几类铁磁流体模型及其数值求解
|
摘要
铁磁流体具有某些特殊的物理性质,它已广泛的应用到化学、机械、材料、磁路设计、医学等领域,对铁磁流体的建模及其数值求解已成为计算流体力学领域中研究的热点.虽然该方面的研究取得了很大的进展,但仍存在许多值得研究的问题.本文围绕铁磁流体化学机械抛光、铁磁流体热传导、铁磁流体R-T不稳定性三类模型问题的建模和数值求解进行研究,获得了若干理论与数值结果.
针对铁磁流体化学机械抛光(CMP)模型问题.首先建立了以铁磁流体作为抛光液且具有离心力的CMP模型,数值求解表明外界磁场的作用可以有效的改变铁磁流体抛光液压力的分布和转角、倾角、膜厚对载荷与转矩的影响.其次,考虑流体的对流效应,通过对二维流体力学方程组进行简化,推导出对应的Reynolds润滑方程,建立了相应的CMP模型；数值分析了转角、倾角、膜厚对压力分布、载荷和转矩的影响,从而表明了所建模型的合理性.在考虑对流效应的基础上,研究了抛光液为铁磁流体的CMP过程,数值分析表明,在该模型中,外界磁场也能有效的改变抛光液流体压力的分布和大小.最后,通过数值实验比较,研究了不同铁磁流体CMP模型中,外界磁场对压力分布的作用,以及在外界磁场作用下,离心力、对流效应对铁磁流体CMP压力分布的作用.这些结果为改进CMP工艺提供了可供参考的依据.
针对两种铁磁流体热传导问题,即二维封闭方腔铁磁流体自然对流和二维平板间铁磁流体热传导,建立它们所满足的模型方程.然后,在取外界磁场梯度一致的情况下,采用SIMPLE算法对这两种模型方程进行数值求解.对封闭方腔铁磁流体的自然对流,当磁场梯度与温度梯度方向相同时,外界磁场会增加封闭方腔内的自然对流和压力,且随着Rayleigh数的增大,磁Rayleigh数对封闭方腔内的自然对流和压力影响增大的趋势减弱.对开口系统的两平板间热传导问题,随着Rayleigh数的增大,磁Rayleigh数对平板间铁磁流体的热传导影响增大的趋势减弱.且当外界磁场梯度平行于开口方向时,随着外界磁场的增大,更易加强流体的热传导.最后,利用扩散抛物化理论,对铁磁流体热传导问题进行了抛物化简化,给出了铁磁流体热传导问题的层次结构理论,并对两平板间热传导问题的抛物化简化方程进行数值求解.通过与简化前的数值结果比较,验证了铁磁流体热传导抛物化简化的合理性.
针对一种铁磁流体R-T不稳定性问题.推导了铁磁流体R-T不稳定问题满足的模型方程.在一致网格下,利用FD-WENO格式和TVD R-K格式对该模型方程建立了相应的离散系统.数值结果表明,当磁场梯度方向与流速方向一致时,外界磁场的作用会加速界面波头向前传播,反之则会减慢界面波头的传播.当磁场梯度方向与流速方向垂直时,则会增加界面的侧向不稳定性,使其对称性遭到破坏,而且在侧面会形成更多的泡沫状漩涡.这些结果对如何通过外界磁场的大小和方向来控制铁磁流体R-T不稳定性的发展具有参考价值.
Some special physical characteristics of ferrofluid make it is widely ap-plied in the fields of chemistry, mechanism, materials science, magnetic circuit design, medicine, etc. So, modeling ferrofluid and its numerical computation have become a hot topic in field of computational mechanics. Although some great progresses on the topic have been achieved, some important problems remain unresolved. This thesis focuses on modeling and numerical compu-tation of three classes of problems, chemical mechanical polishing (CMP) of ferrofluid, heat conduction of ferrofluid and R-T instability of ferrofluid. Contribution of this thesis on them can be described as follows.
As for CMP modeling of ferrofluid, a CMP model with centrifugal force and ferrofluid slurry is installed firstly. The simulation results show that external magnetic fields can effectively change pressure distribution and in-fluences of nominal clearance, rolling angle and pitch angle on the dimen-sionless resultant forces and moments. Considering convective effect of or-dinary fluid, a Reynolds lubrication equation is derived by simplifying the 2-D Navier-Stokes equations, which is used to build the corresponding CMP model. Rationality of the model is verified by analyzing influences of nomi-nal clearance, rolling angle, pitch angle on the dimensionless resultant forces and moments, in term of numerical computation. Based on the convective effects of fluid, CMP process model of ferrofluid is studied. Simulation anal-ysis shows that distribution and magnitude of the pressure will be changed by changing external magnetic field. At last, by comparing with numerical results, roles of external magnetic fields to the pressure distribution are stud-ied in different ferrofluid CMP models; and influences of centrifugal force or convective effect to the pressure distribution are also researches in different external magnetic fields. These results can work as reference for improving CMP technologies.
