Local volume-conserving lattice Boltzmann model for incompressible multiphase flows

Fang Xiong; Lei Wang; Xinyue Liu

REVIEW 3 major objections 6 minor 61 references

Local volume-conserving lattice Boltzmann model for incompressible multiphase flows

T0 review · 3 major / 6 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read A modified Cahn-Hilliard equation with a signed-distance profile correction keeps each phase's volume nearly constant in two-phase flow simulations, where the classical equation lets small droplets shrink.

desk verdict A plausible new LB scheme for a profile-corrected Cahn-Hilliard model, but the key recovery claim is asserted rather than derived and the numerical support is only qualitative. read the letter →

arxiv 2505.10899 v1 pith:BLWFIMFW submitted 2025-05-16 physics.flu-dyn

Fang Xiong , Lei Wang , Xinyue Liu This is my paper · ORCID

classification physics.flu-dyn

keywords latticeBoltzmannmethodCahn-Hilliardequationlocalvolumeconservationphase-fieldlevel-setsigneddistancefunctioninterfacecapturingmultiphaseflowmultiple-relaxation-time

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 addresses a known failure of the Cahn-Hilliard phase-field equation: although it conserves the total mass of the binary mixture, each individual phase can lose volume because the interface drifts from its equilibrium profile and small droplets shrink. To fix this, it adopts a modified Cahn-Hilliard equation that adds a penalty term pulling the interface back to a hyperbolic-tangent profile, and rewrites the sharpening flux using the signed distance function of a level-set formulation so the flux stays smooth. The paper then constructs a multiple-relaxation-time lattice Boltzmann model whose collision, equilibrium, and forcing terms are chosen to recover this modified equation. In numerical tests, including stationary droplets, a single vortex, Rayleigh-Plateau breakup, and droplet deformation in shear, the new model keeps each droplet's volume nearly constant and preserves interface shape more faithfully than the classical-Cahn-Hilliard lattice Boltzmann model, with the largest gains for small droplets.

What carries the argument

The central object is the modified Cahn-Hilliard equation (Eq. (17)) and the penalty flux carried by the signed distance function $\psi$. The penalty energy of Eq. (6) pushes the interface toward the hyperbolic-tangent profile $|\nabla\phi| = 4\phi(1-\phi)/W$; truncating its variation (Eq. (10)) and substituting $\phi(1-\phi) = \frac{1}{4}[1-\tanh^2(2\psi/W)]$ along with $\nabla\phi/|\nabla\phi|=\nabla\psi/|\nabla\psi|$ yields the smooth level-set-based flux that performs the volume-conservation work. The numerical engine is the multiple-relaxation-time lattice Boltzmann update (30)-(32), with equilibrium (31) and forcing (32) chosen so that the scheme is intended to recover Eq. (17) in the continuum limit.

What would settle it

A decisive check is a formal multiscale (Chapman-Enskog-style) expansion of the phase-field distribution update, Eqs. (30)-(32): if the recovered macroscopic equation contains terms beyond Eq. (17), or misses the level-set flux, the volume-conservation advantage would be an artifact of the discrete method. A purely numerical falsifier is a grid-refinement study of the two-droplet test in Section 4.1: if the scheme truly solves Eq. (17), the per-droplet volume error should shrink toward zero as the lattice spacing and time step are reduced at fixed physical parameters; if the error instead saturates at a level set by the dropped term in Eq. (10), the correction is not converging to the stated continuum target.

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Extended reading notes

Core claim

The central claim is that local volume conservation follows from correcting the interface profile, not from adding a global mass-correction term. The paper's target equation, Eq. (17), is the classical Cahn-Hilliard equation plus a penalty term $\lambda M[\nabla^2\phi - \nabla\cdot(W^{-1}(1-\tanh^2(2\psi/W))\nabla\psi/|\nabla\psi|)]$, where $\psi$ is the signed distance function tied to $\phi$ by $\psi = (W/4)\ln(\phi/(1-\phi))$. With this term, the equilibrium profile is enforced as a soft constraint, and the level-set substitution removes the nonlinear sharpening singularity that affects earlier profile-correction formulations. The lattice Boltzmann model of Section 3 is designed so that its macroscopic limit is this modified equation; the simulation section compares it against the classical-Cahn-Hilliard lattice Boltzmann model in four benchmarks and reports consistently smaller volume drift and better interface morphology, most visibly for droplets below the critical radius where the classical model exhibits Ostwald-ripening-like shrinkage.

Load-bearing premise

The claim rests on the assumption that the lattice Boltzmann update of Section 3, with equilibrium (31) and forcing (32), truly reproduces the modified Cahn-Hilliard equation (17) in the continuum limit, including the new level-set-based source term, and that dropping the first piece of the penalty variation in Eq. (10) is harmless; the paper states both but does not prove the recovery.

