基于格子Boltzmann方法的非线性渗流研究
|
摘要
非线性渗流或非达西渗流作为渗流力学中的一个重要分支，由于其在农业、能源、冶金、化工、材料、环境、生命科学以及医学等众多领域的广泛应用而引起国内外学者的普遍关注，并发展成为国际上研究的一个热点问题。通过对大量相关文献的分析可以看出，非线性渗流问题自身的复杂性大大限制了人们对其背后物理机理的认识，以至于对于某些问题至今也未能给出较为合理准确的数学模型或物理公式。目前对非线性渗流的研究主要基于三类方法：一是实验方法，作为一种传统的研究方法，其在渗流力学中仍然发挥着重要的作用，但是，在实际中该方法又受到很多条件的限制，比如实验周期长、花费大、受实验环境的影响等。二是解析或近似方法，这类方法通常会对问题先进行一定的简化或给出一些合理的假设，在此基础上，进一步通过数学的手段来求得非线性渗流的解析解或近似解。目前，这类方法中最具代表的就是近年来发展起来的基于分形理论的解析方法。另外一类就是数值方法，随着计算机技术的飞速发展，该方法越来越受到人们的广泛关注，并已成为解决实际的问题的一种重要手段。与实验方法相比，数值方法依赖于前期建立的数学模型，但它具有周期短、耗费小、不易受环境影响等特点；同时，该方法也可以弥补解析或近似方法所存在的不足。
目前对非线性渗流的数值研究主要是借助于传统的数值方法(如有限差分法，有限体积法，有限元方法等)来数值求解孔隙尺度上的Navier-Stokes方程或表征体元(REV)尺度体积平均的Navier-Stokes方程。作为一种研究非线性渗流问题的有效手段，数值研究近年来备受关注，其应用范围也越来越广泛，但多孔介质的复杂结构使得这些方法在研究渗流问题时会遇到如下困难：
(1)边界处理复杂，破坏数值方法自身的稳定性；
(2)多孔介质中的微尺度效应难以体现；
(3)计算量较大，并行效率低。
鉴于此，发展高效的数值方法，并利用其研究非线性渗流规律，建立完善的数学描述是揭示非线性物理机理的理论基础。
格子Boltzmann方法(Lattice Boltzmann Method，LBM)作为一种新兴的微观方法，它最初起源于格子气自动机(Lattice Gas Automata，LGA)，但后来也被证实可以从连续的Boltzmann方程经过差分离散得到。格子Boltzmann方法与以连续微分方程为基础的宏观计算流体力学方法有着本质的不同，它是基于流体微观模型和细观动理论方程的方法，因此不受连续假设的限制，这为其能够模拟多孔介质中微细管道中的流动、刻画滑脱效应(Klinkenberg效应)提供了坚实的理论基础。此外，与传统的计算流体力学方法(基于Navier-Stokes方程的数值方法，比如，有限差分法、有限体积法、有限元方法等)相比，LBM还具有许多独特的优势，如计算效率高、边界条件容易实现、具有天然并行性等，更为重要的是其自身的微观特性使得它成为研究多孔介质中非连续流动提供了一种新的有效手段。本文首先发展了LBM的相关理论，在此基础上，利用LBM对单相非线性渗流力学中的一些前沿问题进行了探讨。论文的主要工作包括以下两个方面：
(1)在格子Boltzmann方法的相关理论方面，论文首先提出了一种处理微尺度流动的反弹与Maxwell漫反射组合边界条件，并详细分析了该边界条件在单松弛模型与多松弛模型中的离散效应；其次，论文还给出了一种求解Poisson方程的新的格子Boltzmann模型，该模型可以有效克服已有模型的不足。这些理论的发展将为推动LBM在渗流领域的应用奠定必要的基础。
(2)在非线性渗流的数值研究方面，论文首先系统研究了高速非线性渗流，详细分析了低雷诺数下产生非线性现象的物理机理，并进一步并给出了改进的数学模型；其次，基于(1)中的理论结果，论文详细研究了气体通过多孔介质的滑脱效应，利用文中发展的一些理论结果揭示了滑脱效应产生的物理根源；最后，论文还进一步考察了微尺度多孔材料中的电渗流，系统分析了各种因素对电渗流速度分布的影响，这些结果对实际中的工程问题具有重要的指导意义。
总之，本文利用格子Boltzmann方法研究了孔隙尺度和REV尺度的非线性渗流，为推动该方法在渗流领域中的应用作出了许多有益的尝试。同时，针对针对高度非线性的Poisson-Boltzmann方程和复杂的多孔介质，我们分别构造了新的模型和边界条件；此外，我们还对格子Boltzmann方法进行了大量的数值实验，验证了该方法用于模拟渗流问题的可靠性。这些工作为后续研究的开展奠定了必要的基础。
Nonlinear or non-Darcy flow is an important part of flow in porous media,it attainsan increasing attention and becomes a hot topic due to its wide applications in agriculture,energy,metallurgy,chemical engineering,material science,environment engineering,lifesciences,medicine etc.According to a large number literature published previously,it isfound that,owing to the complexity inherent in nonlinear flow in porous media,the physicsmechanics behind the nonlinear phenomena is not understood clearly,and there is no rea-sonable or accurate formula that can be used to describe this nonlinear effect.Up to now,there may be three methods that can be used to study nonlinear flow in porous media,one isexperimental method,which,as a traditional method,still plays an important role in inves-tigating flow in porous media,but there are also some constrains that limit its applicationsin practice,for example,the time and money spent on experiments are usually very large,and what is more,the practical experiments are usually affected by surrounding conditions.The second is analytical or approximate method,coupling with some assumptions or sim-plicities,it can be used to derive some analytical or approximate solution of nonlinear flowin porous media.The last one is numerical method,which becomes an important approachand attains increasing attention in solving many practical problems with the developmentof computer,compared with experimental method,it depends on mathematical model ofproblems,but it also has many merits,for instance,the expense of numerical method is less,and the surrounding effects can be reduced significantly,meanwhile,it also can be used tocircumvent some shortcomings inherent in analytical or approximate methods.
Now,the numerical researches on nonlinear flow in porous media are to solve theNavier-Stokes equations at pore scale or REV scale with the traditional numerical methods(for example,finite difference method,finite volume method,finite element method andso on.).In the past several years,the numerical research,as an effective method to studynonlinear flow,attained great attention,and was applied in more and more fields.However,these numerical methods will face some problems when they are used to simulate flow inporous media:
(1) The treatment on boundary condition is very complex,which leads to the fact thatthe stability of numerical method becomes much worse;
(2) It is a hard work to discover the micro-scale effect for gas flow in porous media;
(3) The computational expense is very large and the parallel efficiency is very low.Therefore,it is desirable to develop some advanced numerical methods,which can be used to investigate nonlinear flow in porous media,and further to built mathematic formula,this is also the theoretical premise to explain nonlinear physical mechanics.
As a new mesoscopic method,Lattice Boltzmann method (LBM) originates from Lat-tice Gas Automata (LGA),but later it is shown that this method also can be derived fromcontinuum Boltzmann equation with proper discrete scheme.Unlike the traditional compu-tational fluid dynamics methods,LBM is derived from kinetic theory,which leads to LBMis not constrained by continuum hypothesis,it is just this reason that LBM can be used tosimulate pore scale flow with slippage effect in porous media,and successfully display slip-page effect.In addition,compared to traditional numerical methods (the methods to solveNavier-Stokes equations,for example,finite difference method,finite volume method,finiteelement method etc.),LBM has some distinguished characteristics,such as high efficiencyof computation,easy implementation for complex boundary conditions,fully parallelismand so on,what is more,the microscopic characteristic makes it be an effective approach tostudy flow with slippage effect in porous media.In this thesis,the theory of LBM is firstdeveloped,and then,some new and open problems of single phase nonlinear flow in porousmedia are studied with LBM.This thesis is composed of two parts:
(1) In term of the theory related to LBM,a new boundary condition,i.e.,combinedboundary condition of bounce back and Maxwell diffusion,is first proposed,simultaneously,a detailed analysis on discrete effects of this boundary condition in single relaxation modeland multi-relaxation model is further presented,then,a novel model,which can be used tocircumvent shortcomings in previous models,is provided to solve the Poisson equation.Wewould like to point out that the development to LBM will provide a necessary premise forfollowing research on flow in porous media.
(2) In term of the numerical research on nonlinear flow in porous media,the high-velocity nonlinear flow in porous media is first studied,in this subsection,we not onlypresent a more precise formula which can be used to describe high-velocity nonlinear flowin porous media,but also give a detailed analysis on physical mechanism for nonlinear phe-nomena which are usually observed at low Reynolds number,then,the slippage effect ofgas through porous media is also investigated,the physical mechanism of slippage effect isrevealed with the aid of the new theory developed in this thesis,finally,the electro-osmoticflow in porous media is also studied,a detailed analysis on effects of many factors on distri-bution of velocity is presented in this part,furthermore,the results derived in this part playan important role in solving problems in practice.
In conclusion,we used LBM to investigate nonlinear flow in porous media at REVand pore scale,and made many valuable efforts to accelerate the applications of LBM inflow through porous media.Simultaneously,we propose a new lattice Boltzmann model to solve nonlinear Poisson-Boltzmann and a new combination boundary condition to complexporous media.In addition,a large number of numerical tests are conducted to verify thismethod in studying flow in porous media.This work can be viewed as a necessary basis forfuture studies.
目前对非线性渗流的数值研究主要是借助于传统的数值方法(如有限差分法，有限体积法，有限元方法等)来数值求解孔隙尺度上的Navier-Stokes方程或表征体元(REV)尺度体积平均的Navier-Stokes方程。作为一种研究非线性渗流问题的有效手段，数值研究近年来备受关注，其应用范围也越来越广泛，但多孔介质的复杂结构使得这些方法在研究渗流问题时会遇到如下困难：
(1)边界处理复杂，破坏数值方法自身的稳定性；
(2)多孔介质中的微尺度效应难以体现；
(3)计算量较大，并行效率低。
鉴于此，发展高效的数值方法，并利用其研究非线性渗流规律，建立完善的数学描述是揭示非线性物理机理的理论基础。
格子Boltzmann方法(Lattice Boltzmann Method，LBM)作为一种新兴的微观方法，它最初起源于格子气自动机(Lattice Gas Automata，LGA)，但后来也被证实可以从连续的Boltzmann方程经过差分离散得到。格子Boltzmann方法与以连续微分方程为基础的宏观计算流体力学方法有着本质的不同，它是基于流体微观模型和细观动理论方程的方法，因此不受连续假设的限制，这为其能够模拟多孔介质中微细管道中的流动、刻画滑脱效应(Klinkenberg效应)提供了坚实的理论基础。此外，与传统的计算流体力学方法(基于Navier-Stokes方程的数值方法，比如，有限差分法、有限体积法、有限元方法等)相比，LBM还具有许多独特的优势，如计算效率高、边界条件容易实现、具有天然并行性等，更为重要的是其自身的微观特性使得它成为研究多孔介质中非连续流动提供了一种新的有效手段。本文首先发展了LBM的相关理论，在此基础上，利用LBM对单相非线性渗流力学中的一些前沿问题进行了探讨。论文的主要工作包括以下两个方面：
(1)在格子Boltzmann方法的相关理论方面，论文首先提出了一种处理微尺度流动的反弹与Maxwell漫反射组合边界条件，并详细分析了该边界条件在单松弛模型与多松弛模型中的离散效应；其次，论文还给出了一种求解Poisson方程的新的格子Boltzmann模型，该模型可以有效克服已有模型的不足。这些理论的发展将为推动LBM在渗流领域的应用奠定必要的基础。
(2)在非线性渗流的数值研究方面，论文首先系统研究了高速非线性渗流，详细分析了低雷诺数下产生非线性现象的物理机理，并进一步并给出了改进的数学模型；其次，基于(1)中的理论结果，论文详细研究了气体通过多孔介质的滑脱效应，利用文中发展的一些理论结果揭示了滑脱效应产生的物理根源；最后，论文还进一步考察了微尺度多孔材料中的电渗流，系统分析了各种因素对电渗流速度分布的影响，这些结果对实际中的工程问题具有重要的指导意义。
总之，本文利用格子Boltzmann方法研究了孔隙尺度和REV尺度的非线性渗流，为推动该方法在渗流领域中的应用作出了许多有益的尝试。同时，针对针对高度非线性的Poisson-Boltzmann方程和复杂的多孔介质，我们分别构造了新的模型和边界条件；此外，我们还对格子Boltzmann方法进行了大量的数值实验，验证了该方法用于模拟渗流问题的可靠性。这些工作为后续研究的开展奠定了必要的基础。
Nonlinear or non-Darcy flow is an important part of flow in porous media,it attainsan increasing attention and becomes a hot topic due to its wide applications in agriculture,energy,metallurgy,chemical engineering,material science,environment engineering,lifesciences,medicine etc.According to a large number literature published previously,it isfound that,owing to the complexity inherent in nonlinear flow in porous media,the physicsmechanics behind the nonlinear phenomena is not understood clearly,and there is no rea-sonable or accurate formula that can be used to describe this nonlinear effect.Up to now,there may be three methods that can be used to study nonlinear flow in porous media,one isexperimental method,which,as a traditional method,still plays an important role in inves-tigating flow in porous media,but there are also some constrains that limit its applicationsin practice,for example,the time and money spent on experiments are usually very large,and what is more,the practical experiments are usually affected by surrounding conditions.The second is analytical or approximate method,coupling with some assumptions or sim-plicities,it can be used to derive some analytical or approximate solution of nonlinear flowin porous media.The last one is numerical method,which becomes an important approachand attains increasing attention in solving many practical problems with the developmentof computer,compared with experimental method,it depends on mathematical model ofproblems,but it also has many merits,for instance,the expense of numerical method is less,and the surrounding effects can be reduced significantly,meanwhile,it also can be used tocircumvent some shortcomings inherent in analytical or approximate methods.
