass conservation is never verified numerically. The continuous argument in Section 2 correctly shows that the no-flux boundary condition (Eq. 8) together with no-penetration (Eq. 9) imply global conservation of ∫φ; however, the discrete operators at the wall use one-sided stencils and first-order linear extrapolation, so discrete mass conservation does not automatically follow from the continuous argument. Neither the equilibrium drop (Section 5.1) nor the Couette flow (Section 5.2) reports any mass-conservation diagnostic, such as the global mass error as a function of time, even though both are inexpensive to instrument. I request that the authors report the global mass error over time for the fully coupled simulations to substantiate the central claim made in the title.
  2. [Section 3 (choice of Model 4A) and Section 5.2] The accuracy claim for moving contact lines rests on the phase-field equilibrium assumption, which is asserted but not quantified. The paper states in Section 3: 'The choice of version A over version B is based on the assumption that the interface remains close to phase field equilibrium. This assumption appears to be valid in our simulations.' The curvature correction of Eq. (27) and the elimination of model error in Section 3.1 are derived for a prescribed equilibrium circular drop, so their validity in dynamic simulations is not established by construction; moreover, the equilibrium drop test of Section 5.1 is partly a consistency check, since its terminal state is the equilibrium configuration for which the slip model was designed to yield zero spurious slip. The moving-contact-line validation in Section 5.2 compares against molecular-dynamics data only visually (Figures 11–12), at a single resolution, and without reporting how close the dynamic interface remains to phase-field equilibrium (e.g., the residual of ε|∇φ| = φ(1−φ)) or the dynamic contact angle. I request: (a) a quantitative measure of interface deviation from equilibrium in the Couette simulations; (b) a quantitative error metric for the comparison with the MD reference data; and (c) a mesh/interface-thickness resolution study for at least one Couette case. These additions would convert the claim that the treatment 'accurately models moving contact line physics' into a demonstrated one and would also test the integrated-slip relation of Eq. (19) dynamically.
  3. [Figures 7, 9, 10; Section 4] The convergence studies refine the mesh size Δ and the interface thickness ε simultaneously according to ε ∼ Δ^{2/3}, so the reported rates (approximately N_x^{-1/2} for the maximum spurious slip and N_x^{-1} for the wetted length error) conflate the reduction of spatial discretization error with the approach to the sharp-interface limit. Because the near-wall phase-field quantities are computed with first-order extrapolation, it is not clear which part of the observed rates is attributable to the spatial discretization. A complementary study at fixed ε/Δ, or at fixed ε, would separate these effects and clarify the mechanism of convergence claimed in Section 3.1.
minor comments (6)
  1. [Section 2, after Eq. (5)] The phrase 'This form is be obtained by computing' should read 'This form can be obtained by computing'.
  2. [Section 3.1.2, Figure 5] The sign convention for ψ = ε ln(φ/(1−φ)) should be stated explicitly: ψ is positive inside the drop, which makes the inequality in Eq. (27) immediately interpretable.
  3. [Figures 7, 9, 10] The specific grid resolutions N_x are not reported in the convergence studies; including them would improve reproducibility.
  4. [Equation (13)] The curvature estimate is presented without derivation; one line showing that κ reduces to the geometric curvature 1/r for a circular equilibrium interface with |∇ψ| = 1 would make the connection to the contour geometry of Eq. (27) immediate.
  5. [References] Reference [42] is cited as an arXiv preprint from 2005; since the published article in J. Fluid Mech. 564 (2006) 333–360, listed here as [47], appears to contain the same molecular-dynamics data, the published version should be cited.
  6. [Section 5.2, Eq. (32)] It is not stated whether the tanh initial interface satisfies the discrete no-flux boundary condition at the walls to machine precision; a brief compatibility statement would be useful.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the moving-contact-line claim is checked against external MD data, and the equilibrium zero-slip property is a stated design goal rather than a hidden prediction.