Editorial extensions

If this is right

Long-time simulations of droplet breakup and coalescence can track small satellite droplets without the artificial shrinkage that the classical Cahn-Hilliard model imposes below a critical radius.
Because the signed-distance flux remains smooth where the original profile-correction flux has $\phi(1-\phi)$ gradients, the interface-capturing scheme can run at thinner interfaces without the precision loss caused by sharpening-flux jumps.
The two-distribution-function architecture is unchanged, so incompressible multiphase lattice Boltzmann solvers can adopt the modified interface equation by replacing only the phase-field distribution and its forcing term.
In surface-tension-dominated regimes, per-phase volume conservation is maintained alongside global mass conservation, which is what allows the stationary-droplet, vortex, and shear benchmarks to report near-constant volumes.

Reading between the lines

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

The penalty term acts as a soft reinitializer: it pushes the phase field toward the tanh profile that the level-set signed-distance function encodes. Viewed this way, the method is a phase-field cousin of conservative level-set reinitialization, and one could test whether explicit reinitialization every few steps at the same cost recovers the same volume-conservation curves.
Because the penalty strength $\lambda$ is fixed at 0.5 in all tests, the reported gains have not been shown at the optimal penalty strength; a sweep over $\lambda$ at fixed interface thickness, mobility, and grid spacing would reveal how much of the improvement comes from the correction term itself and how much from level-set smoothing.
The formulation is limited to two phases; for three or more phases, one signed-distance field per phase and a triple-junction rule for the penalty fluxes would be needed, so the mechanism does not transfer automatically.

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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

Here's the short version: this is a new multiple-relaxation-time lattice Boltzmann discretization of the profile-corrected Cahn-Hilliard equation that Li and Yang introduced in 2024. The paper's selling point is that this LB model preserves the local volume of small droplets much better than the classical-C-H LB model of Liang et al. The numerical pictures support that claim qualitatively, but the paper never shows that the discrete scheme actually converges to the target equation (17) in the hydrodynamic limit. That missing Chapman-Enskog expansion is not a formality; the forcing term (32) is a vector field while the intended source is divergence-form, and as the introduction recounts, He et al. and Lee et al. failed for exactly that reason. I would not trust the claimed improvement until this is worked out.

What is genuinely new: the MRT-LB scheme with the λ-shifted equilibrium and the ψ-based forcing. That is a legitimate new numerical scheme for an existing PDE, and the choice of test cases (stationary droplets, single vortex, Rayleigh-Plateau, shear) covers the relevant failure modes. The comparison with Ref. [31] is appropriate, and the observed interface behavior matches what the modified C-H equation is designed to do.

Soft spots, in order of severity. First, the lack of any multiscale derivation for the phase-field LB equation. The sentence 'Inspired by previous work...' is not a derivation. The reader should ask the authors to supply the standard Chapman-Enskog or asymptotic analysis, or at least show numerically that a known exact solution is recovered to a specified order. Second, Eq. (10) drops the first term of the penalty variation without estimating its size; that term is not zero when the interface deviates from the equilibrium tanh profile, and the whole volume-preservation argument depends partly on that approximation. Third, the numerical evidence is qualitative for three of the four cases; only the stationary-droplet case has volume-loss curves. No grid-convergence study, no error norms, no quantitative long-time volume error for the vortex, Rayleigh-Plateau, or shear cases. The figures are suggestive, but the claim is stated much stronger than the data.

The citation pattern is fine: the relevant LB-C-H history is cited, and the PDE source [50] is clearly identified. The paper is not a derivative duplicate; it is an application-oriented extension.

Who is this for? Researchers in lattice Boltzmann multiphase flow who care about the droplet-shrinkage problem. The paper deserves serious referee time, but the referee should push hard on the missing derivation and request quantitative checks. If those hold, it becomes a solid contribution.

Referee Report

3 major / 6 minor

Summary. This paper proposes a modified Cahn-Hilliard (C-H) equation for incompressible two-phase flows that combines a profile-correction penalty term with the signed-distance function from level-set methods, and then constructs a multiple-relaxation-time lattice Boltzmann (MRT-LB) model to solve it together with the incompressible Navier-Stokes equations. The central claim is that this LB model preserves the local volume of each phase and the interface shape better than the classical-C-H LB model of Liang et al. [31], especially for small droplets. The paper reports four numerical tests (stationary droplets, single vortex, Rayleigh-Plateau instability, and droplet deformation in a shear flow) and concludes that the proposed model exhibits superior volume conservation and interface fidelity.