Now,the numerical researches on nonlinear flow in porous media are to solve theNavier-Stokes equations at pore scale or REV scale with the traditional numerical methods(for example,finite difference method,finite volume method,finite element method andso on.).In the past several years,the numerical research,as an effective method to studynonlinear flow,attained great attention,and was applied in more and more fields.However,these numerical methods will face some problems when they are used to simulate flow inporous media:
(1) The treatment on boundary condition is very complex,which leads to the fact thatthe stability of numerical method becomes much worse;
(2) It is a hard work to discover the micro-scale effect for gas flow in porous media;
(3) The computational expense is very large and the parallel efficiency is very low.Therefore,it is desirable to develop some advanced numerical methods,which can be used to investigate nonlinear flow in porous media,and further to built mathematic formula,this is also the theoretical premise to explain nonlinear physical mechanics.
As a new mesoscopic method,Lattice Boltzmann method (LBM) originates from Lat-tice Gas Automata (LGA),but later it is shown that this method also can be derived fromcontinuum Boltzmann equation with proper discrete scheme.Unlike the traditional compu-tational fluid dynamics methods,LBM is derived from kinetic theory,which leads to LBMis not constrained by continuum hypothesis,it is just this reason that LBM can be used tosimulate pore scale flow with slippage effect in porous media,and successfully display slip-page effect.In addition,compared to traditional numerical methods (the methods to solveNavier-Stokes equations,for example,finite difference method,finite volume method,finiteelement method etc.),LBM has some distinguished characteristics,such as high efficiencyof computation,easy implementation for complex boundary conditions,fully parallelismand so on,what is more,the microscopic characteristic makes it be an effective approach tostudy flow with slippage effect in porous media.In this thesis,the theory of LBM is firstdeveloped,and then,some new and open problems of single phase nonlinear flow in porousmedia are studied with LBM.This thesis is composed of two parts:
(1) In term of the theory related to LBM,a new boundary condition,i.e.,combinedboundary condition of bounce back and Maxwell diffusion,is first proposed,simultaneously,a detailed analysis on discrete effects of this boundary condition in single relaxation modeland multi-relaxation model is further presented,then,a novel model,which can be used tocircumvent shortcomings in previous models,is provided to solve the Poisson equation.Wewould like to point out that the development to LBM will provide a necessary premise forfollowing research on flow in porous media.
(2) In term of the numerical research on nonlinear flow in porous media,the high-velocity nonlinear flow in porous media is first studied,in this subsection,we not onlypresent a more precise formula which can be used to describe high-velocity nonlinear flowin porous media,but also give a detailed analysis on physical mechanism for nonlinear phe-nomena which are usually observed at low Reynolds number,then,the slippage effect ofgas through porous media is also investigated,the physical mechanism of slippage effect isrevealed with the aid of the new theory developed in this thesis,finally,the electro-osmoticflow in porous media is also studied,a detailed analysis on effects of many factors on distri-bution of velocity is presented in this part,furthermore,the results derived in this part playan important role in solving problems in practice.
In conclusion,we used LBM to investigate nonlinear flow in porous media at REVand pore scale,and made many valuable efforts to accelerate the applications of LBM inflow through porous media.Simultaneously,we propose a new lattice Boltzmann model to solve nonlinear Poisson-Boltzmann and a new combination boundary condition to complexporous media.In addition,a large number of numerical tests are conducted to verify thismethod in studying flow in porous media.This work can be viewed as a necessary basis forfuture studies.
引文
[1]沈平平，杨永智．温室气体在石油开采中资源化利用的科学问题．中国基础科学，2006，3：23-81．
[2]石钟慈．第三种科学方法-计算机时代的科学计算．(第一版)．北京：清华大学出版社，2000．
[3]刘儒勋，王志峰．数值模拟方法和运动界面追踪．(第一版)．合肥：中国科学技大学出版社，2001．
[4]苏铭德，黄素逸．计算流体力学基础．(第一版)．北京：清华大学出版社，1997．
[5]张涵信，沈孟育．计算流体力学：差分方法的原理和应用．(第一版)．北京：国防工业出版社，2003．
[6]Karniadakis G.E.,Beskok A.,Aluru N.Microflows and Nanoflows:Fundamentals and Simulations.New York:Springer-Verlag,2005.
[7]Gad-el-Hak M.The MEMS Handbook.Boca Raton:CRC Press,2006.
[8]Wolf-Gladrow D.A.Lattice-Gas Cellular Automata and Lattice Boltzmann Models.Springer-Verlag Berlin Heidelberg,2000.
[9]Succi S.The Lattice Boltzmann Equation for Fluid Dynamics and Beyond.Oxford:Clarendon Press,2001.
[10]郭照立，郑楚光，李青等．流体动力学的格子Boltzmann方法．(第一版)．武汉：湖北科学技术出版社，2002．
[11]Benzi R.,Succi S.,Vergassola M.The lattice Boltzmann equation:theory and applications.Phys.Rep.,1992,222(3):145-197.
[12]Chen S.,Doolen G.Lattice Boltzmann method for fluid flows.Annu.Rev.Fluid Mech.,1998,30:329-364.
[13]He X.,Luo L.S.Theory of lattice Boltzmann method:From the Boltzmann equation to the lattice Boltzmann equation.Phys.Rev.E,1997,56(6):6811-6817.
[14]Shan X.,He X.Discretization of the velocity space in the solution of the Boltzmann equation.Phys.Rev.Lett.,1998,80(1):65-68.
[15]李元香．格子气自动机．(第一版)．北京：清华大学出版社，1994．
[16]Frish U,Hasslacher B,Pomeau Y.Lattice-gas automata for the Navier-Stokes equations.Phys.Rev.Lett.,1986,56:1505-1508.
[17]Chapman S.,Cowling T.G.The Mathematical Theory of Non-Uniform Gases.(3rd edition).Cambridge:Cambridge University Press,1970.
[18]Chopard B.,Droz M.Cellular Automata Modeling of Physical Systems.Beijing:Tsinghua University Press,2003.
[19] Chen H., Chen S., Matthaeus H. Recovery of the Navier-Stokes equations using a lattice gas Boltzmann method. Phys. Rev. A, 1992, 45:R5339-R5342.
[20] Qian Y. H., d'Humieres D., Lallemand P. Lattice BGK models for Navier-Stokes equation. Europhys.Lett., 1992, 17(6):479-484.
[21] Rothman D. H. Cellular-automaton fluids: a model for flow in porous media. Geophysics, 1988, 53(4):509-518.
[22] Succi S., Foti E., Higuera F. Three-dimensional flows in complex geometries with the lattice Boltzmann method. Europhy. Lett., 1989, 10(5):433-438.
[23] Adrover A., Giona M. Predictive model for permeability of correlated porous media. Chem. Eng. J., 1996, 64(1):7-19.
[24] Singh M.,MohantyK. K. Permeability of spatially correlated porous media. Chem. Eng. Sci., 2000, 55(22):5393-5403.
[25] Kim J., Lee J., Lee K. C. Nonlinear correction to Darcy's law for a flow through periodic arrays of elliptic cylinders. Physica A, 2001, 293(1-2):13-20.
[26] Koch D. L., Ladd A. J. C. Moderate Reynolds number flows through periodic and random arrays of aligned cylinders. J. Fluid Mech., 1997, 349:31-66.
[27] Koch D. L., Hill R. J. Inertial effects in suspension and porous media flows. Annu. Rev. Fluid Mech., 2001,33:619-647.
[28] Jeong N., Choi D. H., Lin C. L. Prediction of Darcy-Forchheimer drag for micro-porous structures of complex geometry using the lattice Boltzmann method. J. Micromech. Microeng., 2006, 16:2240-2250.
[29] Tang G. H., Tao W. Q., He Y. L. Gas slippage effect on microscale porous flow using lattice Boltzmann method. Phys. Rev. E, 2005, 72:056301.
[30] Tang G. H., Tao W. Q., He Y. L. Three-dimensional lattice Boltzmann model for gaseous flow in rectangular microducts and microscale porous media. J. Appl. Phys., 2005,97:104918.
[31] Dardis O., McCloskey J. Lattice Boltzmann scheme with real numbered solid density for the simulation of flow in porous media. Phys. Rev. E, 1998, 57(4):4834-4837.
[32] Spaid M. A. A., Phelan F. R. Lattice Boltzmann methods for modeling microscale flow in fibrous porous media. Phys. Fluids, 1997, 9(9):2468-2474.
[33] Spaid M. A. A., Phelan F. R. Modeling void formation dynamics in fibrous porous media with the lattice Boltzmann method. Composites Part A, 1998, 29A(7):749-755.
[34] Martys N. S. Improved approximation of the Brinkman equation using a lattice Boltzmann method. Phys. Fluids, 2001, 13(6):1807-1810.
[35] Guo Z., Zhao T. S. Lattice Boltzmann model for incompressible flows through porous media. Phys. Rev. E, 2002,66(3):36304.
[36] Ladd A. J. C. Numerical simulations of particulate suspensions via a discretized Boltzmann equation. 1. Theoretical foundation. J. Fluid Mech., 1994,271:285-309.
[37] Ladd A. J. C. Numerical simulations of particulate suspensions via a discretized Boltzmann equation. 2. Numerical results. J. Fluid Mech., 1994,271:311-339.
[38] Aidun C. K., Lu Y. Lattice Boltzmann simulation of solid particles suspended in fluid. J. Stat. Phys., 1995, 81(1-2):49-61.
[39] Aidun C. K., Lu Y.,Ding E. J. Direct analysis of particulate suspensions with inertia using the discrete Boltzmann equation. J. Fluid Mech., 1998, 373:287-311.
[40] Qi D. Lattice-Boltzmann simulations of particles in non-zero-Reynolds-number flows. J. Fluid Mech., 1999,385:41-62.
[41] Heemels M. W., Hagen M. H. J., Lowe C. P. Simulating solid colloidal particles using the lattice-Boltzmann method. J. Comput. Phys. 2000, 164(1):48-61.
[42] Qi D. Lattice-Boltzmann simulations of fluidization of rectangular particles. Int. J. Multiphase Flow, 2000,26(3):421-433.
[43] Ladd A. J. C. Short-time motion of colloidal particles: numerical simulation via a fluctuating lattice-Boltzmann equation. Phys. Rev. Lett., 1993, 70(9):1339-1342.
[44] Segre P. N., Behrend O. P., Pusey P. N. Short-time Brownian motion in colloidal suspensions: experiment and simulation. Phys. Rev. E, 1995, 52(5):5070-5083.
[45] Ladd A. J. C. Sedimentation of homogeneous suspensions of non-Brownian spheres. Phys. Fluids, 1997,9(3):491-499.
[46] Benzi R., Succi S. Two-dimensional turbulence with the lattice Boltzmann equation. J. Phys. A, 1990,23(1):1-5.
[47] Succi S., d'Humieres D., Qian Y. H. et al. On the small-scale dynamical behavior of lattice BGK and lattice Boltzmann schemes. J. Scient. Comput., 1993, 8(3):219-230.
[48] Chen S., Wang Z., Shan X. et al. Lattice Boltzmann computational fluid dynamics in three dimensions. J. Stat. Phys., 1992, 68(3-4):379-400.
[49] Martinez D. O., Matthaeus W. H., Chen S. et al. Comparison of spectral method and lattice Boltzmann simulations of two-dimensional hydrodynamics. Phys. Fluids, 1994, 6(3):1285-1298.
[50] Benzi R., Struglia M. V., Tripiccione R. Extended self-similarity in numerical simulations of three-dimensional anisotropic turbulence. Phys. Rev. E, 1996, 53(6):R5565-R5568.
[51] Amati G., Succi S., Piva R. Massively parallel lattice-Boltzmann simulation of turbulent channel flow. Int. J. Modern Phys. C, 1997, 8(4):869-877.
[52] Amati G., Succi S., Benzi R. Turbulent channel flow simulations using a coarse-grained extension of the lattice Boltzmann method. Fluid Dyn. Res., 1997, 19(5):289-302.
[53] Eggels J. G. M. Direct and large-eddy simulation of turbulent fluid flow using the lattice-Boltzmann scheme. Int. J. Heat Fluid Flow, 1996, 17(3):307-323.
[54] Hazi G. Bias in the direct numerical simulation of isotropic turbulence using the lattice Boltzmann method. Phys. Rev. E, 2005, 71(3): 1-7.
[55] Yu H., Girimaji S. S., Luo L. S. Lattice Boltzmann simulations of decaying homogeneous isotropic turbulence. Phys. Rev. E, 2005, 71(1): 1-5.
[56] Chen H., Orszag S. A., Staroselsky I. Y. et al. Expanded analogy between Boltzmann kinetic theory of fluids and turbulence. J. Fluid Mech., 2004, 519:301-314.
