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Connectivity of the diffuse interface and fine structure of minimizers in the Allen-Cahn theory of phase transitions.

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SAO PAULO JOURNAL OF MATHEMATICAL SCIENCES 2025年第1期19卷 1-49页

作者： Fusco, Giorgio Univ Aquila Dipartimento Matemat Pura & Applicata Via Vetoio I-67010 Laquila Italy

In the Allen-Cahn theory of phase transitions, minimizers partition the domain in subregions, the sets where a minimizer is near to one or to another of the zeros of the potential. These subregions that model the phases are separated by a tiny diffuse interface. Understanding the shape of this diffuse interface is an important step toward the description of the structure of minimizers. We assume Dirichlet data and present general conditions on the domain and on the boundary datum ensuring the connectivity of the diffuse interface. Then we restrict to the case of two dimensions and show that the phases can be separated, in a certain optimal way, by a connected network with a well-defined structure. This network is contained in the diffuse interface and is a priori unknown. Under general assumption on the potential and on the Dirichlet datum, we show that, if we assume the connectivity of the phases, then we can obtain precise information on the shape of the network and in turn a detailed description of the fine structure of minimizers. In particular, we can characterize the shape and the size of the various phases and also how they depend on the surface tensions. These results were announced at the 8th IST-IME Meeting in September 2022 in Lisbon.

关键词： Allen-Cahn energy diffuse interface Phase separation Network

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diffuse interface model for incompressible two-phase flows with large density ratios

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JOURNAL OF COMPUTATIONAL PHYSICS 2007年第2期226卷 2078-2095页

作者： Ding, Hang Spelt, Peter D. M. Shu, Chang Imperial Coll Dept Chem Engn London SW7 2AZ England Natl Univ Singapore Dept Mech Engn Singapore 119260 Singapore

We investigate the applicability of an incompressible diffuse interface model for two-phase incompressible fluid flows with large viscosity and density contrasts. diffuse-interface models have been used previously primarily for density-matched fluids, and it remains unclear to what extent such models can be used for fluids of different density, thereby potentially limiting the application of these models. In this paper, the convective Cahn-Hilliard equation and the condition that the velocity field is divergence-free are derived from the conservation law of mass of binary mixtures in a straightforward way, for fluids with large density and viscosity ratios. Differences in the equations of motion with a previously derived quasi-incompressible model are shown to result from the respective assumptions made regarding the relationship between the diffuse fluxes of two species. The convergence properties of the model are investigated for cases with large density ratio. Quantitative comparisons are made with results from previous studies to validate the model and its numerical implementation. Tests show that the variation in volume during the computation is of the order of machine accuracy, which is consistent with our use of a conservative discretization scheme (finite volume methods) for the Cahn-Hilliard equation. Results of the method are compared with previous work for the change in topology of rising bubbles and Rayleigh-Taylor instability. Additional results are presented for head-on droplet collision and the onset of droplet entrainment in stratified flows. (c) 2007 Elsevier Inc. All rights reserved.

关键词： diffuse interface large density ratio multiphase flow bubbles Rayleigh-Taylor instability droplet collision stratified flow

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diffuse interface modelling of reactive multi-phase flows applied to a sub-critical cryogenic jet

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APPLIED MATHEMATICAL MODELLING 2020年 84卷 405-424页

作者： Deng, Xi Boivin, Pierre Aix Marseille Univ Cent Marseille CNRS M2P2 Marseille France Imperial Coll London Dept Aeronaut London SW7 2AZ England

In order to simulate cryogenic H-2 - O-2 jets under subcritical condition, a numerical model is constructed to solve compressible reactive multi-component flows which involve complex multi-physics processes such as moving material interfaces, shock waves, phase transition and combustion. The liquid and reactive gaseous mixture are described by a homogeneous mixture model with diffusion transport for heat, momentum and species. A hybrid thermodynamic closure strategy is proposed to construct an equation of state (EOS) for the mixture. The phase transition process is modeled by a recent fast relaxation method which gradually reaches the thermo-chemical equilibrium without iterative process. A simplified transport model is also implemented to ensure the accurate behavior in the limit of pure fluids and maintain computational efficiency. Last, a 12-step chemistry model is included to account for hydrogen combustion. Then the developed numerical model is solved with the finite volume method where a low dissipation AUSM (advection upstream splitting method) Riemann solver is extended for multi-component flows. A homogeneous reconstruction strategy compatible with the homogeneous mixture model is adopted to prevent numerical oscillations across material interfaces. Having included these elements, the model is validated on a number of canonical configurations, first for multiphase flows, and second for reactive flows. These tests allow recovery of the expected behavior in both the multiphase and reactive limits, and the model capability is further demonstrated on a 2D burning cryogenic H-2 - O-2 jet, in a configuration reminiscent of rocket engine ignition. (C) 2020 Elsevier Inc. All rights reserved.