Significance. The paper addresses a relevant and well-documented limitation of phase-field LB methods: local phase-volume loss for small droplets. The construction in Sec. 2, based on an existing penalty-energy idea [27,50] and the level-set reformulation [17], is clearly presented, and the chosen benchmarks are standard and appropriate. If the MRT-LB system in Sec. 3 is rigorously shown to reproduce Eq. (17), the model could be a useful contribution to interface-capturing LB methods. However, the manuscript currently leaves the load-bearing multiscale derivation as an assertion, and the numerical evidence is largely qualitative. The interface-shape preservation is partly by construction because the penalty term in Eq. (17) explicitly targets the hyperbolic-tangent profile (4); the decisive open question is whether the discrete LB equations actually realize Eq. (17) accurately. These gaps are addressable, but they are central rather than cosmetic.

major comments (3)

[Section 3.2, Eqs. (30)-(32)] The central assertion that the LB system recovers the modified C-H equation (17) is not derived. The text states only that the evolution equation is 'inspired by' Refs. [31,37]. This is load-bearing because Eq. (31) shifts the chemical potential by lambda*phi and Eq. (32) inserts the psi-dependent flux as a forcing term; one must show, via a Chapman-Enskog or multiscale expansion, that the discrete moments produce exactly div(M*grad(hbar)) + lambda*M*[Laplacian(phi) - div((1/W)(1 - tanh^2(2*psi/W))*grad(psi)/|grad(psi)|)] and produce no extraneous terms of order Delta t. In particular, the recovered diffusion of the shifted potential includes div(M*grad(lambda*phi)), and the force in Eq. (32) must cancel the unwanted part and reproduce the intended flux to all required orders. The time-derivative term dt(phi*u) in Eq. (32), approximated by the backward difference in Eq. (38), is a specific source of such discretization errors. The introduction itself (Sec. 1, Refs. [28-30]) documents earlier LB-C-H models that failed to recover the intended equation for exactly this class of reason, so the derivation is not a formality. Without it, the claim that the simulations solve Eq. (17) is unverified.
[Section 2, Eqs. (9)-(10)] The derivation of Eq. (17) drops the first term on the right-hand side of Eq. (9), with the justification that 4*phi*(1-phi)/W - |grad(phi)| tends to zero as the interface approaches the hyperbolic-tangent profile. No error bound, asymptotic estimate, or numerical measurement of this term is given. In the deformed-interface tests of Secs. 4.2-4.4 the interface is not at the equilibrium profile, so the dropped term is not obviously negligible; it contributes to the effective penalty and could modify the local-volume behavior. The authors should either bound this term in the regimes simulated or explicitly state that Eq. (17) is the model to be solved, independent of the gradient-flow derivation from Eq. (8).
[Section 4] The quantitative evidence for the central claim is insufficient. Only Fig. 3(b) in Sec. 4.1 provides volume-evolution curves; the single-vortex (Sec. 4.2), Rayleigh-Plateau (Sec. 4.3), and shear-flow (Sec. 4.4) comparisons consist of qualitative interface snapshots. The paper reports no relative volume-change errors, no grid-convergence study, and no comparison with a well-resolved finite-difference solution of the continuum Eq. (17). A convergence test (e.g., the single-vortex case at time T) and time series of the relative volume error for each case are needed to substantiate the statement that the model 'achieves more precise volume conservation for each phase' (Abstract). As written, the superiority claim is only visually supported for three of the four benchmarks.

minor comments (6)

[Section 4.1, Eq. (41)] The radii r1 and r2 in Eq. (41) are not defined; please specify their values.
[General] The quantity 'local volume' is never defined precisely; please state whether it is the integral of phi over the domain, the area enclosed by the phi = 0.5 contour, or another measure, since the main quantitative claim is expressed through this quantity.
[Section 3.1] The density rho used in Eqs. (20), (21), (28), and (29) is never expressed as a function of the order parameter phi; although all tests use density ratio 1, the method is presented for general incompressible multiphase flows, so the interpolation formula (e.g., rho = rho_l*phi + rho_g*(1-phi)) and the way rho is updated should be stated.
[Section 3.2, Eqs. (36)-(37)] The regularization in Eq. (37) and the clipping of phi to [0,1] before computing psi change the signed-distance function; the effect of these choices on the forcing term (32), especially near the bulk regions, should be discussed or tested.
[General] The boundary conditions (periodic, solid walls, moving walls) are not stated for any of the four simulations, which hampers reproducibility.
[General] The text contains numerous typos, including 'surface tansion', 'casued', 'presssion', 'conservaed', 'latice spacing', and the repeated phrase 'more consistent representation ... more consistently' in the abstract; a careful proofread is needed.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the LB model is constructed to recover the target modified Cahn-Hilliard equation, and the interface-preservation behavior is a designed property of that equation rather than a prediction that is equivalent to its own input.