full rationale

The paper's central new claim is that the no-flux phase-field BC plus the GNBC-based slip treatment (Model 4A, Eq. 30) conserves mass and captures static and moving contact lines. The moving-contact-line part is validated by comparing two-phase Couette flow against molecular dynamics reference data from Qian, Wang, and Sheng [42], with wall-fluid friction parameters taken from that external study; no constant is fitted in the present work to the MD contours. The self-citations ([20], [28], [51], [52]) are prior-model foundations and an early-version note; they do not function as a uniqueness theorem or as a hidden ansatz that forces the result. The curvature correction (Sec. 3.1.2, Eqs. 27-31) is transparently constructed to make the equilibrium profile produce zero spurious slip; the equilibrium-drop test is therefore a consistency check of the discrete solver against the design target, not an independent empirical prediction, but the paper does not claim otherwise. The acknowledged assumption in Sec. 3 that the interface stays close to phase-field equilibrium (choice of version A over B) is an unquantified limitation and a correctness risk for strongly non-equilibrium flows, but it is not a circular step. The derivation chain is self-contained: the model form is derived from the GNBC integral relation (Eq. 19) and the geometry of a diffuse circular contour (Eq. 27), and the dynamic validation is externally grounded.

Assumptions & free parameters 0 free parameters · 8 assumptions · 0 invented entities

The central claim introduces no new physical parameters or entities. The curvature correction is a geometric identity for phase-field equilibrium, not a fitted constant. The main fragility is the phase-field equilibrium assumption underlying Model 4A, and the GNBC hypothesis is carried over from prior molecular dynamics work.

assumptions (8)
  • domain assumption The incompressible Navier-Stokes mixture momentum equation (Eq. 2) uses density and viscosity linear in phi and includes the regularization mass flux S = (rho1 - rho2)R.
    Adopted from [28]; central to the two-phase flow solver.
  • domain assumption The CDI phase-field equation (Eq. 1) with the no-flux wall boundary condition (Eq. 8) conserves total phi, and therefore mass.
    Stated in Section 2; discrete conservation is asserted but not proven.
  • domain assumption The generalized Navier boundary condition hypothesis beta u_slip = sigma_v + sigma_Y_tilde (Eq. 14) holds.
    This hypothesis comes from the molecular dynamics analysis in [41] and is the physical basis of the boundary condition.
  • ad hoc to paper The phase-field equilibrium relation epsilon |grad phi| = phi(1 - phi) holds.
    Used to derive the version A models in Section 3; the authors flag this assumption.
  • standard math For a curved diffuse interface at equilibrium, the contact angle of an arbitrary contour obeys cos(theta_phi0) = (1 + kappa_p psi) cos(theta_0.5) (Eq. 27).
    Exact for concentric circular arcs (Figure 5) and assumed valid locally for general interfaces.
  • domain assumption In 2D, the in-plane curvature kappa_p equals the full EB-model curvature given by Eq. 13.
    From [52], this equivalence underlies the curvature estimate.
  • ad hoc to paper Near-wall derivatives of phi and psi computed with one-sided stencils and first-order linear extrapolation are sufficiently accurate.
    Needed because the second-order CDI model cannot support a second boundary condition; the convergence results support but do not prove this.
  • domain assumption The wall-fluid friction beta is given by the linear interpolation beta = (beta1 - beta2) phi + beta2.
    This follows common practice and mirrors the interpolation of density and viscosity.