关键词： Multiphase flows Reacting flows Phase transition diffuse interface

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diffuse interface simulation of ternary fluids in contact with solid

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JOURNAL OF COMPUTATIONAL PHYSICS 2016年 309卷 37-51页

作者： Zhang, Chun-Yu Ding, Hang Gao, Peng Wu, Yan-Ling Univ Sci & Technol China Dept Modern Mech Hefei 230027 Peoples R China Univ Sci & Technol China State Key Lab Fire Sci Hefei 230027 Peoples R China

In this article we developed a geometrical wetting condition for diffuse-interface simulation of ternary fluid flows with moving contact lines. The wettability of the substrate in the presence of ternary fluid flows is represented by multiple contact angles, corresponding to the different material properties between the respective fluid and the substrate. Displacement of ternary fluid flows on the substrate leads to the occurrence of moving contact point, at which three moving contact lines meet. We proposed a weighted contact angle model, to replace the jump in contact angle at the contact point by a relatively smooth transition of contact angle over a region of 'diffuse contact point' of finite size. Based on this model, we extended the geometrical formulation of wetting condition for two-phase flows with moving contact lines to ternary flows with moving contact lines. Combining this wetting condition, a Navier-Stokes solver and a ternary-fluid model, we simulated two-dimensional spreading of a compound droplet on a substrate, and validated the numerical results of the drop shape at equilibrium by comparing against the analytical solution. We also checked the convergence rate of the simulation by investigating the axisymmetric drop spreading in a capillary tube. Finally, we applied the model to a variety of applications of practical importance, including impact of a circular cylinder into a pool of two layers of different fluids and sliding of a three-dimensional compound droplet in shear flows. (C) 2016 Elsevier Inc. All rights reserved.

关键词： diffuse interface Ternary fluid flows Moving contact lines Wetting conditions

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diffuse interface immersed boundary method for multi-fluid flows with arbitrarily moving rigid bodies

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JOURNAL OF COMPUTATIONAL PHYSICS 2018年 360卷 202-228页

作者： Patel, Jitendra Kumar Natarajan, Ganesh Indian Inst Technol Guwahati Dept Mech Engn Gauhati 781039 Assam India

We present an interpolation-free diffuse interface immersed boundary method for multiphase flows with moving bodies. A single fluid formalism using the volume-of-fluid approach is adopted to handle multiple immiscible fluids which are distinguished using the volume fractions, while the rigid bodies are tracked using an analogous volume-of-solid approach that solves for the solid fractions. The solution to the fluid flow equations are carried out using a finite volume-immersed boundary method, with the latter based on a diffuse interface philosophy. In the present work, we assume that the solids are filled with a "virtual" fluid with density and viscosity equal to the largest among all fluids in the domain. The solids are assumed to be rigid and their motion is solved using Newton's second law of motion. The immersed boundary methodology constructs a modified momentum equation that reduces to the Navier-Stokes equations in the fully fluid region and recovers the no-slip boundary condition inside the solids. An implicit incremental fractional-step methodology in conjunction with a novel hybrid staggered/non-staggered approach is employed, wherein a single equation for normal momentum at the cell faces is solved everywhere in the domain, independent of the number of spatial dimensions. The scalars are all solved for at the cell centres, with the transport equations for solid and fluid volume fractions solved using a high-resolution scheme. The pressure is determined everywhere in the domain (including inside the solids) using a variable coefficient Poisson equation. The solution to momentum, pressure, solid and fluid volume fraction equations everywhere in the domain circumvents the issue of pressure and velocity interpolation, which is a source of spurious oscillations in sharp interface immersed boundary methods. A well-balanced algorithm with consistent mass/momentum transport ensures robust simulations of high density ratio flows with strong body forces. The propos

关键词： diffuse interface Immersed boundary Solid fraction Volume-of-fluid Rigid solids Sedimentation

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diffuse interface modeling of the morphology and rheology of immiscible polymer blends

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PHYSICS OF FLUIDS 2003年第9期15卷 2567-2575页

作者： Keestra, BJ Van Puyvelde, PCJ Anderson, PD Meijer, HEH Eindhoven Univ Technol Fac Mech Engn NL-5600 MB Eindhoven Netherlands Katholieke Univ Leuven Dept Chem Engn B-3001 Louvain Belgium

A diffuse interface model is used to simulate a step-shear experiment of a binary immiscible polymer blend. The gradient theory used in diffuse interface modeling makes it possible to incorporate interfacial tension and governs the process of coalescence and breakup without any additional decision criteria. The interface tensor q, a direct outcome of the model, is used to relate microstructural information to the first-normal stress difference N-1. The results obtained are in qualitative agreement with experiments reported in the literature.(C) 2003 American Institute of Physics.