full rationale

The paper's derivation chain is constructive rather than circular. The modified C-H equation (17) is obtained by adding a penalty energy (6) that explicitly targets the hyperbolic-tangent interface profile (4); the LB equilibrium (31) and forcing (32) are then written to match the terms of Eq. (17). This is model construction, not a derivation of a conclusion from an assumption that already contains that conclusion. The numerical sections test whether the resulting LB scheme actually preserves local volume and interface shape in practice, which is a legitimate check of the implementation. The approximation in Eq. (10), where the first term of the penalty variation is dropped, is an uncontrolled truncation and a potential numerical-modeling risk, but it is not a circular step: the dropped term is not silently replaced by the result the paper claims. The absence of a Chapman-Enskog or multiscale derivation for Eqs. (30)-(32) is a notable verification gap, and the paper's own historical discussion of earlier failed recoveries makes that gap material; however, a missing proof is a correctness concern, not evidence that the claim reduces by definition to its inputs. The only self-citations (e.g., Refs. [23] and [25]) appear in background statements about the usefulness of LB methods and do not carry the load-bearing argument. No fitted parameter is renamed as a prediction, no uniqueness theorem is imported from the authors' prior work, and no known result is merely relabeled. Overall, the central claim has independent content in the numerical comparison against the classical C-H LB model of Ref. [31].

Assumptions & free parameters 4 free parameters · 5 assumptions · 0 invented entities

The central model inherits its profile-correction mechanism and the ad hoc constant lambda from Ref [50]; the new LB layer adds an unproven recovery assumption. No new physical entities are postulated.

free parameters (4)

lambda (penalty constant) = 0.5
Chosen from Ref [50] as the strength of the profile-correction penalty; an ad hoc constant not fitted in this paper.
M (mobility) = 0.1
Set per simulation and standard for LB-Cahn-Hilliard models; controls the diffusion time scale.
W (interface thickness) = 3.0
Set per simulation as the interface width; appears in the profile-correction terms.
xi (level-set regularization) = 1e-10
Introduced in Eq. (37) to avoid the logarithm singularity in the level-set function; a numerical safeguard.

assumptions (5)

standard math The equilibrium profile of the order parameter is the hyperbolic-tangent function (Eq. 4).
For the classical Cahn-Hilliard equation at equilibrium, the one-dimensional interface profile is a hyperbolic tangent.
domain assumption The mapping psi = (W/4) ln(phi/(1-phi)) defines a valid level-set function.
Used to replace the sharpening flux phi(1-phi) and the normal vector Grad(phi)/|Grad(phi)|; the mapping is singular as phi approaches 0 or 1.
ad hoc to paper The first term on the right-hand side of Eq. (9) is negligible, so Eq. (10) holds.
The term lambda(1-2phi)(4phi(1-phi)/W - |Grad(phi)|)/(4W) is dropped without bounds on phi or |Grad(phi)|; this truncation is inherited from the cited profile-correction derivation.
domain assumption The MRT-LB equations (20) and (30) recover the Navier-Stokes equations and the modified Cahn-Hilliard equation respectively.
Assumed by analogy with Ref [31]; no Chapman-Enskog multiscale derivation is presented for the new source term.
domain assumption The isotropic gradient and Laplacian discretizations (39)-(40) are sufficient for the singular level-set terms.
The term Grad(psi)/|Grad(psi)| is evaluated with standard isotropic finite differences; behavior near |Grad(psi)| = 0, which occurs in bulk regions and droplet centers, is not discussed.

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Cite this review

Pith. "Pith review of Local volume-conserving lattice Boltzmann model for incompressible multiphase flows." pith.science (2026). https://pith.science/paper/BLWFIMFW

@misc{pith2026250510899,
  author       = {Pith},
  title        = {Pith review of: Local volume-conserving lattice Boltzmann model for incompressible multiphase flows},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/BLWFIMFW}},
  note         = {Machine review of arXiv:2505.10899}
}

read the original abstract

The Cahn-Hilliard (C-H) equation, as a classical diffusion-interface method of phase-field, has been extensively employed for simulating two-phase fluid dynamics. However, it suffers from a key challenge in the simulation process, specifically the volume conservation of each phase cannot be guaranteed. To address this issue, in this paper, a modified C-H equation for two-phase flow modeling is first introduced, and the basic idea of this model lies in that it combines the profile correction method with the level-set approach, and thus, it effectively improves the deficiency of the classical C-H equation in terms of volume non-conservation of each phase. Based on this modified C-H equation, we further propose an accurate interface-capturing lattice Boltzmann (LB) model. After that, we perform a range of numerical simulations, including two stationary droplets immersed in the gas phase, single vortex, Rayleigh-Plateau fluid instability, and droplet deformation under a shear flow. These simulations illustrate that the proposed LB model has superior performance in maintaining local volume conservation and accurately capturing interfaces. More importantly, compared to the LB model derived from the classical C-H equation, it not only achieves more precise volume conservation for each phase but also provides a more consistent representation of the droplet's interface morphology more consistently, especially in dealing with small droplet problems.