[57] Chen H., Kandasamy S., Orszag S. A. et al. Extended Boltzmann kinetic equation for turbulent flows. Science, 2003, 301:633-636.
[58] Dong Y. H., Sagaut P., Marie S. Inertial consistent subgrid model for large-eddy simulation based on the lattice Boltzmann method. Phys. Fluids, 2008,20:035104.
[59] Dong Y. H., Sagaut P. A study of time correlations in lattice Boltzmann-based large-eddy simulation of isotropic turbulence. Phys. Fluids, 2008, 20:035105.
[60] Succi S., Amati G., Benzi R. Challenges in lattice Boltzmann computing. J. Stat. Phys., 1995, 81:5-16.
[61] Terxeira C. M. Incorporating turbulence models into the lattice-Boltzmann method. Int. J. Modern Phys. C, 1998,9:1159-1175.
[62] Filippova O.,Succi S. et al. Multiscale lattice Boltzmann schemes with turbulence modeling. J. Comp. Phys., 2001, 170:812-829.
[63] Rothman D. H., Keller J. M. Immiscible cellular-automaton fluids. J. Stat. Phys., 1988, 52:1119-1127.
[64] Gunstensen A. K., Rothman D. H., Zaleski S. et al. Lattice Boltzmann model of immiscible fluids. Phys. Rev. E, 1991,43:4320-4327.
[65] Grunau D., Chen S., Eggert K. A lattice Boltzmann model for multiphase fluid flows. Phys. Fluids A, 1993, 5(10):2557-2562.
[66] Kono K. Ishizuka T. Tsuda H. et al. Application of lattice Boltzmann model to multiphase flow with phase transition. Comput. Phys. Commun., 2000, 129:110-120.
[67] D'ortona U. Salin D. Cieplak M. et al. Interfacial phenomena in Boltzmann cellular automata. Europhys. Lett., 1994, 28:317-322.
[68] Shan X., Chen H. Lattice Boltzmann model for simulating flows with multiple phases and components. Phys. Rev. E, 1993,47:1815-1819.
[69] Shan X., Chen H. Simulation of nonideal gases and liquid-gas phase-transitions by the lattice Boltzmann equation. Phys. Rev. E, 1994, 49:2941-2948.
[70] Qian Y, Succi S., Orszag S. Recent advances in lattice Boltzmann computing. Annu. Rev. Comput. Phys., 1995,3:195-242.
[71] Shan X. Pressure tensor calculation in a class of nonideal gas lattice Boltzmann models. Phys. Rev. E, 2008, 77: 066702.
[72] Swift M. R., Osborn W. R., Yeomans J. M. Lattice Boltzmann simulations of nonideal fluids. Phys. Rev. Lett., 1995, 75:830-833.
[73] Swift M. R., Orlandini E., Osborn W. R. et al. Lattice Boltzmann simulations of liquid-gas and binary fluid systems. Phys. Rev. E, 1996, 54(5):5014-5052.
[74] Inamuro T., Konishi N., Ogino F. A Galilean invariant model of the lattice Boltzmann method for multiphase fluid flows using free-energy approach. Computer, Phys. Communications, 2000, 129:32-45.
[75] Luo L. S. Unified theory of lattice Boltzmann models for nonideal gases, Phys. Rev. Lett., 1998, 8:1618-1621.
[76] Luo L. S. Theory of lattice Boltzmann method: lattice Boltzmann models for nonideal gases. Phys. Rev. E, 2000, 62:4982-4996.
[77] He X., Shan X., Doolen G. Discrete Boltzmann equation model for non-ideal gases. Phys. Rev. E, 1998,57:R13-R16.
[78] He X., Chen S., Zhang R. A lattice Boltzmann scheme for incompressible multiphase flow and its application in simulation of Ray leigh-Taylor instability. J. Comput. Phys., 1999, 152:642-663.
[79] He X., Zhang R., Chen S. et al. On the three-dimensional Rayleigh-Taylor instability. Phys. Fluids, 1999,11:1143-1152.
[80] Guo Z., Zhao T. S. Discrete velocity and lattice Boltzmann models for binary mixtures of nonideal fluids. Phys. Rev. E, 2003, 68:035302.
[81] Nie X. B., Doolen G. D., Chen S. Lattice Boltzmann simulations of fluid flows in MEMS. J. Stat. Phys., 2002, 107:279-289.
[82] Succi S. Mesoscopic modeling of slip motion at fluid-solid interfaces with heterogeneous catalysis. Phys. Rev. Lett., 2002, 89:064502.
[83] Lim C. Y.,Shu C., Niu X. D. et al. Application of lattice Boltzmann method to simulate microchannel flows. Phys. Fluids, 2002,14: 2299-2308.
[84] Ansumali S., Karlin I. V. Kinetic boundary conditions in the lattice Boltzmann method. Phys. Rev. E, 2002, 66:026311.
[85] Tang G. H., Tao W. Q, He Y. L. Lattice Boltzmann method for gaseous microflows using kinetic theory boundary conditions. Phys. Fluids, 2005, 17:058101.
[86] Shan X., Yuan X. F., Chen H. Kinetic theory representation of hydrodynamics: a way beyond the Navier-Stokes equation. J. Fluid Mech., 2006, 550:413-441.
[87] Kim S. H., Pitsch H. Slip velocity and Knudsen layer in the lattice Boltzmann method for microscale flows. Phys. Rev. E, 2008, 77:026704.
[88] Tang G. H., Zhang Y. H., Gu X. J. et al. Lattice Boltzmann modelling Knudsen layer effect in non-equilibrium flows. Europhys. Lett., 2008, 83:40008.
[89] Guo Z. Zhao T. S., Shi Y. Physical symmetry, spatial accuracy, and relaxation time of the lattice Boltzmann equation for microgas flows. J. Appl. Phys., 2006, 99:074903.
[90] Zhang Y. H., Gu X. J., Barber R. W. et al. Capturing Knudsen layer phenomena using a lattice Boltzmann model. Phys. Rev. E, 2006, 74:046704.
[91] Guo Z., Shi B., Zhao T. S. et al. Discrete effects on boundary conditions for the lattice Boltzmann equation in simulating microscale gas flows. Phys. Rev. E, 2007,76:056704.
[92] Zheng L., Guo Z. L., Shi B. C. Discrete effects on thermal boundary conditions for the thermal lattice Boltzmann method in simulating microscale gas flows. Europhys. Lett., 2008, 82:44002.
[93] Sofonea V., Sekerka R. F. Diffuse-reflection boundary conditions for a thermal lattice Boltzmann model in two dimensions: Evidence of temperature jump and velocity slip in microchannel. Phys. Rev. E, 2005, 71: 066709.
[94] Shu C, Niu X. D., Chew Y. T. A lattice Boltzmann kinetic model for microflow. J. Stat. Phys., 2005, 121: 239-245.
[95] Dawson S.P., Chen S., Doolen G. D. Lattice Boltzmann computations for reaction-diffusion equations. J. Chem. Phys., 1993, 98:1514-1523.
[96] Chen S., Dawson S. P., Doolen G. D. et al. Lattice methods and their applications to reacting systems. Comp. Chem. Eng., 1995,19:617-646.
[97] Succi S., Bella G., Papetti F. Lattice kinetic theory for numerical combustion. J. Sci. Comput., 1997, 12:395-408.
[98] Filippova O., Hanel D. A novel lattice BGK approach for low Mach number combustion. J. Comput. Phys., 2000, 158:139-160.
[99] Yamamoto K., He X., Doolen G. D. Simulation of combustion field with lattice Boltzmann method. J. Stat. Phys., 2002, 107(1/2):367-383.
[100] Chen S., Chen H., Martinez D. O. et al. Lattice Boltzmann model for simulating of magnetohy-drodynamics. Phys. Rev. Lett., 1991, 67:3776-3779.'
[101] MartinezD. O., Chen S. Lattice Boltzmann magnetohydrodynamics. Phys. Plasmas, 1997,1:1850-1867.
[102] Dellar J. Lattice kinetic scheme for magnetohydrodynamics. J. Comput. Phys., 2002, 179(1):95-126.
[103] Chen X. W., Shi B. C. A new lattice Boltzmann model for incompressible nagnetohydrodynamics. Chin. Phys., 2005, 17(7): 1398-1406.
[104] Bhatnagar P. L., Gross E. P., Krook, M. A model for collision processes in gases. I. Small amplitude processes in charged and neutral one-component systems. Phys. Rev., 1954, 94:511-525.
[105] d'Humieres D. Generalized lattice-Boltzmann equations, in rarefied gas dynamics: theory and simulations. In: Shizgal B. D., Weaver D. P. Progress in Astronautics and Aeronautics. Washington, DC: AIAA, 1992,159:450-458.
[106] Lallemand P., Luo L. S. Theory of the lattice Boltzmann method: Dispersion, dissipation, isotropy, Galilean invarioance and stability. Phys. Rev. E, 2000, 61:6546-6562.
[107] He X., Zou Q., Luo L. S. et al. Analytic solutions of simple flows and analysis of nonslip boundary conditions for the lattice Boltzmann BGK model. J. Stat. Phys., 1997, 87:115-136.
[108] Ziegler D. Boundary conditions for lattice Boltzmann simulations. J. Stat. Phys., 1993, 71:1171-1177.
[109] Zou Q., He X. On pressure and velocity boundary conditions for the lattice Boltzmann BGK model. Phys. Fluids, 1997, 9:1591-1597.
[110] Noble D., Chen S., Georgiadis J. G. et al. A consistent hydrodynamic boundary conditions for lattice Boltzmann method. Phys. Fluids, 1995,7:203-205.
[111] Maier R. S., Bernard R. D., Grunau D. W. Boundary condition for lattice Boltzmann simulations. Phys. Fluids, 1996, 8:1788-1801
[112] Inamuro T., Yoshino M., Ogino F. A non-slip boundary condition for lattice Boltzmann simulations. Phys. Fluids, 1995, 7:2928-2930.
[113] Skordos P. A. Initial and boundary conditions for the lattice Boltzmann method. Phys. Rev. E, 1993,48:4823-4842.
[114] Chen S., Martinez D., Mei R. On boundary conditions in lattice Boltzmann methods. Phys. Fluids, 1996, 8(9):2527-2536.
[115] Guo Z. L., Zheng C.G., Shi B.C. Non-equilibrium extrapolation method for velocity and pressure boundary conditions in the lattice Boltzmann method. Chin. Physics, 2002, 11(4): 366-374.
[116] Bear J. Dynamics of Fluids in Porous Media. New York: American Elsevier, 1972.
[117] Scheidegger A. E. The Physics of Flow through Porous Media. Toronto: University of Toronto, 1974.
[118] Nield D. A., Bejan A. Convection in Porous Media. New York: Springer, 2006.
[119] Lee S. L., Yang J. H. Modeling of Darcy-Forchheimer drag for fluid flow across a bank of circular cylinders. Int. J. Heat Mass Transfer, 1997,40:3149-3155.
[120] Andrade Jr J. S., Costa U. M. S., Almeida M. P. et al. Inertial effects on fluid flow through disordered porous media. Phys. Rev. Lett., 1999, 82:5249-5252.
[121] Ghaddar C. K. On the permeability of unidirectional fibrous media: A parallel computational approach. Phys. Fluids, 1995, 7:2563-2586.
[122] Mei C. C., Auriault J. L. The effect of weak inertia on flow through a porous medium. J. Fluid Mech., 1991, 222:647-663.
[123] Coulaud O., Morel P., Caltagirone J. P. Numerical modeling of nonlinear effects in laminar flow through a porous medium. J. Fluid Mech., 1988,190:393-407.
[124] Skjetne E., Auriault J. L. New insights on steady, non-linear flow in porous media. Eur. J. Mech. E/Fluids, 1999, 18:131-145.
[125] Skjetne E., Hansen A., Gudmundsson J. S. High-velocity flow in a rough fracture. J. Fluid Mech., 1999,383:1-28.
[126] Rojas S., Koplik J. Nonlinear flow in porous media. Phys. Rev. E, 1998, 58:4776-4782.
[127] Zimmerman R. W., Al-Yaarubi A., Pain C. C. et al. Non-linear regimes of fluid flow in rock fractures. Int. J. Rock Mech. Min. Sci., 2004, 413:163-169.
[128] Firdaouss M., Guermond J. L., Quere P. L. Nonlinear corrections to Darcy's law at low Reynolds numbers. J. Fluid Mech., 1997, 343:331-350.
[129] Fourar M., Radilla G., Lenormand R. et al. On the non-linear behavior of a laminar single-phase flow through two and three-dimensional porous media. Adv. Water Res., 2004, 27:669-677.
[130] Lucas Y., Panfilov M., Bues M. High velocity flow through fractured and porous media: the role of flow non-periodicity. Euro. J. Mech. B/Fluids, 2007,26:295-303.
[131] Panfilov M., Fourar M. Physical splitting of nonlinear effects in high-velocity stable flow through porous media. Adv. Water Res. 2006,29:30-41.
[132] Hill R., Koch D. L., Ladd A. J. C. The first effects of fluid inertia on flows in ordered and random arrays of spheres. J. Fluid Mech., 2001,448:213-241.
[133] Ferreol B., Rothman D. H. Lattice-Boltzmann simulations of flow through Fontainebleau sandstone. Transport Porous Med., 1995,20:3-20.
[134] Heijs A. W. J., Lowe C. P. Numerical evaluation of the permeability and the Kozeny constant for two types of porous media. Phys. Rev. E, 1995, 51:4346-4352.