关键词： .breakup polymer blends interfacial tension Moldenaers morphology development diffuse interface rst-normal stress

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diffuse interface model of volume nucleation in glasses

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THERMOCHIMICA ACTA 1996年第0期280卷 83-100页

作者： Granasy, L Research Institute for Solid State Physics H-1525 Budapest POB 49 Hungary

A diffuse interface theory (DIT) of nucleation is applied for the crystallization of various oxide glasses showing volume nucleation. It is demonstrated that, in contrast with the classical theory which often yields ''anomalous'' nucleation prefactors, the DIT is consistent with the experiments. A method is outlined for distinguishing homogeneous and volumetric heterogeneous nucleation mechanisms.

关键词： diffuse interface glass model nucleation

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diffuse interface energies capturing the Euler number: Relaxation and renomalization

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COMMUNICATIONS IN MATHEMATICAL SCIENCES 2007年第1期5卷 233-242页

作者： Du, Qiang Liu, Chun Ryham, Rolf Wang, Xiaoqiang Penn State Univ Dept Math University Pk PA 16802 USA Rice Univ Dept Math Houston TX 77005 USA Florida State Univ Dept Math Tallahassee FL 32306 USA Florida State Univ Sch Computat Sci Tallahassee FL 32306 USA

We introduce a set of new interfacial energies for approximating the Euler number of level surfaces in the phase field (diffuse-interface) representation. These new formulae have simpler forms than those studied earlier in [Q. Du, C. Liu and X. Wang, Retrieving topological information for phase field models, SIAM J. Appl. Math., 65, 1913-1932, 2005], and do not contain higher order derivatives of the phase field function. Theoretical justifications are provided via formal asymptotic analysis, and practical validations are performed through numerical experiments. Relaxation and renormalization schemes are also developed to improve the robustness of the new energy functionals.

关键词： Euler number topological index implicitly defined surfaces phase field diffuse interface interfacial energy surface tension bending energy vesicle membrane

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diffuse interface Modeling for Electromigration Induced Void Growth 31

Diffuse Interface Modeling for Electromigration Induced Void...

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31st Symposium on Microelectronics Technology and Devices (SBMicro)

作者： Queiroz, L. M. G. M. de Orio, R. L. Univ Estadual Campinas FEEC Sch Elect & Comp Engn DSIFSemicond Instruments & Photon Dept Av Albert Einstein 400 Cid Univ Zeferino Vaz BR-13083852 Campinas SP Brazil

ISBN: (纸本)9781509027880

Modeling and simulation of the temporal evolution of electromigration voids are critical to evaluate the reliability of interconnects of modern integrated circuits. In this work, the diffuse interface model is applied to investigate two different mechanisms of the void evolution due to electromigration. Two-dimensional numerical simulations have been performed and interconnect resistance increase has been monitored. The results recovered the main features of the electromigration induced failure observed in experiments.

关键词： electromigration diffuse interface reliability microelectronic interconnection

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Unconditionally Energy Stable Linear Schemes for the diffuse interface Model with Peng-Robinson Equation of State

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JOURNAL OF SCIENTIFIC COMPUTING 2018年第2期75卷 993-1015页

作者： Li, Hongwei Ju, Lili Zhang, Chenfei Peng, Qiujin Shandong Normal Univ Sch Math & Stat Jinan 250014 Shandong Peoples R China Univ South Carolina Dept Math Columbia SC 29208 USA Renmin Univ China Inst Math Sci Beijing 100872 Peoples R China

In this paper, we investigate numerical solution of the diffuse interface model with Peng-Robinson equation of state, that describes real states of hydrocarbon fluids in the petroleum industry. Due to the strong nonlinearity of the source terms in this model, how to design appropriate time discretizations to preserve the energy dissipation law of the system at the discrete level is a major challenge. Based on the "Invariant Energy Quadratization" approach and the penalty formulation, we develop efficient first and second order time stepping schemes for solving the single-component two-phase fluid problem. In both schemes the resulted temporal semi-discretizations lead to linear systems with symmetric positive definite spatial operators at each time step. We rigorously prove their unconditional energy stabilities in the time discrete sense. Various numerical simulations in 2D and 3D spaces are also presented to validate accuracy and stability of the proposed linear schemes and to investigate physical reliability of the target model by comparisons with laboratory data.

关键词： diffuse interface Linear scheme Peng-Robinson equation of state Invariant energy quadratization Energy stability Penalty formulation

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