Figures

Figures reproduced from arXiv: 2505.10899 by the authors.

**Figure 1.** Illustration of a circular droplet contracting within a stationary medium. Therefore, while the classical C-H model maintains the overall volume of the binary system, it fails to preserve the local volume. To address this issue, a straightforward approach is to ensure that the diffuse interface matches the hyperbolic tangent profile described in Eq. (4). Based on this, Li et al. [27, 50] inroduced an improved C-H eq… view at source ↗

**Figure 2.** A depiction of the variable ϕ and the signed distance function ψ Following previous works [17, 41, 52, 53], we modify the interface-capturing model by utilizing the relationship defined in Eq. (15) to substitute the non-linear sharpening flux term ϕ(1 − ϕ) from Eq. (11) with ψ. As shown in [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗

**Figure 3.** Comparison of the current model with the classical C-H model for local volume preservation, (a) interface profile configuration, (b) droplets local volume evolution over time. Fig. (3a) displays the interface profiles of the current model and the LB model in Ref. [31] at a specific in time. From the observations, the droplets calculated by the LB model for classical C-H equation in Ref. [31] exhibit obvious contract… view at source ↗

Figures from the paper (5 more)

**Figure 4.** Figure 4: Interfacial shape profiles of the disk at different times, times in order T/4, T/2, 3T/4, T, (a) the LB model in Ref. [31], (b) the present model [PITH_FULL_IMAGE:figures/full_fig_p013_4.png]

**Figure 5.** Figure 5: Comparison of the interface profiles between the present model and the LB model in Ref. [31] with the initial state at different times (a) T/2, (b) T. 4.3. Rayleigh-Plateau fluid instability Rayleigh-Plateau fluid instability is a type of interfacial instability caused…

**Figure 6.** Figure 6: Profiles of Rayleigh-Plateau fluid instability, (a) LB model in Ref. [31], (b) the present model [PITH_FULL_IMAGE:figures/full_fig_p015_6.png]

**Figure 7.** Figure 7: Diagram of a droplet subjected to shear flow. Within the domain of phase field methods, volume diffusion often leads to contraction and coarsening phenomena. If the interface profile strays from its equilibrium condition, the volume within the interface will diffuse in…

**Figure 8.** Figure 8: Evolution of droplet interfacial profile over time, (a) the LB model in Ref. [31], (b) the present model [PITH_FULL_IMAGE:figures/full_fig_p017_8.png]

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Reference graph

Works this paper leans on

61 extracted references · 61 canonical work pages

[31]

Liang, B.C

H. Liang, B.C. Shi, Z.L. Guo, Z. Chai, Phase-field-based multiple-relaxation-time lattice Boltzmann model for incompressible multiphase flows, Phys. Rev. E 89 (5) (2014) 053320

work page 2014
[50]

J. Li, J. Yang, Local volume-conservation-improved di ffuse interface model for simulation of Rayleigh-Plateau fluid instability, Comput. Phys. Commun. 296 (2024) 109050

work page 2024
[17]

Jain, Accurate conservative phase-field method for simulation of two-phase flows, J

S.S. Jain, Accurate conservative phase-field method for simulation of two-phase flows, J. Comput. Phys. 469 (2022) 111529

work page 2022
[1]

Gonnermann, M

H.M. Gonnermann, M. Manga, S.A. Fagents, Dynamics of magma ascent in the volcanic conduit, Modeling volcanic processes: The physics and mathematics of volcanism, Cambridge University Press (2013) 55-84

work page 2013
[2]

Dhir, Boiling heat transfer, Annu

V .K. Dhir, Boiling heat transfer, Annu. Rev. Fluid Mech. 30 (1) (1998) 365-401

work page 1998
[3]

Hart, A review of technologies for transporting heavy crude oil and bitumen via pipelines, J

A. Hart, A review of technologies for transporting heavy crude oil and bitumen via pipelines, J. Pet. Explor. Prod. Technol. 4 (2014) 327-336

work page 2014
[4]

Thorsen, S.J

T. Thorsen, S.J. Maerkl, S.R. Quake, Microfluidic large-scale integration, Science 298 (5593) (2002) 580-584

work page 2002
[5]

Cristini, Y .C

V . Cristini, Y .C. Tan, Theory and numerical simulation of droplet dynamics in complex flows—a review, Lab Chip 4 (4) (2004) 257-264

work page 2004

Show all 61 references

[6]

Elbar, A

C. Elbar, A. Poiatti, Weak solutions and sharp interface limit of the anisotropic Cahn-Hilliard equation with disparate mobility and inhomoge- neous potential, arXiv preprint arXiv 2502.11849 (2025)