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Cite this review

Pith. "Pith review of A mass-conserving contact line treatment for second-order conservative phase field methods based on the generalized Navier boundary condition." pith.science (2026). https://pith.science/paper/5XOMVY7T

@misc{pith2026241216843,
  author       = {Pith},
  title        = {Pith review of: A mass-conserving contact line treatment for second-order conservative phase field methods based on the generalized Navier boundary condition},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/5XOMVY7T}},
  note         = {Machine review of arXiv:2412.16843}
}
read the original abstract

A mass-conserving contact line treatment for second-order conservative phase field methods is presented and applied to the conservative diffuse interface (CDI) model. The treatment centers on a no-flux boundary condition for the phase field along with a slip boundary condition for the velocity that is based on the generalized Navier boundary condition (GNBC). Since the CDI model is a second-order partial differential equation, it does not permit a second (contact angle) boundary condition, in contrast to the popular fourth-order Cahn-Hilliard model. As such, we use one-sided stencils and extrapolations from the interior of the domain to compute phase-field-related quantities on and near the wall. Additionally, we propose novel modifications to the GNBC on the continuous and discrete levels that reduce spurious slip velocity when the contact angle achieves its equilibrium value. The proposed treatment is validated with the equilibrium drop and two-phase Couette flow test cases.

Figures

Figures reproduced from arXiv: 2412.16843 by the authors.

Figure 1
Figure 1. Schematic of the surface tension forces acting on a contact line, in a sharp interface representation. The [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. A prescribed equilibrium phase field ϕ(x, y) representing a drop that forms a θ = 60◦ contact angle with the wall. The drop is circular, corresponding to capillary equilibrium, and the interface is at phase field equilibrium. As a result, the contours of the phase field are concentric circular arcs. The thick black line is the ϕ = 0.5 contour, while the inner dashed line is the ϕ = 0.9 contour and the outer dashed l… view at source ↗
Figure 3
Figure 3. Contact line slip u Y slip(x), computed using Model 1A, as a function of x, for the prescribed equilibrium phase field. The two vertical lines mark the locations where the ϕ = 0.5 contour intersects the wall. There is a significant discrepancy between numerical and exact, and the exact slip is non-zero, indicating both discretization and model error. The gradients of ϕ can then be written in terms of the gradient of… view at source ↗
Figures from the paper (9 more)
Figure 4
Figure 4. Figure 4: Blue: Contact angle θ(x) = cos−1 ((−∂ϕ/∂y)/|∇ϕ|) as a function of wall position x, for the prescribed equilibrium phase field. The contact angle is calculated exactly using symbolic math tools. The horizontal line marks θ = θeq = 60◦ . Orange: Phase field ϕ(x) as a fun…
Figure 5
Figure 5. Figure 5: Schematic of how the contact angle of an arbitrary contour [PITH_FULL_IMAGE:figures/full_fig_p010_5.png]
Figure 6
Figure 6. Figure 6: Contact line slip u Y slip(x), computed using Model 4A, as a function of x. The two vertical lines mark the locations where the ϕ = 0.5 contour intersects the wall. Compared to [PITH_FULL_IMAGE:figures/full_fig_p012_6.png]
Figure 7
Figure 7. Figure 7: Convergence plot of maximum spurious slip [PITH_FULL_IMAGE:figures/full_fig_p013_7.png]
Figure 8
Figure 8. Figure 8: Time series of the evolution of the phase field for the equilibrium drop test case. The black lines mark the [PITH_FULL_IMAGE:figures/full_fig_p015_8.png]
Figure 9
Figure 9. Figure 9: Convergence plot of maximum spurious slip [PITH_FULL_IMAGE:figures/full_fig_p016_9.png]
Figure 10
Figure 10. Figure 10: Convergence plot of the relative wetted length error [PITH_FULL_IMAGE:figures/full_fig_p016_10.png]
Figure 11
Figure 11. Figure 11: Time series of the evolution of the phase field for the symmetric two-phase Couette flow test case. The [PITH_FULL_IMAGE:figures/full_fig_p017_11.png]
Figure 12
Figure 12. Figure 12: Time series of the evolution of the phase field for the asymmetric two-phase Couette flow test case. The [PITH_FULL_IMAGE:figures/full_fig_p017_12.png]

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Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

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    physics.flu-dyn 2026-07 conditional novelty 5.0 of 10

    A local geometric ghost-node wetting boundary condition for conservative Allen-Cahn lattice Boltzmann simulations imposes contact angles accurately and matches droplet spreading, impact, and orifice-passage benchmarks.