[135] Koponen A., Kandhai D., Hellen E. Permeability of three-dimensional random fiber webs. Phys. Rev. Lett., 1998,80:716-719.
[136] Pan C., Luo L. S., Miller C. T. An evaluation of lattice Boltzmann schemes for porous medium flow simulation. Comput. Fluids, 2006, 35:898-909.
[137] Jin G., Patzek T. W, Silin D. B. Direct prediction of the absolute permeability of unconsolidated and consolidated reservoir rock. Paper SPE90084. In: SPE Annual Technical Conference and Exhibition. Houston: Society of Petroleum Engineers, Inc., 2004. 1-15.
[138] Chai Z., Shi B., Zheng L. Simulating high Reynolds number flow in two-dimensional lid-driven cavity by multi-relaxation-time lattice Boltzmann method. Chin. Phys., 2006,15(8):1855-1863.
[139] Ginzburg I., d'Humieres D. Multireflection boundary conditions for lattice Boltzmann models. Phys. Rev. E, 2003, 68:066614.
[140] Shankar P. N., Deshpande M.D. Fluid mechanics in the driven cavity. Annu. Rev. Fluid Mech., 2000, 32:93-136.
[141] Ghia U., Guia K. N., Shin C. T. High-Re solutions for incompressible flow using the Navier-Stokes equations and a multigrid method. J. Comput. Phys., 1982,48:387-411.
[142] Hou S. L., Chen S. Y., Zou Q. S. et al. Simulation of cavity flow by the lattice Boltzmann method. J. Comput. Phys., 1995,118:329-3471.
[143] Nallasamy M., Prasad K. On cavity flow at high Reynolds numbers. J. Fluid Mech., 1977,79:391-414.
[144] Zhou X. Y., Shi B. C., Wang N. C. Numerical simulation fo LBGK model for high Reynolds number flow. Chin. Phys., 2004, 13:712-719.
[145] Ram(?)ak M., (?)kerget L. A subdomain boundary element method for high-Reynolds laminar flow using stream-vorticity formulation. Int. J. Numer. Meth. Fluids, 2004,46:815-847.
[146] Erturek E., Corke T. C, G(?)kc(?)l C. Numerical solutions of 2-D steady incompressible driven cavity flow at high Reynolds numbers. Int. J. Numer. Meth. Fluids, 2005,48:747-774.
[147] Hou S. L. Lattice Boltzmann method for incompressible viscous flow: PhD thesis. Kansas State University, 1995.
[148] Jackson G. W., James D. F., Jackson G. W. et al. The permeability of fibrous porous media. Can. J. Chem. Eng., 1986, 64:364-374.
[149] Sangani A. S., Acrivos A. Slow flow past periodic arrays of cylinders with applicationto heat transfer. Int. J. Multphase flow, 1982, 8:193-206.
[150] Gad-el-Hak M. The fluid mechanics of microdevices-the Freeman scholar lecture. ASME Trans. J. Fluids Eng., 1999,121:5-33.
[151] Stone H. A., Stroock A. D., Ajdari A. Engineering flows in small devices: Microfluidics toward a Lab-on-a-Chip. Annu. Rev. Fluid Mech., 2004, 36:381-411.
[152] Ho C. M., Tai Y. C. Micro-Electro-Mechanicalsystems (MEMS) and fluid flows. Annu. Rev. Fluid Mech., 1998,30:579-612.
[153] Squires T. M., Quake S. R. Microfluidics: Fluid physics at the nanoliter scale. Rev. Mod. Phys., 2005,77:977-1026.
[154] Abramson A. R., Tien C. L. Recent developments in microscale thermophysical engineering. Microscale Thermophys. Eng., 1999, 3:229-244.
[155] Obot N. T. Toward a better understanding of friction and heat/mass transfer in microchannels-a literature review. Microscale Thermophys. Eng., 2002, 6:155-173.
[156] Guo Z. Y., Li Z. X. Size effect on single-phase channel flow and heat transfer at microscale. Int. J. Heat Fluid Flow, 2003, 24:284-298.
[157] Wu P., Little W. A. Measurement of friction factors for the flow of gases in very fine channels used for microminiature Joule-Thomson refrigerators. Cryogenics, 1983, 23:273-277.
[158] Pfahler J., Harley J., Bau H. et al. Liquid and gas transport in small channels. ASME Dyn. Syst. Control Div. 1990,19:149-157.
[159] Pfahler J., Harley J., Bau H. et al. Gas and liquid flow in small channels. ASME Dyn. Syst. Control Div., 1991,32:49-60.
[160] Choi S. B., Barron R. F., Warrington R. Q. Fluid flow and heat transfer in micro tubes. ASME Dyn. Syst. Control Div., 1991, 32:123-133.
[161] Yu D., Warrington R., Barron R. et al. An experimental and theoretical investigation of fluid flow and heat transfer in microtubes. In: Flecher L. S., Aihara F. Proceedings of the 1995 ASME/JSME Thermal Engineering Joint Conference. New York: ASME, 1995. 523-530.
[162] Turner S. E., Lam L. C, Faghri M. et al. Experimental investigation of gas flow in microchannels. Trans. ASME J. Heat Transfer, 2004, 126:753-763.
[163] Harley J., Huang Y., Bau H. et al. Gas flow in micro-channels. J. Fluid Mech., 1995, 284:257-274.
[164] Papautsky I., Brazzle J., Ameel T. et al. Laminar fluid behavior in microchannels using micropolar fluid theory. Sens. Actuators A, 1999,73:101-108.
[165] Sugiyama W., Sawadaa T., Nakamori K. Rarefied gas flow between two flat plates with two dimensional surface roughness. Vacuum, 1996,47(6-8):791-794.
[166] Usami M., Fujimoto T., Kato S. Mass-flow reduction of rarefied gas by roughness of a slit surface. Trans. Jpn. Soc. Mech. Eng., Ser. B, 1988, 54:1042-1050.
[167] Sun H., Faghri M. Effect of surface roughness on nitrogen flow in a microchannel using the direct simulation Monte Carlo method. Numer. Heat Transfer Part A, 2003,43:1-8.
[168] Ji Y., Yuan K., Chung J. N. Numerical simulation of wall roughness on gaseous flow and heat transfer in a microchannel. Int. J. Heat Mass Transfer, 2006,49:1329-1339.
[169] Croce G., D'Agaro P., Filippo A. Compressibility and rarefaction effects on pressure drop in rough microchannels. Heat Transfer Eng., 2007, 28:688-695.
[170] Zhang T., Li L. Numerical simulation of roughness effect on gaseous flow and heat transfer in microchannels. In: Pornsinsirirak T. N. Proceedings of the 2nd IEEE International Conference on Nano/Micro Engineered and Molecular Systems. Bangkok: IEEE, 2007. 136-161.
[171] Cao B. Y, Chen M., Guo Z. Y. Effect of surface roughness on gas flow in microchannels by molecular dynamics simulation. Int. J. Eng. Sci., 2006,44:927-937.
[172] Arkilic E. B., Schmidt M. A., Breuer K. S. Gaseous slip flow in long microchannels. J. Microelec-tromech. Syst., 1997, 6:167-178.
[173] Adzumi H. On the flow of gases through a porous wall. Bull. Chem. Soc. Japan, 1937, 12(6):304-312.
[174] Klinkenberg L. J. The Permeability of porous media to liquid and gases. API Drilling and production practice, 1941, 200-213.
[175] Jones S. C. A rapid accurate unsteady-state Klinkenberg permeameter. SPE J., 1972, 3535:383-393.
[176] Jones F. O., Owens W. W. A laboratory study of low permeability gas sands. SPE J., 1979, 7551:1-10.
[177] Sampath K., Keighin C. W. Factors affecting ga s slippage in tight sandstones. SPE J., 1981, 9872:409-415.
[178] Wu Y. S., Pruess K., Persoff P. Gas flow in porous media with Klinkenberg effects. Trans. Porous Media., 1998,32:117-137.
[179] Kaluarachchi J. Analytical solution to two-dimensional axisymmetric gas flow with Klinkenberg effect. J. Environ. Eng., 1995,121(5):417-420.
[180] Grove D. M., Ford M. G. Evidence for permeability minima in low-pressure gas flow through porous media. Nature, 1958, 182:999-1000.
[181] De Socio L. M., Marino L. Gas flow in a permeable medium. J. Fluid Mech., 2006, 557:119-133.
[182] Florence F. A., Rushing J. A., Newsham K. E. et al. Improved permeability prediction relations for low permeability sands. SPE J., 2007, 107954:1-18.
[183] Tanikawa W., Shimamoto T. Klinkenberg effect for gas permeability and its comparison to water permeability for porous sedimentary rocks. Hydrol. Earth Syst. Sci. Discuss., 2006, 3:1315-1338.
[184] Woudberg S., Du Plessis J. P. Predicting the permeability of very low porosity sandstones. Trans. Porous Media, 2008, 73:39-55.
[185] Pavan V., Oxarango L. A new momentum equation for gas flow in porous media: The Klinkenberg effect seen through the kinetic theory. J. Stat. Phys., 2007, 126(2):355-389.
[186] Skjetne E., Auriault J. L. Homogenization of wall slip in the Darcy flow through porous media. Trans. Porous Media, 1999, 36:293-306.
[187] Chastanet J. Royer P., Auriault J. L. Does Klinkenberg's law survive upscaling? Trans. Porous Media, 2004, 56:171-198.
[188] Parthlban C, Patll P. R. Convection in a porous medium with velocity slip and temperature jump boundary conditions. Heat Mass Transfer, 1996, 32:27-31.
[189] Haddad O. M., Abuzaid M. M., Al-Nimr M. A. Developing free-convection gas flow in a vertical open-ended microchannel filled with porous media. Nume. Heat Transfer, Part A, 2005, 48:693-710.
[190] Nield D. A., Kuznetsov A. V. Forced convection with slip-flow in a channel or duct occupied by a hyper-porous medium saturated by a rarefied gas. Trans. Porous Media, 2006, 64:161-170.
[191] Skjetne E. High Velocity flow in porous media: analytical, numerical and experimental studies: PhD Thesis. Norwegian University of Science and Technology, 1995.
[192] Liu Y. W., Zhou F. X., Yan G. W. Lattice Boltzmann simulations of the Klinkenberg effect in porous media. Chin. J. Comput. Phys., 2003, 20(2):157-160.
[193] Verhaeghe F, Luo L. S., Blanpain B. Lattice Boltzmann modeling of microchannel flow in slip flow regime. J. Comput. Phys., 2009, 228:147-157.
[194] Guo Z, Zheng C. Analysis of lattice Boltzmann equation for microscale gas flows: Relaxation times, boundary conditions and the Knudsen layer. Int. J. Comput. Dyn., 2008,7:465-473.
[195] Guo Z., Shi B., Zheng C. An extended Navier-Stokes formulation for gas flows in the Knudsen layer near a wall. Europhys. Lett., 2007, 80:24001.
[196] Guo Z., Zheng C., Shi B. Lattice Boltzmann equation with multiple effective relaxation times for gaseous microscale flow. Phys. Rev. E, 2008, 77:036707.
[197] Collins R. E. Flow of fluids through porous materials. Tulsa: The Petroleum Publishing Company, 1976.
[198] Burgreen D., Nakache F. R. Electrokinetic flow in ultrafine capillary slits. J. Phys. Chem., 1964 68:1084-1091.
[199] Rice C. L., Whitehead R. Electrokinetic flow in a narrow cylindrical capillary. J. Phys. Chem., 1965,69:4017-4024.
[200] Qu W., Li D. Model for overlapped EDL fields. J. Colloid Interface Sci., 2000,224:397-407.
[201] Patankar N. A., Hu H. H. Numerical simulation of electroosmotic flow. Anal. Chem., 1998, 70:1870-1881.
[202] Bianchi F., Ferrigno R., Girault H. H. Finite element simulation of an Electroosmotic-Driven flow division at a T-junction of microscale dimensions. Anal.Chem., 2000,72:1987-1993.
[203] Mitchell M. J., Qiao R., Aluru N. R. Meshless analysis of steady-state electro-osmotic transport. J. Microelectromechanical Syst., 2002, 9:435-449.
[204] Jin Y., Luo G. A. Numerical calculation of the electroosmotic flow at the cross region in microfluidic chips. Electrophoresis, 2003, 24:1242-1252.
[205] Zhao T. S., Liao Q. Thermal effects on electroosmotic pumping of liquids in microchannels. J. Micromech. Microeng., 2002, 12:962-970.
[206] Fan Z. H., Harrison D. J. Micromachining of capillary electrophoresis injectors and separators on glass chips and evaluation of flow at capillary intersections. Anal. Chem., 177,66:177-184.
[207] Santiago J. G. Electroosmotic Flows in Microchannels with Finite Inertial and Pressure Forces. Anal. Chem., 2001, 73:2353-2365.
[208] Soderman O., Jonsson B. Electro-osmosis: Velocity profiles in different geometries with both temporal and spatial resolution. J. Chem. Phys., 1996,105:10300 - 10311.
[209] Keh H. J., Tseng H. C. Transient Electrokinetic Flow in Fine Capillaries. J. Colloid Interface Sci., 2001, 242:450-459.
[210] Yang C., Ng C.B., Chan V. Transient analysis of electroosmotic flow in a slit microchannel. J. Collid Interface Sci., 2002 248:524-527.