2025 arXiv
[7]

Steinke, S.G

M.E. Steinke, S.G. Kandlikar, An experimental investigation of flow boiling characteristics of water in parallel microchannels, J. Heat Transfer 126 (4) (2004) 518-526

2004
[8]

Leshansky, S

A.M. Leshansky, S. Afkhami, M.C. Jullien, P. Tabeling, Obstructed breakup of slender drops in a microfluidic T junction, Phys. Rev. Lett. 108 (26) (2012) 264502

2012
[9]

Link, S.L

D.R. Link, S.L. Anna, D.A. Weitz, H.A. Stone, Geometrically mediated breakup of drops in microfluidic devices, Phys. Rev. Lett. 92 (5) (2004) 054503

2004
[10]

Jullien, M.J

M.C. Jullien, M.J. Tsang Mui Ching, C. Cohen, L. Menetrier, P. Tabeling, Droplet breakup in microfluidic T-junctions at small capillary numbers, Phys. Fluids 21 (7) (2009)

2009
[11]

Yue, J.J

P. Yue, J.J. Feng, C. Liu, J. Shen, A di ffuse-interface method for simulating two-phase flows of complex fluids, J. Fluid Mech. 515 (2004) 293-317

2004
[12]

H. Hao, X. Li, C. Jiang, H. Tan, An interfacial profile-preserving approach for phase field modeling of incompressible two-phase flows, Int. J. Multiph. Flow 174 (2024) 104750

2024
[13]

Hirt, B.D

C.W. Hirt, B.D. Nichols, V olume of fluid method (VOF) for the dynamics of free boundaries, J. Comput. Phys. 39 (1981) 201–225

1981
[14]

Sussman, P

M. Sussman, P. Smereka, Axisymmetric free boundary problems, J. Fluid Mech. 341 (1997) 269-294

1997
[15]

Unverdi, G

S.O. Unverdi, G. Tryggvason, A front-tracking method for viscous, incompressible, multi-fluid flows, J. Comput. Phys. 100 (1) (1992) 25-37

1992
[16]

Jacqmin, Calculation of two-phase Navier–Stokes flows using phase-field modeling, J

D. Jacqmin, Calculation of two-phase Navier–Stokes flows using phase-field modeling, J. Comput. Phys. 155 (1) (1999) 96-127

1999
[18]

Zhang, Z

C. Zhang, Z. Guo, H. Liang, High-order lattice-Boltzmann model for the Cahn-Hilliard equation, Phys. Rev. E 99 (4) (2019) 043310

2019
[19]

Sethian, P

J.A. Sethian, P. Smereka, Level set methods for fluid interfaces, Annu. Rev. Fluid Mech. 35 (1) (2003) 341-372

2003
[20]

Bonhomme, J

R. Bonhomme, J. Magnaudet, F. Duval, B. Piar, Inertial dynamics of air bubbles crossing a horizontal fluid–fluid interface, J. Fluid Mech. 707 (2012) 405-443

2012
[21]

H. Wang, X. Yuan, H. Liang, Z. Chai, B. Shi, A brief review of the phase-field-based lattice Boltzmann method for multiphase flows, Capillarity 2 (3) (2019) 32-52

2019
[22]

Cahn, J.E

J.W. Cahn, J.E. Hilliard, Free energy of a nonuniform system. I. Interfacial free energy, J. Chem. Phys. 28 (2) (1958) 258-267

1958
[23]

L. Wang, K. He, H. Wang, Phase-field-based lattice Boltzmann model for simulating thermocapillary flows, Phys. Rev. E 108 (5) (2023) 055306

2023
[24]

Krüger, H

T. Krüger, H. Kusumaatmaja, A. Kuzmin, O. Shardt, G. Silva, E.M. Viggen, The lattice Boltzmann method, Springer International Publishing 10 (978-3) (2017) 4-15

2017
[25]

Xiong, L

F. Xiong, L. Wang, J. Huang, K. Luo, A thermodynamically consistent phase-field lattice Boltzmann method for two-phase electrohydrody- namic flows, J. Sci. Comput. 103 (1) (2025) 1-32

2025
[26]

Q.Z. Li, Z.L. Lu, Z. Chen, C. Shu, Y .Y . Liu, T.Q. Guo, A simplified lattice Boltzmann model for two-phase electro-hydrodynamics flows and its application to simulations of droplet deformation in electric field, Appl. Math. Model. 122 (2023) 99-126

2023
[27]

Y . Li, J.I. Choi, J. Kim, A phase-field fluid modeling and computation with interfacial profile correction term, Commun. Nonlinear Sci. Numer. Simul. 30 (1-3) (2016) 84-100. 19

2016
[28]

