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Maxwell-Stefan-theory-based lattice Boltzmann model for diffusion in multicomponent mixtures

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Physical Review E 2019年第2期99卷 023312-023312页

作者： Zhenhua Chai Xiuya Guo Lei Wang Baochang Shi School of Mathematics and Statistics Huazhong University of Science and Technology Wuhan 430074 China Hubei Key Laboratory of Engineering Modeling and Scientific Computing Huazhong University of Science and Technology Wuhan 430074 China School of Mathematics and Physics China University of Geosciences Wuhan 430074 China

The phenomena of diffusion in multicomponent (more than two components) mixtures are universal in both science and engineering, and from the mathematical point of view, they are usually described by the Maxwell-Stefan (MS)-theory-based diffusion equations where the molar average velocity is assumed to be zero. In this paper, we propose a multiple-relaxation-time lattice Boltzmann (LB) model for the mass diffusion in multicomponent mixtures and also perform a Chapman-Enskog analysis to show that the MS continuum equations can be correctly recovered from the developed LB model. In addition, considering the fact that the MS-theory-based diffusion equations are just a diffusion type of partial differential equations, we can also adopt much simpler lattice structures to reduce the computational cost of present LB model. We then conduct some simulations to test this model and find that the results are in good agreement with the previous work. Besides, the reverse diffusion, osmotic diffusion, and diffusion barrier phenomena are also captured. Finally, compared to the kinetic-theory-based LB models for multicomponent gas diffusion, the present model does not include any complicated interpolations, and its collision process can still be implemented locally. Therefore, the advantages of single-component LB method can also be preserved in present LB model.

关键词： Lattice-Boltzmann methods

Generalized lattice boltzmann method: modeling, analysis, and elements

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arXiv 2019年

作者： Shi, Baochang Chai, Zhenhua School of Mathematics and Statistics Huazhong University of Science and Technology Wuhan430074 China Hubei Key Laboratory of Engineering Modeling and Scientific Computing Huazhong University of Science and Technology Wuhan430074 China

In this paper, we first present a unified framework for the modelling of generalized lattice Boltzmann method (GLBM). We then conduct a comparison of the four popular analysis methods (Chapman-Enskog analysis, Maxwell iteration, direct Taylor expansion and recurrence equations approaches) that have been used to obtain the macroscopic Navier-Stokes equations and nonlinear convection-diffusion equations from the GLBM, and show that from mathematical point of view, these four analysis methods are equivalent to each other. Finally, we give some elements that are needed in the implementation of the GLBM, and also find that some available LB models can be obtained from this GLBM. Copyright © 2019, The Authors. All rights reserved.

关键词： Nonlinear equations

On Coupling Lemma and Stochastic Properties with Unbounded Observables for 1-d Expanding Maps

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arXiv 2020年

作者： Chen, Jianyu Zhang, Hong-Kun Zhang, Yiwei School of Mathematical Sciences Center for Dynamical Systems and Differential Equations Soochow University SuzhouJiangsu215006 China Department of Mathematics and Statistics University of Massachusetts Amherst AmherstMA01003 United States School of Mathematics and Statistics Center for Mathematical Sciences Hubei Key Laboratory of Engineering Modeling and Scientific Computing Hua-Zhong University of Sciences and Technology Wuhan430074 China

In this paper, we establish a coupling lemma for standard families in the setting of piecewise expanding interval maps with countably many branches. Our method merely requires that the expanding map satisfies Chernov’s one-step expansion at q-scale and eventually covers a magnet interval. Therefore, our approach is particularly powerful for maps whose inverse Jacobian has low regularity and those who does not satisfy the big image property. The main ingredients of our coupling method are two crucial lemmas: the growth lemma in terms of the characteristic Z function and the covering ratio lemma over the magnet interval. We first prove the existence of an absolutely continuous invariant measure. What is more important, we further show that the growth lemma enables the liftablity of the Lebesgue measure to the associated Hofbauer tower, and the resulting invariant measure on the tower admits a decomposition of Pesin-Sinai type. Furthermore, we obtain the exponential decay of correlations and the almost sure invariance principle (which is a functional version of the central limit theorem). For the first time, we are able to make a direct relation between the mixing rates and the Z function, see (2.7). The novelty of our results relies on establishing the regularity of invariant density, as well as verifying the stochastic properties for a large class of unbounded observables. Finally, we verify our assumptions for several well known examples that were previously studied in the literature, and unify results to these examples in our framework. Copyright © 2020, The Authors. All rights reserved.