[211] Qiao R., Aluru N. R. Transient analysis of electro-osmotic transport by a reduced-order modeling approach.Int. J. Numer. Meth. Engrg., 2003, 56:1023-1050.
[212] Tian F., Li B., Kwok D. Y. Lattice Boltzmann simulation of electroosmotic flows in micro- and nanochannels. In: Proceedings of the 2004 International Conference on MEMS, NANO and Smart Systems (ICMENS' 04). Alberta: IEEE, 2004. 294-299.
[213] He X., Li N. Lattice Boltzmann simulation of electrochemical systems. Comput. Phys. Commun., 2000, 129:158-166.
[214] Guo Z., Zhao T. S., Shi Y. A lattice Boltzmann algorithm for electro-osmotic flows in microfluidic devices. J. Chem. Phys., 2005, 122:144907.
[215] Wang J., Wang M., Li Z. Lattice Poisson-Boltzmann simulations of electro-osmotic flows in microchannels. J. Collid Interface Sci., 2006, 296:729-736.
[216] Hlushkou D., Kandhai D., Tallarek U. Coupled lattice-Boltzmann and finite-difference simulation of electroosmosis in microfluidic channels. Int. J. Numer. Methods Fluids, 2004,46:507-532.
[217] Guo Z., Shi B., Wang N. Lattice BGK model for incompressible Navier-Stokes equation. J. Comput. Phys., 2000,165:288-306.
[218] Chai Z., Shi B. Simulation of electro-osmotic flow in microchannel with lattice Boltzmann method. Phys. Lett. A, 2007, 364:183-188.
[219] Chai Z., Shi B. A novel lattice Boltzmann model for the Poisson equation. Appl. Math. Model., 2008,32:2050-2058.
[220] Masliyah J. Electrokinetic Transport Phenomena. Alberta: Alberta Oil Sands Technology and Research Authority, 1994.
[221] Guo Z., Zheng C., Shi B. Discrete lattice effects on the forcing term in the lattice Boltzmann method. Phys. Rev. E, 2002, 65:046308.
[222] Chai Z., Guo Z., Shi B. Study of electro-osmotic flows in microchannels packed with variable porosity media via lattice Boltzmann method. J. Appl. Phys., 2007, 101:104913.
[223] Siyyam H. I. An accurate solution of the Poisson equation by the finite difference-Chebyshev-Tau method. Appl. Math. Mech., 2001, 22(8):0935-0939.
[224] Boschitsch A. H., Fenley M. O. Hybrid boundary element and finite difference method for solving the nonlinear Poisson-Boltzmann equation. J. Comput. Chem., 2004, 25(7):935-955.
[225] Cai Z., Kim S., Kim S. et al. A finite element method using singular functions for Poisson equations: Mixed boundary conditions. Comput. Methods Appl. Mech. Engrg., 2006, 195:2635-2648.
[226] Kikuchi F., Saito H. Remarks on a posteriori error estimation for finite element solutions. J. Comput. Appl. Math., 2007, 199(2):329-336.
[227] Liao S. J. General boundary element method for Poisson equation with spatially varying conductivity. Eng. Anal. Bound. Elem., 1998, 21:23-38.
[228] Guo Z. L., Shi B. C, Wang N. C. Fully Lagrangian and lattice Boltzmann methods for the advection-diffusion equation. J. Sci. Comput., 1999, 14(3):291-300.
[229] Deng B., Shi B. C., Wang G. C. A new lattice Bhatnagar-Gross-Krook model for convection-diffusion equation with a source term. Chin. Phys. Lett., 2005, 22(2):267-270.
[230] Shi B., Deng B., Du R. et al. A new scheme for source term in LBGK model for convection-diffusion equation. Comput. Math. Appl., 2008, 55:1568-1575.
[231] Chen S., Dawson S. P., Doolen G. D. et al. Lattice methods and their applications to reacting systems. Computers Chem. Engny, 1995,19(6/7):617-646.
[232] Yu X., Shi B. A lattice Bhatnaar-Gross-Krook model for a class of the generalized Burgers equation. Chin. Phys., 2006, 15(07): 1441-1449.
[233] Yu X., Shi B. A lattice Boltzmann model for reaction dynamic systems with time systems. Appl. Math. Comput., 2006, 181:958-965.
[234] Yan G. A Lattice Boltzmann Equation for Waves. J. Comput. Phys., 2000, 161:61-69.
[235] Hirabayashi M., Chen Y., Ohashi H. The lattice BGK model for the Poisson equation. JSME Int. J. Ser. B, 2001,44(1):45-52.
[236] Zhang C. Y., Tan H. L., Liu M. R. et al. A lattice Boltzmann model and simulation of KdV-Burgers equation. Commun. Theor. Phys, 2004, 42(2):281-284.
[237] Ma C. A New Lattice Boltzmann Model for KdV-Burgers Equation. Chin. Phys. Lett., 2005, 29(9):2313-2315.
[238] Chai Z., Shi B., Zheng L. A unified lattice Boltzmann model for some nonlinear partial differential equations. Chaos, Solitons and Fractals, 2008, 36:874-882.
[239] Dawson S. P., Chen S., Doolen D. G. Lattice Boltzmann Computations for reaction-diffusion equations. J. Chem. Phys., 1993, 98(2): 1514-1523.
[240] Chen S., T(?)lke J., Geller S. et al. Lattice Boltzmann model for incompressible axisymmetric flows. Phys. Rev. E, 2008, 78:046703.
[241] Coelho D., Shapiro M., Thovert J. F.et al. Electroosmotic phenomena in porous media. J. Colloid Interface Sci., 1996,181:169-190.
[242] Kang Y., Yang C., Huang X. AC electroosmosis in microchannels packed with a porous medium. J. Micromech. Microeng., 2004, 14:1249-1257.
[243] Scales N., Taita N. Modeling electroosmotic and pressure-driven flows in porous microfluidic devices: Zeta potential and porosity changes near the channel walls. J. Chem. Phys., 2006, 125:094714.
[244] Wang M., Wang J., Chen S. et al. Electrokinetic pumping effects of charged porous media in microchannels using the lattice Poisson-Boltzmann method. J. Colloid Interface Sci., 2006, 304:246-253.
[245] Qiao R., Aluru N. R. Ion concentrations and velocity profiles in nanochannel electroosmotic flows. J. Chem. Phys., 2003,118:4692-4701.
[246] Nithiarasu P., Seetharamu K. N., Sundararajan T. Natural convective heat transfer in a fluid saturated variable porosity medium. Int. J. Heat Mass Transfer, 1997,40:3955-3967.
[247] Liu S., J. Masliyah. Single fluid flow in porous media. Chem. Eng. Commun., 1996,148-150:653-732.
[248] Zeng S., Chen C. H., Mikkelsen Jr J. C. et al. Fabrication and characterization of electroosmotic micropumps. Sens. Actuators B, 2001, 79:107-114.
[249] Ergun S. Fluid flow through packed columns. Chem. Eng. Prog., 1952,48:89-94.
[250] Vafai K. Convective flow and heat transfer in variable-porosity media. J. Fluid Mech., 1984, 147:233-259.
[251] Liapis A. I., Grimes B. A. Modeling the velocity field of the electroosmotic flow in charged capillaries and in capillary columns packed with charged particles: Interstitial and intraparticle velocities in capillary electro-chromatography systems. J. Chromatogr. A, 2000, 877(1-2): 181-215.
[252] Tallarek U., Rapp E., Seidel-Morgenstern A. et al. Electroosmotic flow phenomena in packed capillaries: From the interstitial velocities to intraparticle and boundary layer mass transfer. J. Phys. Chem. B, 2002,106:12709-12721.
[253] Iwatsu R., Hyun J. M., Kuwahara K. Mixed convection in a driven cavity with a stable vertical temperature gradient. Int. J. Heat Mass Transfer, 1993, 36:1601-1608.
[254] Oztop H. F. Combined convection heat transfer in a porous lid-driven enclosure due to heater with finite length. Int. Commun. Heat Mass Transf., 2006,33:772-779.
[255] Khanafer K. M., Chamkha A. J. Mixed convection flow in a lid-driven enclosure filled with a fluid-saturated porous medium. Int. J. Heat Mass Transfer, 1999,42:2465-2481.
[256] Al-Amiri A. M. Analysis of momentum and energy transfer in a lid-driven cavity filled with a porous medium. Int. J. Heat Mass Transfer, 2000,43:3513-3527.
[257] Jue T. C. Analysis of flows driven by a torsionally-oscillatory lid in a fluid-saturated porous enclosure with thermal stable stratification. Int. J. Therm. Sci., 2002,41:795-804.
[258] Guo Z., Zhao T. S. A lattice Boltzmann model for convection heat transfer in porous media. Numerical Heat Transfer Part B, 2005, 47:157-177.
[2]石钟慈．第三种科学方法-计算机时代的科学计算．(第一版)．北京：清华大学出版社，2000．
[3]刘儒勋，王志峰．数值模拟方法和运动界面追踪．(第一版)．合肥：中国科学技大学出版社，2001．
[4]苏铭德，黄素逸．计算流体力学基础．(第一版)．北京：清华大学出版社，1997．
[5]张涵信，沈孟育．计算流体力学：差分方法的原理和应用．(第一版)．北京：国防工业出版社，2003．
[6]Karniadakis G.E.,Beskok A.,Aluru N.Microflows and Nanoflows:Fundamentals and Simulations.New York:Springer-Verlag,2005.
[7]Gad-el-Hak M.The MEMS Handbook.Boca Raton:CRC Press,2006.
[8]Wolf-Gladrow D.A.Lattice-Gas Cellular Automata and Lattice Boltzmann Models.Springer-Verlag Berlin Heidelberg,2000.
[9]Succi S.The Lattice Boltzmann Equation for Fluid Dynamics and Beyond.Oxford:Clarendon Press,2001.
[10]郭照立，郑楚光，李青等．流体动力学的格子Boltzmann方法．(第一版)．武汉：湖北科学技术出版社，2002．
[11]Benzi R.,Succi S.,Vergassola M.The lattice Boltzmann equation:theory and applications.Phys.Rep.,1992,222(3):145-197.
[12]Chen S.,Doolen G.Lattice Boltzmann method for fluid flows.Annu.Rev.Fluid Mech.,1998,30:329-364.
[13]He X.,Luo L.S.Theory of lattice Boltzmann method:From the Boltzmann equation to the lattice Boltzmann equation.Phys.Rev.E,1997,56(6):6811-6817.
[14]Shan X.,He X.Discretization of the velocity space in the solution of the Boltzmann equation.Phys.Rev.Lett.,1998,80(1):65-68.
[15]李元香．格子气自动机．(第一版)．北京：清华大学出版社，1994．
[16]Frish U,Hasslacher B,Pomeau Y.Lattice-gas automata for the Navier-Stokes equations.Phys.Rev.Lett.,1986,56:1505-1508.
[17]Chapman S.,Cowling T.G.The Mathematical Theory of Non-Uniform Gases.(3rd edition).Cambridge:Cambridge University Press,1970.
[18]Chopard B.,Droz M.Cellular Automata Modeling of Physical Systems.Beijing:Tsinghua University Press,2003.
[19] Chen H., Chen S., Matthaeus H. Recovery of the Navier-Stokes equations using a lattice gas Boltzmann method. Phys. Rev. A, 1992, 45:R5339-R5342.
[20] Qian Y. H., d'Humieres D., Lallemand P. Lattice BGK models for Navier-Stokes equation. Europhys.Lett., 1992, 17(6):479-484.
[21] Rothman D. H. Cellular-automaton fluids: a model for flow in porous media. Geophysics, 1988, 53(4):509-518.
[22] Succi S., Foti E., Higuera F. Three-dimensional flows in complex geometries with the lattice Boltzmann method. Europhy. Lett., 1989, 10(5):433-438.
[23] Adrover A., Giona M. Predictive model for permeability of correlated porous media. Chem. Eng. J., 1996, 64(1):7-19.
[24] Singh M.,MohantyK. K. Permeability of spatially correlated porous media. Chem. Eng. Sci., 2000, 55(22):5393-5403.
[25] Kim J., Lee J., Lee K. C. Nonlinear correction to Darcy's law for a flow through periodic arrays of elliptic cylinders. Physica A, 2001, 293(1-2):13-20.
[26] Koch D. L., Ladd A. J. C. Moderate Reynolds number flows through periodic and random arrays of aligned cylinders. J. Fluid Mech., 1997, 349:31-66.
[27] Koch D. L., Hill R. J. Inertial effects in suspension and porous media flows. Annu. Rev. Fluid Mech., 2001,33:619-647.
[28] Jeong N., Choi D. H., Lin C. L. Prediction of Darcy-Forchheimer drag for micro-porous structures of complex geometry using the lattice Boltzmann method. J. Micromech. Microeng., 2006, 16:2240-2250.
[29] Tang G. H., Tao W. Q., He Y. L. Gas slippage effect on microscale porous flow using lattice Boltzmann method. Phys. Rev. E, 2005, 72:056301.
[30] Tang G. H., Tao W. Q., He Y. L. Three-dimensional lattice Boltzmann model for gaseous flow in rectangular microducts and microscale porous media. J. Appl. Phys., 2005,97:104918.
[31] Dardis O., McCloskey J. Lattice Boltzmann scheme with real numbered solid density for the simulation of flow in porous media. Phys. Rev. E, 1998, 57(4):4834-4837.