X. He, S. Chen, R. Zhang, A lattice Boltzmann scheme for incompressible multiphase flow and its application in simulation of Rayleigh- Taylor instability, J. Comput. Phys. 152 (2) (1999) 642-663

1999
[29]

Lee, C.L

T. Lee, C.L. Lin, A stable discretization of the lattice Boltzmann equation for simulation of incompressible two-phase flows at high density ratio, J. Comput. Phys. 206 (1) (2005) 16-47

2005
[30]

Zheng, C

H.W. Zheng, C. Shu, Y .T. Chew, A lattice Boltzmann model for multiphase flows with large density ratio, J. Comput. Phys. 218 (1) (2006) 353-371

2006
[32]

Zheng, C

H.W. Zheng, C. Shu, Y .T. Chew, J.H. Sun, Three-dimensional lattice Boltzmann interface capturing method for incompressible flows, Int. J. Numer. Methods Fluids 56 (9) (2008) 1653-1671

2008
[33]

Y .Q. Zu, S. He, Phase-field-based lattice Boltzmann model for incompressible binary fluid systems with density and viscosity contrasts, Phys. Rev. E 87 (4) (2013) 043301

2013
[34]

L. Ju, P. Liu, B. Yan, J. Bao, S. Sun, Z. Guo, A well-balanced lattice Boltzmann model for binary fluids based on the incompressible phase-field theory, arXiv preprint arXiv, 2311.10827 (2023)

2023 arXiv
[35]

Geier, A

M. Geier, A. Fakhari, T. Lee, Conservative phase-field lattice Boltzmann model for interface tracking equation, Phys. Rev. E 91 (6) (2015) 063309

2015
[36]

Fakhari, M.H

A. Fakhari, M.H. Rahimian, Phase-field modeling by the method of lattice Boltzmann equations, Phys. Rev. E 81 (3) (2010) 036707

2010
[37]

Liang, R

H. Liang, R. Wang, Y . Wei, J. Xu, Lattice Boltzmann method for interface capturing, Phys. Rev. E 107 (2) (2023) 025302

2023
[38]

Y . Wang, C. Shu, J.Y . Shao, J. Wu, X.D. Niu, A mass-conserved diffuse interface method and its application for incompressible multiphase flows with large density ratio, J. Comput. Phys. 290 (2015) 336-351

2015
[39]

Mirjalili, S.S

S. Mirjalili, S.S. Jain, M. Dodd, Interface-capturing methods for two-phase flows: An overview and recent developments, Center for Turbu- lence Research Annual Research Briefs, 2017 (117-135) (2017) 13

2017
[40]

Chiu, Y .T

P.H. Chiu, Y .T. Lin, A conservative phase field method for solving incompressible two-phase flows, J. Comput. Phys. 230 (1) (2011) 185-204

2011
[41]

Chiu, A coupled phase field framework for solving incompressible two-phase flows, J

P.H. Chiu, A coupled phase field framework for solving incompressible two-phase flows, J. Comput. Phys. 392 (2019) 115-140

2019
[42]

Kendon, M.E

V .M. Kendon, M.E. Cates, I. Pagonabarraga, J.C. Desplat, P. Bladon, Inertial e ffects in three-dimensional spinodal decomposition of a symmetric binary fluid mixture: a lattice Boltzmann study, J. Fluid Mech. 440 (2001) 147-203

2001
[43]

Zhang, Q

J. Zhang, Q. Du, Numerical studies of discrete approximations to the Allen–Cahn equation in the sharp interface limit, SIAM J. Sci. Comput. 31 (4) (2009) 3042-3063

2009
[44]

Maraldi, L

M. Maraldi, L. Molari, D. Grandi, A unified thermodynamic framework for the modelling of di ffusive and displacive phase transitions, Int. J. Eng. Sci. 50 (1) (2012) 31-45

2012
[45]

Bertozzi, S

A. Bertozzi, S. Esedoglu, A. Gillette, Analysis of a two-scale Cahn–Hilliard model for binary image inpainting, Multiscale Model. Simul. 6 (3) (2007) 913-936

2007
[46]

Hilhorst, J

D. Hilhorst, J. Kampmann, T.N. Nguyen, K.G. Van Der Zee, Formal asymptotic limit of a di ffuse-interface tumor-growth model, Math. Models Meth. Appl. Sci. 25 (06) (2015) 1011-1043

2015
[47]

Wise, J.S

S.M. Wise, J.S. Lowengrub, H.B. Frieboes, V . Cristini, Three-dimensional multispecies nonlinear tumor growth — I: model and numerical method, J. Theor. Biol. 253 (3) (2008) 524-543

2008
[48]

Soligo, A

G. Soligo, A. Roccon, A. Soldati, Mass-conservation-improved phase field methods for turbulent multiphase flow simulation, Acta Mech. 230 (2019) 683-696