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Works this paper leans on

61 extracted references · 56 canonical work pages · cited by 1 Pith paper

  1. [42]

    Molecular hydrodynamics of the moving contact line in two-phase immiscible flows

    T. Qian, X.-P. Wang, P. Sheng, Molecular hydrodynamics of the moving contact line in two- phase immiscible flows, arXiv preprint cond-mat/0510403 (2005)

    work page Pith review arXiv 2005

  2. [1]

    J. H. Snoeijer, B. Andreotti, Moving contact lines: scales, regimes, and dynamical transitions, Annual review of fluid mechanics 45 (2013) 269–292

    work page 2013

  3. [2]

    Y. Sui, H. Ding, P. D. Spelt, Numerical simulations of flows with moving contact lines, Annual Review of Fluid Mechanics 46 (2014) 97–119

    work page 2014

  4. [3]

    Mohammad Karim, A review of physics of moving contact line dynamics models and its applications in interfacial science, Journal of Applied Physics 132 (2022)

    A. Mohammad Karim, A review of physics of moving contact line dynamics models and its applications in interfacial science, Journal of Applied Physics 132 (2022)

    work page 2022

  5. [4]

    C. Huh, L. E. Scriven, Hydrodynamic model of steady movement of a solid/liquid/fluid contact line, Journal of colloid and interface science 35 (1971) 85–101

    work page 1971

  6. [5]

    E. B. Dussan V., The moving contact line: the slip boundary condition, Journal of Fluid Mechanics 77 (1976) 665–684. doi:10.1017/S0022112076002838

    work page doi:10.1017/s0022112076002838 1976

  7. [6]

    Dussan, On the spreading of liquids on solid surfaces: static and dynamic contact lines, Annual Review of Fluid Mechanics 11 (1979) 371–400

    E. Dussan, On the spreading of liquids on solid surfaces: static and dynamic contact lines, Annual Review of Fluid Mechanics 11 (1979) 371–400

    work page 1979

  8. [7]

    Hocking, A moving fluid interface

    L. Hocking, A moving fluid interface. part 2. the removal of the force singularity by a slip flow, Journal of Fluid Mechanics 79 (1977) 209–229

    work page 1977

Show all 61 references
  1. [8]

    C. Huh, S. Mason, The steady movement of a liquid meniscus in a capillary tube, Journal of fluid mechanics 81 (1977) 401–419

    1977

  2. [9]

    M.-Y. Zhou, P. Sheng, Dynamics of immiscible-fluid displacement in a capillary tube, Physical Review Letters 64 (1990) 882. 18

    1990

  3. [10]

    Mirjalili, S

    S. Mirjalili, S. S. Jain, M. Dodd, Interface-capturing methods for two-phase flows: An overview and recent developments, Center for Turbulence Research Annual Research Briefs (2017) 117– 135

    2017

  4. [11]

    McKee, M

    S. McKee, M. F. Tomé, V. G. Ferreira, J. A. Cuminato, A. Castelo, F. Sousa, N. Mangiavacchi, The mac method, Computers & Fluids 37 (2008) 907–930

    2008

  5. [12]

    Tryggvason, R

    G. Tryggvason, R. Scardovelli, S. Zaleski, Direct numerical simulations of gas–liquid multiphase flows, Cambridge university press, 2011

    2011

  6. [13]

    Prosperetti, Computational Methods for Multiphase Flow, Cambridge university press, 2009

    A. Prosperetti, Computational Methods for Multiphase Flow, Cambridge university press, 2009

    2009

  7. [14]

    Scardovelli, S

    R. Scardovelli, S. Zaleski, Interface reconstruction with least-square fit and split eulerian– lagrangian advection, International Journal for Numerical Methods in Fluids 41 (2003) 251– 274