关键词： Inverse problems

Corrigendum to ‘Comparative study of natural convection melting inside a cubic cavity using an improved two-relaxation-time lattice Boltzmann model’ [International Journal of Heat and Mass Transfer, 143 (2019) 118449]

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International Journal of Heat and Mass Transfer 2021年 166卷

作者： Yong Zhao Lei Wang Zhenhua Chai Baochang Shi School of Mathematics and Statistics Huazhong University of Science and Technology Wuhan 430074 China Hubei Key Laboratory of Engineering Modeling and Scientific Computing Huazhong University of Science and Technology Wuhan 430074 China School of Mathematics and Physics China University of Geosciences Wuhan 430074 China

Mixed bounce-back boundary scheme of the general propagation lattice Boltzmann method for advection-diffusion equations

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Physical Review E 2019年第6期99卷 063316-063316页

作者： Xiuya Guo Zhenhua Chai Shengyong Pang Yong Zhao Baochang Shi School of Mathematics and Statistics Huazhong University of Science and Technology Wuhan 430074 China Hubei Key Laboratory of Engineering Modeling and Scientific Computing Huazhong University of Science and Technology Wuhan 430074 China State Key Laboratory of Material Processing and Die & Mould Technology Huazhong University of Science and Technology Wuhan 430074 China

In this work, a mixed bounce-back boundary scheme of general propagation lattice Boltzmann (GPLB) model is proposed for isotropic advection-diffusion equations (ADEs) with Robin boundary condition, and a detailed asymptotic analysis is also conducted to show that the present boundary scheme for the straight walls has a second-order accuracy in space. In addition, several numerical examples, including the Helmholtz equation in a square domain, the diffusion equation with sinusoidal concentration gradient, one-dimensional transient ADE with Robin boundary and an ADE with a source term, are also considered. The results indicate that the numerical solutions agree well with the analytical ones, and the convergence rate is close to 2.0. Furthermore, through adjusting the two parameters in the GPLB model properly, the present boundary scheme can be more accurate than some existing lattice Boltzmann boundary schemes.

关键词： Lattice-Boltzmann methods

Unique continuation properties for one dimensional higher order Schrödinger equations

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arXiv 2019年

作者： Huang, Tianxiao Huang, Shanlin Zheng, Quan Sun Yat-sen University Guangdong Zhuhai519082 China School of Mathematics and Statistics Hubei Key Laboratory of Engineering Modeling and Scientific Computing Huazhong University of Science and Technology Hubei Wuhan430074 China School of Mathematics and Statistics Huazhong University of Science and Technology Hubei Wuhan430074 China

We study two types of unique continuation properties for the higher order Schrödinger equation with potential i∂tu = (−∆x)mu + V(t, x)u, (t, x) ∈ 1+n, 2 ≤ m ∈ N+. The first one says if u has certain exponential decay at two times, then u ≡ 0, and this result is sharp by constructing critical non-trivial solutions. The second one says if u ≡ 0 in an arbitrary half-space of 1+n, then u ≡ 0 identically. The uniqueness theorems are given when n = 1, but we also prove partial results when n ∈ N+ for their own interests. Possibility or obstacles to proving these unique continuation properties in higher spatial dimensions are also discussed. Copyright © 2019, The Authors. All rights reserved.

关键词： Geometry

The Most Probable Transition Paths of Stochastic Dynamical Systems: A Sufficient and Necessary Characterization

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arXiv 2021年

作者： Huang, Yuanfei Huang, Qiao Duan, Jinqiao School of Mathematics and Statistics Center for Mathematical Sciences Hubei Key Laboratory of Engineering Modeling and Scientific Computing Huazhong University of Science and Technology Wuhan430074 China Department of Statistics and Data Science National University of Singapore 6 Science Drive 2 Singapore117546 Singapore School of Data Science City University of Hong Kong Kowloon Hong Kong Department of Mathematics Faculty of Sciences University of Lisbon Campo Grande Edifício C6 LisboaPT-1749-016 Portugal Division of Mathematical Sciences School of Physical and Mathematical Sciences Nanyang Technological University 21 Nanyang Link Singapore637371 Singapore The Dongguan Key Laboratory for Data Science and Intelligent Medicine Department of Mathematics Department of Physics Great Bay University Guangdong Dongguan523000 China