2019
[49]

P. Yue, C. Zhou, J.J. Feng, Spontaneous shrinkage of drops and mass conservation in phase-field simulations, J. Comput. Phys. 223 (1) (2007) 1-9

2007
[51]

Bertozzi, S

A.L. Bertozzi, S. Esedoglu, A. Gillette, Inpainting of binary images using the Cahn–Hilliard equation, IEEE Trans. Image Process. 16 (1) 20 2006 285-291

2006
[52]

Wacławczyk, A consistent solution of the re-initialization equation in the conservative level-set method, J

T. Wacławczyk, A consistent solution of the re-initialization equation in the conservative level-set method, J. Comput. Phys. 299 (2015) 487-525

2015
[53]

Chiodi, O

R. Chiodi, O. Desjardins, A reformulation of the conservative level set reinitialization equation for accurate and robust simulation of complex multiphase flows, J. Comput. Phys. 343 (2017) 186-200

2017
[54]

Y .H. Qian, D. d’Humières, P. Lallemand, Lattice BGK models for Navier-Stokes equation, Europhys. Lett. 17 (6) (1992) 479

1992
[55]

Ginzburg, F

I. Ginzburg, F. Verhaeghe, D. d’Humieres, Two-relaxation-time lattice Boltzmann scheme: About parametrization, velocity, pressure and mixed boundary conditions, Commun. Comput. Phys. 3 (2) (2008) 427-478

2008
[56]

Lallemand, L.S

P. Lallemand, L.S. Luo, Theory of the lattice Boltzmann method: Dispersion, dissipation, isotropy, Galilean invariance, and stability, Phys. Rev. E 61 (6) (2000) 6546

2000
[57]

Z. Chai, B. Shi, Z. Guo, A multiple-relaxation-time lattice Boltzmann model for general nonlinear anisotropic convection-diffusion equations, J. Sci. Comput. 69 (2016) 355-390

2016
[58]

Thampi, S

S.P. Thampi, S. Ansumali, R. Adhikari, S. Succi, Isotropic discrete Laplacian operators from lattice hydrodynamics, J. Comput. Phys. 234 (2013) 1-7

2013
[59]

J. Bao, Z. Guo, Phase-field lattice Boltzmann model with singular mobility for quasi-incompressible two-phase flows, Phys. Rev. E 109 (2) (2024) 025302

2024
[60]

H.G. Lee, J. Yang, J. Park, J. Kim, E ffect of Space Dimensions on Equilibrium Solutions of Cahn-Hilliard and Conservative Allen-Cahn equations, Numer. Math., Theory, Methods Appl. 13 (3) (2020) 644-664

2020
[61]

Cahn, On spinodal decomposition, Acta Metall

J.W. Cahn, On spinodal decomposition, Acta Metall. 9 (9) (1961) 795-801. 21

1961

Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.

[1] [31]

Liang, B.C

H. Liang, B.C. Shi, Z.L. Guo, Z. Chai, Phase-field-based multiple-relaxation-time lattice Boltzmann model for incompressible multiphase flows, Phys. Rev. E 89 (5) (2014) 053320

work page 2014

[2] [50]

J. Li, J. Yang, Local volume-conservation-improved di ffuse interface model for simulation of Rayleigh-Plateau fluid instability, Comput. Phys. Commun. 296 (2024) 109050

work page 2024

[3] [17]

Jain, Accurate conservative phase-field method for simulation of two-phase flows, J

S.S. Jain, Accurate conservative phase-field method for simulation of two-phase flows, J. Comput. Phys. 469 (2022) 111529

work page 2022

[4] [1]

Gonnermann, M

H.M. Gonnermann, M. Manga, S.A. Fagents, Dynamics of magma ascent in the volcanic conduit, Modeling volcanic processes: The physics and mathematics of volcanism, Cambridge University Press (2013) 55-84

work page 2013

[5] [2]

Dhir, Boiling heat transfer, Annu

V .K. Dhir, Boiling heat transfer, Annu. Rev. Fluid Mech. 30 (1) (1998) 365-401

work page 1998

[6] [3]

Hart, A review of technologies for transporting heavy crude oil and bitumen via pipelines, J

A. Hart, A review of technologies for transporting heavy crude oil and bitumen via pipelines, J. Pet. Explor. Prod. Technol. 4 (2014) 327-336

work page 2014

[7] [4]

Thorsen, S.J

T. Thorsen, S.J. Maerkl, S.R. Quake, Microfluidic large-scale integration, Science 298 (5593) (2002) 580-584

work page 2002

[8] [5]

Cristini, Y .C

V . Cristini, Y .C. Tan, Theory and numerical simulation of droplet dynamics in complex flows—a review, Lab Chip 4 (4) (2004) 257-264

work page 2004