Mass conservative and energy stable finite difference methods for the quasi-incompressible Navier–Stokes–Cahn–Hilliard system: Primitive variable and projection-type schemes

Guo, Zhu; Lin, Ping; Lowengrub, John; Wise, Steven M.

doi:10.1016/j.cma.2017.08.011

Cited by 87 publications

(61 citation statements)

References 51 publications

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“…[50][51][52][53][54] On account of convection, a portion of the free energy is transformed into kinetic energy. [50][51][52][53][54] On account of convection, a portion of the free energy is transformed into kinetic energy.…”

Section: The-state-of-the-art Phase-field Model To Simulate Phase Sepmentioning

confidence: 99%

Progress Report on Phase Separation in Polymer Solutions

et al. 2019

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The ORCID identification number(s) for the author(s) of this article can be found under https://doi.org/10.1002/adma.201806733.Polymeric porous media (PPM) are widely used as advanced materials, such as sound dampening foams, lithium-ion batteries, stretchable sensors, and biofilters. The functionality, reliability, and durability of these materials have a strong dependence on the microstructural patterns of PPM. One underlying mechanism for the formation of porosity in PPM is phase separation, which engenders polymer-rich and polymer-poor (pore) phases. Herein, the phase separation in polymer solutions is discussed from two different aspects: diffusion and hydrodynamic effects. For phase separation governed by diffusion, two novel morphological transitions are reviewed: "cluster-topercolation" and "percolation-to-droplets," which are attributed to an effect that the polymer-rich and the solvent-rich phases reach the equilibrium states asynchronously. In the case dictated by hydrodynamics, a deterministic nature for the microstructural evolution during phase separation is scrutinized. The deterministic nature is caused by an interfacial-tensiongradient (solutal Marangoni force), which can lead to directional movement of droplets as well as hydrodynamic instabilities during phase separation. Polymeric Porous Media

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Section: The-state-of-the-art Phase-field Model To Simulate Phase Sepmentioning

confidence: 99%

Progress Report on Phase Separation in Polymer Solutions

et al. 2019

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“…On the other hand, for the mass averaged velocity method, the mass conservation is assured instead of incompressibility. This naturally yields the quasiincompressible Navier-Stoker-Cahn-Hilliard (q-NSCH) model [34,42], which in fact leads to a slightly compressible mixture only inside the interfacial region.…”

Section: Introductionmentioning

confidence: 99%

An energy stable C0 finite element scheme for a quasi-incompressible phase-field model of moving contact line with variable density

Shen

Huang

Lin

et al. 2020

Journal of Computational Physics

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In this paper, we focus on modeling and simulation of two-phase flow problems with moving contact lines and variable density. A thermodynamically consistent phase-field model with General Navier Boundary Condition is developed based on the concept of quasi-incompressibility and the energy variational method. A mass conserving C0 finite element scheme is proposed to solve the PDE system. Energy stability is achieved at the fully discrete level. Various numerical results confirm that the proposed scheme for both P1 element and P2 element are energy stable.

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“…This is because non-energy-stable schemes may introduce truncation errors that destroy the physical law numerically. Thus, developing energy stable numerical algorithms is necessary for accurately resolving the dynamics of gradient flow models [20,21,23,30,44].Over the years, the development of numerical algorithms has been done primarily on a specific gradient flow model, exploiting its specific mathematical properties and structures. The noticeable ones include the Allen-Cahn and Cahn-Hilliard equation [1,9,11,20,24,32,36,42] as well as the molecular beam epitaxy model [6,12,[26][27][28]41].…”

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Arbitrarily high-order unconditionally energy stable SAV schemes for gradient flow models

Gong

Zhao

Wang

2020

Computer Physics Communications

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We present a paradigm for developing arbitrarily high order, linear, unconditionally energy stable numerical algorithms for gradient flow models. We apply the energy quadratization (EQ) technique to reformulate the general gradient flow model into an equivalent gradient flow model with a quadratic free energy and a modified mobility. Given solutions up to t n = n∆t with ∆t the time step size, we linearize the EQ-reformulated gradient flow model in (t n , t n+1 ] by extrapolation. Then we employ an algebraically stable Runge-Kutta method to discretize the linearized model in (t n , t n+1 ]. Then we use the Fourier pseudo-spectral method for the spatial discretization to match the order of accuracy in time. The resulting fully discrete scheme is linear, unconditionally energy stable, uniquely solvable, and may reach arbitrarily high order. Furthermore, we present a family of linear schemes based on predictioncorrection methods to complement the new linear schemes. Some benchmark numerical examples are given to demonstrate the accuracy and efficiency of the schemes.solutions [13]. This is because non-energy-stable schemes may introduce truncation errors that destroy the physical law numerically. Thus, developing energy stable numerical algorithms is necessary for accurately resolving the dynamics of gradient flow models [20,21,23,30,44].Over the years, the development of numerical algorithms has been done primarily on a specific gradient flow model, exploiting its specific mathematical properties and structures. The noticeable ones include the Allen-Cahn and Cahn-Hilliard equation [1,9,11,20,24,32,36,42] as well as the molecular beam epitaxy model [6,12,[26][27][28]41]. As a result, most numerical algorithms developed for a specific gradient flow model can hardly be applied to another gradient flow model with a different free energy and mobility. The statusquo did not change until the energy quaratization (EQ) approach was introduced a few years ago [39,44], which turns out to be rather general so that it can be readily applied to general gradient flow models with no restrictions on the specific form of the mobility and free energy [16,40] (so long as it is bounded below, which is usually the case). Based on the idea of EQ, the scalar auxiliary variable (SAV) approach was introduced later [31], where each time step only linear systems with constant coefficients need to be solved. Several other extensions of EQ and SAV approaches have been explored further. For instance, regularization terms [5] and stabilization terms [38] are added, and the modified energy quadratization technique [22,25] are introduced to improve the EQ methodology. We notice that most existing schemes related to the EQ methodology are up to 2nd order accurate in time. Some higher-order ETD schemes have been introduced to solve the MBE model and Cahn-Hilliard models recently, but their theoretical proofs of energy stability are still missing. Shen et al.[30] remarked on higher-order BDF schemes using the SAV approach, but no rigorous theoretical p...

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Mass conservative and energy stable finite difference methods for the quasi-incompressible Navier–Stokes–Cahn–Hilliard system: Primitive variable and projection-type schemes

Cited by 87 publications

References 51 publications

Progress Report on Phase Separation in Polymer Solutions

Progress Report on Phase Separation in Polymer Solutions

An energy stable C0 finite element scheme for a quasi-incompressible phase-field model of moving contact line with variable density

Arbitrarily high-order unconditionally energy stable SAV schemes for gradient flow models

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