    2003

  8. [15]

    J. E. Pilliod Jr, E. G. Puckett, Second-order accurate volume-of-fluid algorithms for tracking material interfaces, Journal of Computational Physics 199 (2004) 465–502

    2004

  9. [16]

    Zhang, C

    D. Zhang, C. Jiang, D. Liang, Z. Chen, Y. Yang, Y. Shi, A refined volume-of-fluid algorithm for capturing sharp fluid interfaces on arbitrary meshes, Journal of Computational Physics 274 (2014) 709–736

    2014

  10. [17]

    Olsson, G

    E. Olsson, G. Kreiss, S. Zahedi, A conservative level set method for two phase flow ii, Journal of Computational Physics 225 (2007) 785–807

    2007

  11. [18]

    Osher, R

    S. Osher, R. Fedkiw, K. Piechor, Level set methods and dynamic implicit surfaces, Appl. Mech. Rev. 57 (2004) B15–B15

    2004

  12. [19]

    Chiu, Y.-T

    P.-H. Chiu, Y.-T. Lin, A conservative phase field method for solving incompressible two-phase flows, Journal of Computational Physics 230 (2011) 185–204

    2011

  13. [20]

    Mirjalili, C

    S. Mirjalili, C. B. Ivey, A. Mani, A conservative diffuse interface method for two-phase flows with provable boundedness properties, Journal of Computational Physics 401 (2020) 109006

    2020

  14. [21]

    Ten Eikelder, K

    M. Ten Eikelder, K. Van Der Zee, I. Akkerman, D. Schillinger, A unified framework for navier– stokes cahn–hilliard models with non-matching densities, Mathematical Models and Methods in Applied Sciences 33 (2023) 175–221

    2023

  15. [22]

    J. Shen, X. Yang, Decoupled, energy stable schemes for phase-field models of two-phase incom- pressible flows, SIAM Journal on Numerical Analysis 53 (2015) 279–296

    2015

  16. [23]

    M. A. Khanwale, K. Saurabh, M. Fernando, V. M. Calo, H. Sundar, J. A. Rossmanith, B. Gana- pathysubramanian, A fully-coupled framework for solving cahn-hilliard navier-stokes equations: Second-order, energy-stable numerical methods on adaptive octree based meshes, Computer Phy...

    2022

  17. [24]

    M. A. Khanwale, K. Saurabh, M. Ishii, H. Sundar, J. A. Rossmanith, B. Ganapathysubrama- nian, A projection-based, semi-implicit time-stepping approach for the cahn-hilliard navier- stokes equations on adaptive octree meshes, Journal of Computational Physics 475 (2023) 111874. 19

    2023

  18. [25]

    Huang, G

    Z. Huang, G. Lin, A. M. Ardekani, Consistent and conservative scheme for incompressible two-phase flows using the conservative allen-cahn model, Journal of Computational Physics 420 (2020) 109718

    2020

  19. [26]

    S. M. Allen, J. W. Cahn, A microscopic theory for antiphase boundary motion and its appli- cation to antiphase domain coarsening, Acta metallurgica 27 (1979) 1085–1095

    1979

  20. [27]

    Gomez, Y

    H. Gomez, Y. Leng, T. Hu, S. Mukherjee, V. Calo, Phase-field modeling for flow simulation, in: Frontiers in Computational Fluid-Structure Interaction and Flow Simulation: Research from Lead Investigators Under Forty-2023, Springer, 2023, pp. 79–117

    2023

  21. [28]

    Mirjalili, A

    S. Mirjalili, A. Mani, Consistent, energy-conserving momentum transport for simulations of two-phase flows using the phase field equations, Journal of Computational Physics 426 (2021) 109918

    2021

  22. [29]