The most probable transition paths of a stochastic dynamical system are the global minimizers of the Onsager–Machlup action functional and can be described by a necessary but not sufficient condition, the Euler–Lagrange equation (a second-order differential equation with initial-terminal conditions) from a variational principle. This work is devoted to showing a sufficient and necessary characterization for the most probable transition paths of stochastic dynamical systems with Brownian noise. We prove that, under appropriate conditions, the most probable transition paths are completely determined by a first-order ordinary differential equation. The equivalence is established by showing that the Onsager–Machlup action functional of the original system can be derived from the corresponding Markovian bridge process. For linear stochastic systems and the nonlinear Hongler’s model, the first-order differential equations determining the most probable transition paths are shown analytically to imply the Euler–Lagrange equations of the Onsager–Machlup functional. For general nonlinear systems, the determining first-order differential equations can be approximated, in a short time or for the small noise case. Some numerical experiments are presented to illustrate our results. Copyright © 2021, The Authors. All rights reserved.

关键词： Dynamical systems

A generalized lattice boltzmann model for fluid flow system and its application in two-phase flows

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arXiv 2019年

作者： Yuan, Xiaolei Chai, Zhenhua Wang, Huili Shi, Baochang School of Mathematics and Statistics Huazhong University of Science and Technology Wuhan430074 China Hubei Key Laboratory of Engineering Modeling and Scientific Computing Huazhong University of Science and Technology Wuhan430074 School of Mathematics and Computer Science Wuhan Textile University Wuhan430073

In this paper, a generalized lattice Boltzmann (LB) model with a mass source is proposed to solve both incompressible and nearly incompressible Navier-Stokes (N-S) equations. This model can be used to deal with single-phase and twophase ows problems with a mass source term. From this generalized model, we can not only get some existing models, but also derive new models. Moreover, for the incompressible model derived, a modified pressure scheme is introduced to calculate the pressure, and then to ensure the accuracy of the model. In this work, we will focus on a two-phase ow system, and in the frame work of our generalized LB model, a new phase-field-based LB model is developed for incompressible and quasi-incompressible two-phase ows. A series of numerical simulations of some classic physical problems, including a spinodal decomposition, a static droplet, a layered Poiseuille ow, and a bubble rising ow under buoyancy, are performed to validate the developed model. Besides, some comparisons with previous quasi-incompressible and incompressible LB models are also carried out, and the results show that the present model is accurate in the study of two-phase ows. Finally, we also conduct a comparison between quasiincompressible and incompressible LB models for two-phase ow problems, and find that in some cases, the proposed quasi-incompressible LB model performs better than incompressible LB models. Copyright © 2019, The Authors. All rights reserved.

关键词： Two phase flow

Phase-field-based lattice boltzmann model for immiscible incompressible n-phase flows

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arXiv 2019年

作者： Yuan, Xiaolei Liang, Hong Chai, Zhenhua Shi, Baochang School of Mathematics and Statistics Huazhong University of Science and Technology Wuhan430074 China Hubei Key Laboratory of Engineering Modeling and Scientific Computing Huazhong University of Science and Technology Wuhan430074 China Department of Physics Hangzhou Dianzi University Hangzhou310018 China

In this paper, we develop an efficient lattice Boltzmann (LB) model for simulating immiscible incompressible N-phase ows (N ≥ 2) based on the Cahn-Hilliard phase field theory. In order to facilitate the design of LB model and reduce the calculation of the gradient term, the governing equations of the N-phase system are reformulated, and they satisfy the conservation of mass, momentum and the second law of thermodynamics. In the present model, (N -1) LB equations are employed to capture the interface, and another LB equation is used to solve the Navier-Stokes (N-S) equations, where a new distribution function for the total force is delicately designed to reduce the calculation of the gradient term. The developed model is first validated by two classical benchmark problems, including the tests of static droplets and the spreading of a liquid lens, the simulation results show that the current LB model is accurate and efficient for simulating incompressible N-phase uid ows. To further demonstrate the capability of the LB model, two numerical simulations, including dynamics of droplet collision for four uid phases and dynamics of droplets and interfaces for five uid phases, are performed to test the developed model. The results show that the present model can successfully handle complex interactions among N (N ≥ 2) immiscible incompressible ows. Copyright © 2019, The Authors. All rights reserved.

关键词： Navier Stokes equations