[32] Spaid M. A. A., Phelan F. R. Lattice Boltzmann methods for modeling microscale flow in fibrous porous media. Phys. Fluids, 1997, 9(9):2468-2474.
[33] Spaid M. A. A., Phelan F. R. Modeling void formation dynamics in fibrous porous media with the lattice Boltzmann method. Composites Part A, 1998, 29A(7):749-755.
[34] Martys N. S. Improved approximation of the Brinkman equation using a lattice Boltzmann method. Phys. Fluids, 2001, 13(6):1807-1810.
[35] Guo Z., Zhao T. S. Lattice Boltzmann model for incompressible flows through porous media. Phys. Rev. E, 2002,66(3):36304.
[36] Ladd A. J. C. Numerical simulations of particulate suspensions via a discretized Boltzmann equation. 1. Theoretical foundation. J. Fluid Mech., 1994,271:285-309.
[37] Ladd A. J. C. Numerical simulations of particulate suspensions via a discretized Boltzmann equation. 2. Numerical results. J. Fluid Mech., 1994,271:311-339.
[38] Aidun C. K., Lu Y. Lattice Boltzmann simulation of solid particles suspended in fluid. J. Stat. Phys., 1995, 81(1-2):49-61.
[39] Aidun C. K., Lu Y.,Ding E. J. Direct analysis of particulate suspensions with inertia using the discrete Boltzmann equation. J. Fluid Mech., 1998, 373:287-311.
[40] Qi D. Lattice-Boltzmann simulations of particles in non-zero-Reynolds-number flows. J. Fluid Mech., 1999,385:41-62.
[41] Heemels M. W., Hagen M. H. J., Lowe C. P. Simulating solid colloidal particles using the lattice-Boltzmann method. J. Comput. Phys. 2000, 164(1):48-61.
[42] Qi D. Lattice-Boltzmann simulations of fluidization of rectangular particles. Int. J. Multiphase Flow, 2000,26(3):421-433.
[43] Ladd A. J. C. Short-time motion of colloidal particles: numerical simulation via a fluctuating lattice-Boltzmann equation. Phys. Rev. Lett., 1993, 70(9):1339-1342.
[44] Segre P. N., Behrend O. P., Pusey P. N. Short-time Brownian motion in colloidal suspensions: experiment and simulation. Phys. Rev. E, 1995, 52(5):5070-5083.
[45] Ladd A. J. C. Sedimentation of homogeneous suspensions of non-Brownian spheres. Phys. Fluids, 1997,9(3):491-499.
[46] Benzi R., Succi S. Two-dimensional turbulence with the lattice Boltzmann equation. J. Phys. A, 1990,23(1):1-5.
[47] Succi S., d'Humieres D., Qian Y. H. et al. On the small-scale dynamical behavior of lattice BGK and lattice Boltzmann schemes. J. Scient. Comput., 1993, 8(3):219-230.
[48] Chen S., Wang Z., Shan X. et al. Lattice Boltzmann computational fluid dynamics in three dimensions. J. Stat. Phys., 1992, 68(3-4):379-400.
[49] Martinez D. O., Matthaeus W. H., Chen S. et al. Comparison of spectral method and lattice Boltzmann simulations of two-dimensional hydrodynamics. Phys. Fluids, 1994, 6(3):1285-1298.
[50] Benzi R., Struglia M. V., Tripiccione R. Extended self-similarity in numerical simulations of three-dimensional anisotropic turbulence. Phys. Rev. E, 1996, 53(6):R5565-R5568.
[51] Amati G., Succi S., Piva R. Massively parallel lattice-Boltzmann simulation of turbulent channel flow. Int. J. Modern Phys. C, 1997, 8(4):869-877.
[52] Amati G., Succi S., Benzi R. Turbulent channel flow simulations using a coarse-grained extension of the lattice Boltzmann method. Fluid Dyn. Res., 1997, 19(5):289-302.
[53] Eggels J. G. M. Direct and large-eddy simulation of turbulent fluid flow using the lattice-Boltzmann scheme. Int. J. Heat Fluid Flow, 1996, 17(3):307-323.
[54] Hazi G. Bias in the direct numerical simulation of isotropic turbulence using the lattice Boltzmann method. Phys. Rev. E, 2005, 71(3): 1-7.
[55] Yu H., Girimaji S. S., Luo L. S. Lattice Boltzmann simulations of decaying homogeneous isotropic turbulence. Phys. Rev. E, 2005, 71(1): 1-5.
[56] Chen H., Orszag S. A., Staroselsky I. Y. et al. Expanded analogy between Boltzmann kinetic theory of fluids and turbulence. J. Fluid Mech., 2004, 519:301-314.
[57] Chen H., Kandasamy S., Orszag S. A. et al. Extended Boltzmann kinetic equation for turbulent flows. Science, 2003, 301:633-636.
[58] Dong Y. H., Sagaut P., Marie S. Inertial consistent subgrid model for large-eddy simulation based on the lattice Boltzmann method. Phys. Fluids, 2008,20:035104.
[59] Dong Y. H., Sagaut P. A study of time correlations in lattice Boltzmann-based large-eddy simulation of isotropic turbulence. Phys. Fluids, 2008, 20:035105.
[60] Succi S., Amati G., Benzi R. Challenges in lattice Boltzmann computing. J. Stat. Phys., 1995, 81:5-16.
[61] Terxeira C. M. Incorporating turbulence models into the lattice-Boltzmann method. Int. J. Modern Phys. C, 1998,9:1159-1175.
[62] Filippova O.,Succi S. et al. Multiscale lattice Boltzmann schemes with turbulence modeling. J. Comp. Phys., 2001, 170:812-829.
[63] Rothman D. H., Keller J. M. Immiscible cellular-automaton fluids. J. Stat. Phys., 1988, 52:1119-1127.
[64] Gunstensen A. K., Rothman D. H., Zaleski S. et al. Lattice Boltzmann model of immiscible fluids. Phys. Rev. E, 1991,43:4320-4327.
[65] Grunau D., Chen S., Eggert K. A lattice Boltzmann model for multiphase fluid flows. Phys. Fluids A, 1993, 5(10):2557-2562.
[66] Kono K. Ishizuka T. Tsuda H. et al. Application of lattice Boltzmann model to multiphase flow with phase transition. Comput. Phys. Commun., 2000, 129:110-120.
[67] D'ortona U. Salin D. Cieplak M. et al. Interfacial phenomena in Boltzmann cellular automata. Europhys. Lett., 1994, 28:317-322.
[68] Shan X., Chen H. Lattice Boltzmann model for simulating flows with multiple phases and components. Phys. Rev. E, 1993,47:1815-1819.
[69] Shan X., Chen H. Simulation of nonideal gases and liquid-gas phase-transitions by the lattice Boltzmann equation. Phys. Rev. E, 1994, 49:2941-2948.
[70] Qian Y, Succi S., Orszag S. Recent advances in lattice Boltzmann computing. Annu. Rev. Comput. Phys., 1995,3:195-242.
[71] Shan X. Pressure tensor calculation in a class of nonideal gas lattice Boltzmann models. Phys. Rev. E, 2008, 77: 066702.
[72] Swift M. R., Osborn W. R., Yeomans J. M. Lattice Boltzmann simulations of nonideal fluids. Phys. Rev. Lett., 1995, 75:830-833.
[73] Swift M. R., Orlandini E., Osborn W. R. et al. Lattice Boltzmann simulations of liquid-gas and binary fluid systems. Phys. Rev. E, 1996, 54(5):5014-5052.
[74] Inamuro T., Konishi N., Ogino F. A Galilean invariant model of the lattice Boltzmann method for multiphase fluid flows using free-energy approach. Computer, Phys. Communications, 2000, 129:32-45.
[75] Luo L. S. Unified theory of lattice Boltzmann models for nonideal gases, Phys. Rev. Lett., 1998, 8:1618-1621.
[76] Luo L. S. Theory of lattice Boltzmann method: lattice Boltzmann models for nonideal gases. Phys. Rev. E, 2000, 62:4982-4996.
[77] He X., Shan X., Doolen G. Discrete Boltzmann equation model for non-ideal gases. Phys. Rev. E, 1998,57:R13-R16.
[78] He X., Chen S., Zhang R. A lattice Boltzmann scheme for incompressible multiphase flow and its application in simulation of Ray leigh-Taylor instability. J. Comput. Phys., 1999, 152:642-663.
[79] He X., Zhang R., Chen S. et al. On the three-dimensional Rayleigh-Taylor instability. Phys. Fluids, 1999,11:1143-1152.
[80] Guo Z., Zhao T. S. Discrete velocity and lattice Boltzmann models for binary mixtures of nonideal fluids. Phys. Rev. E, 2003, 68:035302.
[81] Nie X. B., Doolen G. D., Chen S. Lattice Boltzmann simulations of fluid flows in MEMS. J. Stat. Phys., 2002, 107:279-289.
[82] Succi S. Mesoscopic modeling of slip motion at fluid-solid interfaces with heterogeneous catalysis. Phys. Rev. Lett., 2002, 89:064502.
[83] Lim C. Y.,Shu C., Niu X. D. et al. Application of lattice Boltzmann method to simulate microchannel flows. Phys. Fluids, 2002,14: 2299-2308.
[84] Ansumali S., Karlin I. V. Kinetic boundary conditions in the lattice Boltzmann method. Phys. Rev. E, 2002, 66:026311.
[85] Tang G. H., Tao W. Q, He Y. L. Lattice Boltzmann method for gaseous microflows using kinetic theory boundary conditions. Phys. Fluids, 2005, 17:058101.
[86] Shan X., Yuan X. F., Chen H. Kinetic theory representation of hydrodynamics: a way beyond the Navier-Stokes equation. J. Fluid Mech., 2006, 550:413-441.
[87] Kim S. H., Pitsch H. Slip velocity and Knudsen layer in the lattice Boltzmann method for microscale flows. Phys. Rev. E, 2008, 77:026704.
[88] Tang G. H., Zhang Y. H., Gu X. J. et al. Lattice Boltzmann modelling Knudsen layer effect in non-equilibrium flows. Europhys. Lett., 2008, 83:40008.
[89] Guo Z. Zhao T. S., Shi Y. Physical symmetry, spatial accuracy, and relaxation time of the lattice Boltzmann equation for microgas flows. J. Appl. Phys., 2006, 99:074903.
[90] Zhang Y. H., Gu X. J., Barber R. W. et al. Capturing Knudsen layer phenomena using a lattice Boltzmann model. Phys. Rev. E, 2006, 74:046704.
[91] Guo Z., Shi B., Zhao T. S. et al. Discrete effects on boundary conditions for the lattice Boltzmann equation in simulating microscale gas flows. Phys. Rev. E, 2007,76:056704.
[92] Zheng L., Guo Z. L., Shi B. C. Discrete effects on thermal boundary conditions for the thermal lattice Boltzmann method in simulating microscale gas flows. Europhys. Lett., 2008, 82:44002.
[93] Sofonea V., Sekerka R. F. Diffuse-reflection boundary conditions for a thermal lattice Boltzmann model in two dimensions: Evidence of temperature jump and velocity slip in microchannel. Phys. Rev. E, 2005, 71: 066709.
[94] Shu C, Niu X. D., Chew Y. T. A lattice Boltzmann kinetic model for microflow. J. Stat. Phys., 2005, 121: 239-245.
[95] Dawson S.P., Chen S., Doolen G. D. Lattice Boltzmann computations for reaction-diffusion equations. J. Chem. Phys., 1993, 98:1514-1523.
[96] Chen S., Dawson S. P., Doolen G. D. et al. Lattice methods and their applications to reacting systems. Comp. Chem. Eng., 1995,19:617-646.
[97] Succi S., Bella G., Papetti F. Lattice kinetic theory for numerical combustion. J. Sci. Comput., 1997, 12:395-408.
[98] Filippova O., Hanel D. A novel lattice BGK approach for low Mach number combustion. J. Comput. Phys., 2000, 158:139-160.
[99] Yamamoto K., He X., Doolen G. D. Simulation of combustion field with lattice Boltzmann method. J. Stat. Phys., 2002, 107(1/2):367-383.
[100] Chen S., Chen H., Martinez D. O. et al. Lattice Boltzmann model for simulating of magnetohy-drodynamics. Phys. Rev. Lett., 1991, 67:3776-3779.'
[101] MartinezD. O., Chen S. Lattice Boltzmann magnetohydrodynamics. Phys. Plasmas, 1997,1:1850-1867.
[102] Dellar J. Lattice kinetic scheme for magnetohydrodynamics. J. Comput. Phys., 2002, 179(1):95-126.
[103] Chen X. W., Shi B. C. A new lattice Boltzmann model for incompressible nagnetohydrodynamics. Chin. Phys., 2005, 17(7): 1398-1406.
[104] Bhatnagar P. L., Gross E. P., Krook, M. A model for collision processes in gases. I. Small amplitude processes in charged and neutral one-component systems. Phys. Rev., 1954, 94:511-525.
[105] d'Humieres D. Generalized lattice-Boltzmann equations, in rarefied gas dynamics: theory and simulations. In: Shizgal B. D., Weaver D. P. Progress in Astronautics and Aeronautics. Washington, DC: AIAA, 1992,159:450-458.
[106] Lallemand P., Luo L. S. Theory of the lattice Boltzmann method: Dispersion, dissipation, isotropy, Galilean invarioance and stability. Phys. Rev. E, 2000, 61:6546-6562.