    Rossi, S

    D. Rossi, S. Di Giorgio, S. Pirozzoli, Comparative analysis of volume of fluid and phase– fieldmethods for numerical simulations of two-phase flows, SSRN preprint: 5037935 (2024). doi:10.2139/ssrn.5037935

    2024 doi

  23. [30]

    S. S. Jain, A. Mani, P. Moin, A conservative diffuse-interface method for compressible two-phase flows, Journal of Computational Physics 418 (2020) 109606

    2020

  24. [31]

    Collis, S

    H. Collis, S. Mirjalili, A. Mani, Diffuse interface treatment in generalized curvilinear coordinates with grid-adapting interface thickness, arXiv preprint arXiv:2411.18770 (2024)

    2024 arXiv

  25. [32]

    Mirjalili, S

    S. Mirjalili, S. S. Jain, A. Mani, A computational model for interfacial heat and mass transfer in two-phase flows using a phase field method, International Journal of Heat and Mass Transfer 197 (2022) 123326

    2022

  26. [33]

    Seppecher, Moving contact lines in the cahn-hilliard theory, International journal of engi- neering science 34 (1996) 977–992

    P. Seppecher, Moving contact lines in the cahn-hilliard theory, International journal of engi- neering science 34 (1996) 977–992

    1996

  27. [34]

    Jacqmin, Contact-line dynamics of a diffuse fluid interface, Journal of fluid mechanics 402 (2000) 57–88

    D. Jacqmin, Contact-line dynamics of a diffuse fluid interface, Journal of fluid mechanics 402 (2000) 57–88

    2000

  28. [35]

    H.-Y. Chen, D. Jasnow, J. Vinals, Interface and contact line motion in a two phase fluid under shear flow, Physical review letters 85 (2000) 1686

    2000

  29. [36]

    P. Yue, C. Zhou, J. J. Feng, Sharp-interface limit of the cahn–hilliard model for moving contact lines, Journal of Fluid Mechanics 645 (2010) 279–294

    2010

  30. [37]

    P. Yue, J. J. Feng, Wall energy relaxation in the cahn–hilliard model for moving contact lines, Physics of Fluids 23 (2011)

    2011

  31. [38]

    Huang, G

    Z. Huang, G. Lin, A. M. Ardekani, Implementing contact angle boundary conditions for second- order phase-field models of wall-bounded multiphase flows, Journal of Computational Physics 471 (2022) 111619

    2022

  32. [39]

    Scapin, A

    N. Scapin, A. Shahmardi, W. Chan, S. Jain, S. Mirjalili, M. Pelanti, L. Brandt, A mass- conserving pressure-based method for two-phase flows with phase change, in: Center for Turbulence Research Proceedings of the Summer Program, 2022, pp. 195–204. URL:https: //web.stanford.ed...

    2022

  33. [40]

    M. Shen, B. Q. Li, Comparison of contact angle models in two-phase flow simulations using a conservative phase field equation, International Journal for Numerical Methods in Fluids (2024)

    2024

  34. [41]

    Qian, X.-P

    T. Qian, X.-P. Wang, P. Sheng, Molecular scale contact line hydrodynamics of immiscible flows, Physical Review E 68 (2003) 016306

    2003

  35. [43]

    Koplik, J

    J. Koplik, J. R. Banavar, J. F. Willemsen, Molecular dynamics of poiseuille flow and moving contact lines, Physical review letters 60 (1988) 1282

    1988

  36. [44]

    Koplik, J

    J. Koplik, J. R. Banavar, J. F. Willemsen, Molecular dynamics of fluid flow at solid surfaces, Physics of Fluids A: Fluid Dynamics 1 (1989) 781–794

    1989

  37. [45]

    P. A. Thompson, M. O. Robbins, Simulations of contact-line motion: slip and the dynamic contact angle, Physical review letters 63 (1989) 766

    1989

  38. [46]

    P. A. Thompson, W. Brinckerhoff, M. O. Robbins, Microscopic studies of static and dynamic contact angles, Journal of Adhesion Science and Technology 7 (1993) 535–554