[107] He X., Zou Q., Luo L. S. et al. Analytic solutions of simple flows and analysis of nonslip boundary conditions for the lattice Boltzmann BGK model. J. Stat. Phys., 1997, 87:115-136.
[108] Ziegler D. Boundary conditions for lattice Boltzmann simulations. J. Stat. Phys., 1993, 71:1171-1177.
[109] Zou Q., He X. On pressure and velocity boundary conditions for the lattice Boltzmann BGK model. Phys. Fluids, 1997, 9:1591-1597.
[110] Noble D., Chen S., Georgiadis J. G. et al. A consistent hydrodynamic boundary conditions for lattice Boltzmann method. Phys. Fluids, 1995,7:203-205.
[111] Maier R. S., Bernard R. D., Grunau D. W. Boundary condition for lattice Boltzmann simulations. Phys. Fluids, 1996, 8:1788-1801
[112] Inamuro T., Yoshino M., Ogino F. A non-slip boundary condition for lattice Boltzmann simulations. Phys. Fluids, 1995, 7:2928-2930.
[113] Skordos P. A. Initial and boundary conditions for the lattice Boltzmann method. Phys. Rev. E, 1993,48:4823-4842.
[114] Chen S., Martinez D., Mei R. On boundary conditions in lattice Boltzmann methods. Phys. Fluids, 1996, 8(9):2527-2536.
[115] Guo Z. L., Zheng C.G., Shi B.C. Non-equilibrium extrapolation method for velocity and pressure boundary conditions in the lattice Boltzmann method. Chin. Physics, 2002, 11(4): 366-374.
[116] Bear J. Dynamics of Fluids in Porous Media. New York: American Elsevier, 1972.
[117] Scheidegger A. E. The Physics of Flow through Porous Media. Toronto: University of Toronto, 1974.
[118] Nield D. A., Bejan A. Convection in Porous Media. New York: Springer, 2006.
[119] Lee S. L., Yang J. H. Modeling of Darcy-Forchheimer drag for fluid flow across a bank of circular cylinders. Int. J. Heat Mass Transfer, 1997,40:3149-3155.
[120] Andrade Jr J. S., Costa U. M. S., Almeida M. P. et al. Inertial effects on fluid flow through disordered porous media. Phys. Rev. Lett., 1999, 82:5249-5252.
[121] Ghaddar C. K. On the permeability of unidirectional fibrous media: A parallel computational approach. Phys. Fluids, 1995, 7:2563-2586.
[122] Mei C. C., Auriault J. L. The effect of weak inertia on flow through a porous medium. J. Fluid Mech., 1991, 222:647-663.
[123] Coulaud O., Morel P., Caltagirone J. P. Numerical modeling of nonlinear effects in laminar flow through a porous medium. J. Fluid Mech., 1988,190:393-407.
[124] Skjetne E., Auriault J. L. New insights on steady, non-linear flow in porous media. Eur. J. Mech. E/Fluids, 1999, 18:131-145.
[125] Skjetne E., Hansen A., Gudmundsson J. S. High-velocity flow in a rough fracture. J. Fluid Mech., 1999,383:1-28.
[126] Rojas S., Koplik J. Nonlinear flow in porous media. Phys. Rev. E, 1998, 58:4776-4782.
[127] Zimmerman R. W., Al-Yaarubi A., Pain C. C. et al. Non-linear regimes of fluid flow in rock fractures. Int. J. Rock Mech. Min. Sci., 2004, 413:163-169.
[128] Firdaouss M., Guermond J. L., Quere P. L. Nonlinear corrections to Darcy's law at low Reynolds numbers. J. Fluid Mech., 1997, 343:331-350.
[129] Fourar M., Radilla G., Lenormand R. et al. On the non-linear behavior of a laminar single-phase flow through two and three-dimensional porous media. Adv. Water Res., 2004, 27:669-677.
[130] Lucas Y., Panfilov M., Bues M. High velocity flow through fractured and porous media: the role of flow non-periodicity. Euro. J. Mech. B/Fluids, 2007,26:295-303.
[131] Panfilov M., Fourar M. Physical splitting of nonlinear effects in high-velocity stable flow through porous media. Adv. Water Res. 2006,29:30-41.
[132] Hill R., Koch D. L., Ladd A. J. C. The first effects of fluid inertia on flows in ordered and random arrays of spheres. J. Fluid Mech., 2001,448:213-241.
[133] Ferreol B., Rothman D. H. Lattice-Boltzmann simulations of flow through Fontainebleau sandstone. Transport Porous Med., 1995,20:3-20.
[134] Heijs A. W. J., Lowe C. P. Numerical evaluation of the permeability and the Kozeny constant for two types of porous media. Phys. Rev. E, 1995, 51:4346-4352.
[135] Koponen A., Kandhai D., Hellen E. Permeability of three-dimensional random fiber webs. Phys. Rev. Lett., 1998,80:716-719.
[136] Pan C., Luo L. S., Miller C. T. An evaluation of lattice Boltzmann schemes for porous medium flow simulation. Comput. Fluids, 2006, 35:898-909.
[137] Jin G., Patzek T. W, Silin D. B. Direct prediction of the absolute permeability of unconsolidated and consolidated reservoir rock. Paper SPE90084. In: SPE Annual Technical Conference and Exhibition. Houston: Society of Petroleum Engineers, Inc., 2004. 1-15.
[138] Chai Z., Shi B., Zheng L. Simulating high Reynolds number flow in two-dimensional lid-driven cavity by multi-relaxation-time lattice Boltzmann method. Chin. Phys., 2006,15(8):1855-1863.
[139] Ginzburg I., d'Humieres D. Multireflection boundary conditions for lattice Boltzmann models. Phys. Rev. E, 2003, 68:066614.
[140] Shankar P. N., Deshpande M.D. Fluid mechanics in the driven cavity. Annu. Rev. Fluid Mech., 2000, 32:93-136.
[141] Ghia U., Guia K. N., Shin C. T. High-Re solutions for incompressible flow using the Navier-Stokes equations and a multigrid method. J. Comput. Phys., 1982,48:387-411.
[142] Hou S. L., Chen S. Y., Zou Q. S. et al. Simulation of cavity flow by the lattice Boltzmann method. J. Comput. Phys., 1995,118:329-3471.
[143] Nallasamy M., Prasad K. On cavity flow at high Reynolds numbers. J. Fluid Mech., 1977,79:391-414.
[144] Zhou X. Y., Shi B. C., Wang N. C. Numerical simulation fo LBGK model for high Reynolds number flow. Chin. Phys., 2004, 13:712-719.
[145] Ram(?)ak M., (?)kerget L. A subdomain boundary element method for high-Reynolds laminar flow using stream-vorticity formulation. Int. J. Numer. Meth. Fluids, 2004,46:815-847.
[146] Erturek E., Corke T. C, G(?)kc(?)l C. Numerical solutions of 2-D steady incompressible driven cavity flow at high Reynolds numbers. Int. J. Numer. Meth. Fluids, 2005,48:747-774.
[147] Hou S. L. Lattice Boltzmann method for incompressible viscous flow: PhD thesis. Kansas State University, 1995.
[148] Jackson G. W., James D. F., Jackson G. W. et al. The permeability of fibrous porous media. Can. J. Chem. Eng., 1986, 64:364-374.
[149] Sangani A. S., Acrivos A. Slow flow past periodic arrays of cylinders with applicationto heat transfer. Int. J. Multphase flow, 1982, 8:193-206.
[150] Gad-el-Hak M. The fluid mechanics of microdevices-the Freeman scholar lecture. ASME Trans. J. Fluids Eng., 1999,121:5-33.
[151] Stone H. A., Stroock A. D., Ajdari A. Engineering flows in small devices: Microfluidics toward a Lab-on-a-Chip. Annu. Rev. Fluid Mech., 2004, 36:381-411.
[152] Ho C. M., Tai Y. C. Micro-Electro-Mechanicalsystems (MEMS) and fluid flows. Annu. Rev. Fluid Mech., 1998,30:579-612.
[153] Squires T. M., Quake S. R. Microfluidics: Fluid physics at the nanoliter scale. Rev. Mod. Phys., 2005,77:977-1026.
[154] Abramson A. R., Tien C. L. Recent developments in microscale thermophysical engineering. Microscale Thermophys. Eng., 1999, 3:229-244.
[155] Obot N. T. Toward a better understanding of friction and heat/mass transfer in microchannels-a literature review. Microscale Thermophys. Eng., 2002, 6:155-173.
[156] Guo Z. Y., Li Z. X. Size effect on single-phase channel flow and heat transfer at microscale. Int. J. Heat Fluid Flow, 2003, 24:284-298.
[157] Wu P., Little W. A. Measurement of friction factors for the flow of gases in very fine channels used for microminiature Joule-Thomson refrigerators. Cryogenics, 1983, 23:273-277.
[158] Pfahler J., Harley J., Bau H. et al. Liquid and gas transport in small channels. ASME Dyn. Syst. Control Div. 1990,19:149-157.
[159] Pfahler J., Harley J., Bau H. et al. Gas and liquid flow in small channels. ASME Dyn. Syst. Control Div., 1991,32:49-60.
[160] Choi S. B., Barron R. F., Warrington R. Q. Fluid flow and heat transfer in micro tubes. ASME Dyn. Syst. Control Div., 1991, 32:123-133.
[161] Yu D., Warrington R., Barron R. et al. An experimental and theoretical investigation of fluid flow and heat transfer in microtubes. In: Flecher L. S., Aihara F. Proceedings of the 1995 ASME/JSME Thermal Engineering Joint Conference. New York: ASME, 1995. 523-530.
[162] Turner S. E., Lam L. C, Faghri M. et al. Experimental investigation of gas flow in microchannels. Trans. ASME J. Heat Transfer, 2004, 126:753-763.
[163] Harley J., Huang Y., Bau H. et al. Gas flow in micro-channels. J. Fluid Mech., 1995, 284:257-274.
[164] Papautsky I., Brazzle J., Ameel T. et al. Laminar fluid behavior in microchannels using micropolar fluid theory. Sens. Actuators A, 1999,73:101-108.
[165] Sugiyama W., Sawadaa T., Nakamori K. Rarefied gas flow between two flat plates with two dimensional surface roughness. Vacuum, 1996,47(6-8):791-794.
[166] Usami M., Fujimoto T., Kato S. Mass-flow reduction of rarefied gas by roughness of a slit surface. Trans. Jpn. Soc. Mech. Eng., Ser. B, 1988, 54:1042-1050.
[167] Sun H., Faghri M. Effect of surface roughness on nitrogen flow in a microchannel using the direct simulation Monte Carlo method. Numer. Heat Transfer Part A, 2003,43:1-8.
[168] Ji Y., Yuan K., Chung J. N. Numerical simulation of wall roughness on gaseous flow and heat transfer in a microchannel. Int. J. Heat Mass Transfer, 2006,49:1329-1339.
[169] Croce G., D'Agaro P., Filippo A. Compressibility and rarefaction effects on pressure drop in rough microchannels. Heat Transfer Eng., 2007, 28:688-695.
[170] Zhang T., Li L. Numerical simulation of roughness effect on gaseous flow and heat transfer in microchannels. In: Pornsinsirirak T. N. Proceedings of the 2nd IEEE International Conference on Nano/Micro Engineered and Molecular Systems. Bangkok: IEEE, 2007. 136-161.
[171] Cao B. Y, Chen M., Guo Z. Y. Effect of surface roughness on gas flow in microchannels by molecular dynamics simulation. Int. J. Eng. Sci., 2006,44:927-937.
[172] Arkilic E. B., Schmidt M. A., Breuer K. S. Gaseous slip flow in long microchannels. J. Microelec-tromech. Syst., 1997, 6:167-178.
[173] Adzumi H. On the flow of gases through a porous wall. Bull. Chem. Soc. Japan, 1937, 12(6):304-312.
[174] Klinkenberg L. J. The Permeability of porous media to liquid and gases. API Drilling and production practice, 1941, 200-213.
[175] Jones S. C. A rapid accurate unsteady-state Klinkenberg permeameter. SPE J., 1972, 3535:383-393.
[176] Jones F. O., Owens W. W. A laboratory study of low permeability gas sands. SPE J., 1979, 7551:1-10.
[177] Sampath K., Keighin C. W. Factors affecting ga s slippage in tight sandstones. SPE J., 1981, 9872:409-415.
[178] Wu Y. S., Pruess K., Persoff P. Gas flow in porous media with Klinkenberg effects. Trans. Porous Media., 1998,32:117-137.
[179] Kaluarachchi J. Analytical solution to two-dimensional axisymmetric gas flow with Klinkenberg effect. J. Environ. Eng., 1995,121(5):417-420.
[180] Grove D. M., Ford M. G. Evidence for permeability minima in low-pressure gas flow through porous media. Nature, 1958, 182:999-1000.
[181] De Socio L. M., Marino L. Gas flow in a permeable medium. J. Fluid Mech., 2006, 557:119-133.
[182] Florence F. A., Rushing J. A., Newsham K. E. et al. Improved permeability prediction relations for low permeability sands. SPE J., 2007, 107954:1-18.
[183] Tanikawa W., Shimamoto T. Klinkenberg effect for gas permeability and its comparison to water permeability for porous sedimentary rocks. Hydrol. Earth Syst. Sci. Discuss., 2006, 3:1315-1338.