    1993

  39. [47]

    Qian, X.-P

    T. Qian, X.-P. Wang, P. Sheng, A variational approach to moving contact line hydrodynamics, Journal of Fluid Mechanics 564 (2006) 333–360

    2006

  40. [48]

    Gerbeau, T

    J.-F. Gerbeau, T. Lelievre, Generalized navier boundary condition and geometric conservation law for surface tension, Computer Methods in Applied Mechanics and Engineering 198 (2009) 644–656

    2009

  41. [49]

    Yamamoto, T

    Y. Yamamoto, T. Ito, T. Wakimoto, K. Katoh, Numerical simulations of spontaneous capillary rises with very low capillary numbers using a front-tracking method combined with generalized navier boundary condition, International Journal of Multiphase Flow 51 (2013) 22–32

    2013

  42. [50]

    Smuda, F

    M. Smuda, F. Kummer, The extended discontinuous galerkin method adapted for moving contact line problems via the generalized navier boundary condition, International Journal for Numerical Methods in Fluids 93 (2021) 2921–2945

    2021

  43. [51]

    Brown, S

    R. Brown, S. Mirjalili, M. Khanwale, B. Ganapathysubramanian, A. Mani, A generalized navier boundary condition for modeling contact lines using second-order conservative phase- field methods, Center for Turbulence Research Annual Research Briefs (2022) 193–201

    2022

  44. [52]

    Mirjalili, M

    S. Mirjalili, M. A. Khanwale, A. Mani, Assessment of an energy-based surface tension model for simulation of two-phase flows using second-order phase field methods, Journal of Computational Physics 474 (2023) 111795

    2023

  45. [53]

    M. E. Gurtin, Generalized ginzburg-landau and cahn-hilliard equations based on a microforce balance, Physica D: Nonlinear Phenomena 92 (1996) 178–192

    1996

  46. [54]

    Abels, H

    H. Abels, H. Garcke, G. Grün, Thermodynamically consistent, frame indifferent diffuse interface models for incompressible two-phase flows with different densities, Mathematical Models and Methods in Applied Sciences 22 (2012) 1150013. 21

    2012

  47. [55]

    Chiodi, O

    R. Chiodi, O. Desjardins, A reformulation of the conservative level set reinitialization equation for accurate and robust simulation of complex multiphase flows, Journal of computational physics 343 (2017) 186–200

    2017

  48. [56]

    R. K. Shukla, Nonlinear preconditioning for efficient and accurate interface capturing in sim- ulation of multicomponent compressible flows, Journal of Computational Physics 276 (2014) 508–540

    2014

  49. [57]

    Wacławczyk, A consistent solution of the re-initialization equation in the conservative level- set method, Journal of Computational Physics 299 (2015) 487–525

    T. Wacławczyk, A consistent solution of the re-initialization equation in the conservative level- set method, Journal of Computational Physics 299 (2015) 487–525

    2015

  50. [58]

    S. S. Jain, Accurate conservative phase-field method for simulation of two-phase flows, Journal of Computational Physics 469 (2022) 111529

    2022

  51. [59]

    Jacqmin, Calculation of two-phase navier–stokes flows using phase-field modeling, Journal of computational physics 155 (1999) 96–127

    D. Jacqmin, Calculation of two-phase navier–stokes flows using phase-field modeling, Journal of computational physics 155 (1999) 96–127

    1999

  52. [60]

    Mirjalili, C

    S. Mirjalili, C. B. Ivey, A. Mani, Comparison between the diffuse interface and volume of fluid methods for simulating two-phase flows, International Journal of Multiphase Flow 116 (2019) 221–238

    2019

  53. [61]

    Mirjalili, A

    S. Mirjalili, A. Mani, A conservative second order phase field model for simulation of n-phase flows, Journal of Computational Physics 498 (2024) 112657. 22

    2024

Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.