[184] Woudberg S., Du Plessis J. P. Predicting the permeability of very low porosity sandstones. Trans. Porous Media, 2008, 73:39-55.
[185] Pavan V., Oxarango L. A new momentum equation for gas flow in porous media: The Klinkenberg effect seen through the kinetic theory. J. Stat. Phys., 2007, 126(2):355-389.
[186] Skjetne E., Auriault J. L. Homogenization of wall slip in the Darcy flow through porous media. Trans. Porous Media, 1999, 36:293-306.
[187] Chastanet J. Royer P., Auriault J. L. Does Klinkenberg's law survive upscaling? Trans. Porous Media, 2004, 56:171-198.
[188] Parthlban C, Patll P. R. Convection in a porous medium with velocity slip and temperature jump boundary conditions. Heat Mass Transfer, 1996, 32:27-31.
[189] Haddad O. M., Abuzaid M. M., Al-Nimr M. A. Developing free-convection gas flow in a vertical open-ended microchannel filled with porous media. Nume. Heat Transfer, Part A, 2005, 48:693-710.
[190] Nield D. A., Kuznetsov A. V. Forced convection with slip-flow in a channel or duct occupied by a hyper-porous medium saturated by a rarefied gas. Trans. Porous Media, 2006, 64:161-170.
[191] Skjetne E. High Velocity flow in porous media: analytical, numerical and experimental studies: PhD Thesis. Norwegian University of Science and Technology, 1995.
[192] Liu Y. W., Zhou F. X., Yan G. W. Lattice Boltzmann simulations of the Klinkenberg effect in porous media. Chin. J. Comput. Phys., 2003, 20(2):157-160.
[193] Verhaeghe F, Luo L. S., Blanpain B. Lattice Boltzmann modeling of microchannel flow in slip flow regime. J. Comput. Phys., 2009, 228:147-157.
[194] Guo Z, Zheng C. Analysis of lattice Boltzmann equation for microscale gas flows: Relaxation times, boundary conditions and the Knudsen layer. Int. J. Comput. Dyn., 2008,7:465-473.
[195] Guo Z., Shi B., Zheng C. An extended Navier-Stokes formulation for gas flows in the Knudsen layer near a wall. Europhys. Lett., 2007, 80:24001.
[196] Guo Z., Zheng C., Shi B. Lattice Boltzmann equation with multiple effective relaxation times for gaseous microscale flow. Phys. Rev. E, 2008, 77:036707.
[197] Collins R. E. Flow of fluids through porous materials. Tulsa: The Petroleum Publishing Company, 1976.
[198] Burgreen D., Nakache F. R. Electrokinetic flow in ultrafine capillary slits. J. Phys. Chem., 1964 68:1084-1091.
[199] Rice C. L., Whitehead R. Electrokinetic flow in a narrow cylindrical capillary. J. Phys. Chem., 1965,69:4017-4024.
[200] Qu W., Li D. Model for overlapped EDL fields. J. Colloid Interface Sci., 2000,224:397-407.
[201] Patankar N. A., Hu H. H. Numerical simulation of electroosmotic flow. Anal. Chem., 1998, 70:1870-1881.
[202] Bianchi F., Ferrigno R., Girault H. H. Finite element simulation of an Electroosmotic-Driven flow division at a T-junction of microscale dimensions. Anal.Chem., 2000,72:1987-1993.
[203] Mitchell M. J., Qiao R., Aluru N. R. Meshless analysis of steady-state electro-osmotic transport. J. Microelectromechanical Syst., 2002, 9:435-449.
[204] Jin Y., Luo G. A. Numerical calculation of the electroosmotic flow at the cross region in microfluidic chips. Electrophoresis, 2003, 24:1242-1252.
[205] Zhao T. S., Liao Q. Thermal effects on electroosmotic pumping of liquids in microchannels. J. Micromech. Microeng., 2002, 12:962-970.
[206] Fan Z. H., Harrison D. J. Micromachining of capillary electrophoresis injectors and separators on glass chips and evaluation of flow at capillary intersections. Anal. Chem., 177,66:177-184.
[207] Santiago J. G. Electroosmotic Flows in Microchannels with Finite Inertial and Pressure Forces. Anal. Chem., 2001, 73:2353-2365.
[208] Soderman O., Jonsson B. Electro-osmosis: Velocity profiles in different geometries with both temporal and spatial resolution. J. Chem. Phys., 1996,105:10300 - 10311.
[209] Keh H. J., Tseng H. C. Transient Electrokinetic Flow in Fine Capillaries. J. Colloid Interface Sci., 2001, 242:450-459.
[210] Yang C., Ng C.B., Chan V. Transient analysis of electroosmotic flow in a slit microchannel. J. Collid Interface Sci., 2002 248:524-527.
[211] Qiao R., Aluru N. R. Transient analysis of electro-osmotic transport by a reduced-order modeling approach.Int. J. Numer. Meth. Engrg., 2003, 56:1023-1050.
[212] Tian F., Li B., Kwok D. Y. Lattice Boltzmann simulation of electroosmotic flows in micro- and nanochannels. In: Proceedings of the 2004 International Conference on MEMS, NANO and Smart Systems (ICMENS' 04). Alberta: IEEE, 2004. 294-299.
[213] He X., Li N. Lattice Boltzmann simulation of electrochemical systems. Comput. Phys. Commun., 2000, 129:158-166.
[214] Guo Z., Zhao T. S., Shi Y. A lattice Boltzmann algorithm for electro-osmotic flows in microfluidic devices. J. Chem. Phys., 2005, 122:144907.
[215] Wang J., Wang M., Li Z. Lattice Poisson-Boltzmann simulations of electro-osmotic flows in microchannels. J. Collid Interface Sci., 2006, 296:729-736.
[216] Hlushkou D., Kandhai D., Tallarek U. Coupled lattice-Boltzmann and finite-difference simulation of electroosmosis in microfluidic channels. Int. J. Numer. Methods Fluids, 2004,46:507-532.
[217] Guo Z., Shi B., Wang N. Lattice BGK model for incompressible Navier-Stokes equation. J. Comput. Phys., 2000,165:288-306.
[218] Chai Z., Shi B. Simulation of electro-osmotic flow in microchannel with lattice Boltzmann method. Phys. Lett. A, 2007, 364:183-188.
[219] Chai Z., Shi B. A novel lattice Boltzmann model for the Poisson equation. Appl. Math. Model., 2008,32:2050-2058.
[220] Masliyah J. Electrokinetic Transport Phenomena. Alberta: Alberta Oil Sands Technology and Research Authority, 1994.
[221] Guo Z., Zheng C., Shi B. Discrete lattice effects on the forcing term in the lattice Boltzmann method. Phys. Rev. E, 2002, 65:046308.
[222] Chai Z., Guo Z., Shi B. Study of electro-osmotic flows in microchannels packed with variable porosity media via lattice Boltzmann method. J. Appl. Phys., 2007, 101:104913.
[223] Siyyam H. I. An accurate solution of the Poisson equation by the finite difference-Chebyshev-Tau method. Appl. Math. Mech., 2001, 22(8):0935-0939.
[224] Boschitsch A. H., Fenley M. O. Hybrid boundary element and finite difference method for solving the nonlinear Poisson-Boltzmann equation. J. Comput. Chem., 2004, 25(7):935-955.
[225] Cai Z., Kim S., Kim S. et al. A finite element method using singular functions for Poisson equations: Mixed boundary conditions. Comput. Methods Appl. Mech. Engrg., 2006, 195:2635-2648.
[226] Kikuchi F., Saito H. Remarks on a posteriori error estimation for finite element solutions. J. Comput. Appl. Math., 2007, 199(2):329-336.
[227] Liao S. J. General boundary element method for Poisson equation with spatially varying conductivity. Eng. Anal. Bound. Elem., 1998, 21:23-38.
[228] Guo Z. L., Shi B. C, Wang N. C. Fully Lagrangian and lattice Boltzmann methods for the advection-diffusion equation. J. Sci. Comput., 1999, 14(3):291-300.
[229] Deng B., Shi B. C., Wang G. C. A new lattice Bhatnagar-Gross-Krook model for convection-diffusion equation with a source term. Chin. Phys. Lett., 2005, 22(2):267-270.
[230] Shi B., Deng B., Du R. et al. A new scheme for source term in LBGK model for convection-diffusion equation. Comput. Math. Appl., 2008, 55:1568-1575.
[231] Chen S., Dawson S. P., Doolen G. D. et al. Lattice methods and their applications to reacting systems. Computers Chem. Engny, 1995,19(6/7):617-646.
[232] Yu X., Shi B. A lattice Bhatnaar-Gross-Krook model for a class of the generalized Burgers equation. Chin. Phys., 2006, 15(07): 1441-1449.
[233] Yu X., Shi B. A lattice Boltzmann model for reaction dynamic systems with time systems. Appl. Math. Comput., 2006, 181:958-965.
[234] Yan G. A Lattice Boltzmann Equation for Waves. J. Comput. Phys., 2000, 161:61-69.
[235] Hirabayashi M., Chen Y., Ohashi H. The lattice BGK model for the Poisson equation. JSME Int. J. Ser. B, 2001,44(1):45-52.
[236] Zhang C. Y., Tan H. L., Liu M. R. et al. A lattice Boltzmann model and simulation of KdV-Burgers equation. Commun. Theor. Phys, 2004, 42(2):281-284.
[237] Ma C. A New Lattice Boltzmann Model for KdV-Burgers Equation. Chin. Phys. Lett., 2005, 29(9):2313-2315.
[238] Chai Z., Shi B., Zheng L. A unified lattice Boltzmann model for some nonlinear partial differential equations. Chaos, Solitons and Fractals, 2008, 36:874-882.
[239] Dawson S. P., Chen S., Doolen D. G. Lattice Boltzmann Computations for reaction-diffusion equations. J. Chem. Phys., 1993, 98(2): 1514-1523.
[240] Chen S., T(?)lke J., Geller S. et al. Lattice Boltzmann model for incompressible axisymmetric flows. Phys. Rev. E, 2008, 78:046703.
[241] Coelho D., Shapiro M., Thovert J. F.et al. Electroosmotic phenomena in porous media. J. Colloid Interface Sci., 1996,181:169-190.
[242] Kang Y., Yang C., Huang X. AC electroosmosis in microchannels packed with a porous medium. J. Micromech. Microeng., 2004, 14:1249-1257.
[243] Scales N., Taita N. Modeling electroosmotic and pressure-driven flows in porous microfluidic devices: Zeta potential and porosity changes near the channel walls. J. Chem. Phys., 2006, 125:094714.
[244] Wang M., Wang J., Chen S. et al. Electrokinetic pumping effects of charged porous media in microchannels using the lattice Poisson-Boltzmann method. J. Colloid Interface Sci., 2006, 304:246-253.
[245] Qiao R., Aluru N. R. Ion concentrations and velocity profiles in nanochannel electroosmotic flows. J. Chem. Phys., 2003,118:4692-4701.
[246] Nithiarasu P., Seetharamu K. N., Sundararajan T. Natural convective heat transfer in a fluid saturated variable porosity medium. Int. J. Heat Mass Transfer, 1997,40:3955-3967.
[247] Liu S., J. Masliyah. Single fluid flow in porous media. Chem. Eng. Commun., 1996,148-150:653-732.
[248] Zeng S., Chen C. H., Mikkelsen Jr J. C. et al. Fabrication and characterization of electroosmotic micropumps. Sens. Actuators B, 2001, 79:107-114.
[249] Ergun S. Fluid flow through packed columns. Chem. Eng. Prog., 1952,48:89-94.
[250] Vafai K. Convective flow and heat transfer in variable-porosity media. J. Fluid Mech., 1984, 147:233-259.
[251] Liapis A. I., Grimes B. A. Modeling the velocity field of the electroosmotic flow in charged capillaries and in capillary columns packed with charged particles: Interstitial and intraparticle velocities in capillary electro-chromatography systems. J. Chromatogr. A, 2000, 877(1-2): 181-215.
[252] Tallarek U., Rapp E., Seidel-Morgenstern A. et al. Electroosmotic flow phenomena in packed capillaries: From the interstitial velocities to intraparticle and boundary layer mass transfer. J. Phys. Chem. B, 2002,106:12709-12721.
[253] Iwatsu R., Hyun J. M., Kuwahara K. Mixed convection in a driven cavity with a stable vertical temperature gradient. Int. J. Heat Mass Transfer, 1993, 36:1601-1608.
[254] Oztop H. F. Combined convection heat transfer in a porous lid-driven enclosure due to heater with finite length. Int. Commun. Heat Mass Transf., 2006,33:772-779.
[255] Khanafer K. M., Chamkha A. J. Mixed convection flow in a lid-driven enclosure filled with a fluid-saturated porous medium. Int. J. Heat Mass Transfer, 1999,42:2465-2481.
[256] Al-Amiri A. M. Analysis of momentum and energy transfer in a lid-driven cavity filled with a porous medium. Int. J. Heat Mass Transfer, 2000,43:3513-3527.
[257] Jue T. C. Analysis of flows driven by a torsionally-oscillatory lid in a fluid-saturated porous enclosure with thermal stable stratification. Int. J. Therm. Sci., 2002,41:795-804.
[258] Guo Z., Zhao T. S. A lattice Boltzmann model for convection heat transfer in porous media. Numerical Heat Transfer Part B, 2005, 47:157-177.