As for two kinds of ferrofluid heat conduction problems, nature con-vection in 2-D closed square cavity and heat conduction between two plates, model equations about them are established. When gradient of external mag-netic field is uniform, the two kinds of equations are solved by the SIMPLE algorithm. When direction of gradient of external magnetic field is the same as that of gradient of temperature, the numerical results show that the ex-ternal magnetic field can increase the convection and the pressure in the 2-D closed square cavity, and the trends of increase of the influence on the con-vection and the pressure is decreased with increase of the magnetic Rayleigh number. For the heat conduction between two plates, a kind of open system, the numerical results show that if the magnetic Rayleigh is increased, trends of increase of the influence on the thermal conductivity is decreased too. And if direction of gradient of external magnetic field is parallel to that of the plates, external magnetic can accelerate the heat conduction. Based on these model equations and the diffusion parabolized theory, the heat conduc-tion equations of ferrofluid are simplified and the grade structure theory of the equations is also derived. As an example, the heat conduction model be-tween two plates is calculated numerically. The numerical results show that the simplified equations are more reasonable, compared with the previous ones.
As for the R-T instability problem of ferrofluid, the simplified model equations are derived firstly. Under uniform mesh, the corresponding discrete systems are installed by using weighted essentially non-oscillatory (WENO) scheme and TVD R-K method. Simulation results show that, when direction of the external magnetic filed is the same as that of gravity, velocities of the interface arc increased by the external magnetic field, vice versa. When the direction of the external magnetic field is perpendicular to that of gravity, the case become that instability of the interface is increased, symmetry of the interface is destructed, and more foam like spiral vortexes emerge on the boundaries. These results are very helpful for controlling R-T instability of ferrofluid by adjusting magnitude and direction of external magnetic field.
针对铁磁流体化学机械抛光(CMP)模型问题.首先建立了以铁磁流体作为抛光液且具有离心力的CMP模型,数值求解表明外界磁场的作用可以有效的改变铁磁流体抛光液压力的分布和转角、倾角、膜厚对载荷与转矩的影响.其次,考虑流体的对流效应,通过对二维流体力学方程组进行简化,推导出对应的Reynolds润滑方程,建立了相应的CMP模型；数值分析了转角、倾角、膜厚对压力分布、载荷和转矩的影响,从而表明了所建模型的合理性.在考虑对流效应的基础上,研究了抛光液为铁磁流体的CMP过程,数值分析表明,在该模型中,外界磁场也能有效的改变抛光液流体压力的分布和大小.最后,通过数值实验比较,研究了不同铁磁流体CMP模型中,外界磁场对压力分布的作用,以及在外界磁场作用下,离心力、对流效应对铁磁流体CMP压力分布的作用.这些结果为改进CMP工艺提供了可供参考的依据.
针对两种铁磁流体热传导问题,即二维封闭方腔铁磁流体自然对流和二维平板间铁磁流体热传导,建立它们所满足的模型方程.然后,在取外界磁场梯度一致的情况下,采用SIMPLE算法对这两种模型方程进行数值求解.对封闭方腔铁磁流体的自然对流,当磁场梯度与温度梯度方向相同时,外界磁场会增加封闭方腔内的自然对流和压力,且随着Rayleigh数的增大,磁Rayleigh数对封闭方腔内的自然对流和压力影响增大的趋势减弱.对开口系统的两平板间热传导问题,随着Rayleigh数的增大,磁Rayleigh数对平板间铁磁流体的热传导影响增大的趋势减弱.且当外界磁场梯度平行于开口方向时,随着外界磁场的增大,更易加强流体的热传导.最后,利用扩散抛物化理论,对铁磁流体热传导问题进行了抛物化简化,给出了铁磁流体热传导问题的层次结构理论,并对两平板间热传导问题的抛物化简化方程进行数值求解.通过与简化前的数值结果比较,验证了铁磁流体热传导抛物化简化的合理性.
针对一种铁磁流体R-T不稳定性问题.推导了铁磁流体R-T不稳定问题满足的模型方程.在一致网格下,利用FD-WENO格式和TVD R-K格式对该模型方程建立了相应的离散系统.数值结果表明,当磁场梯度方向与流速方向一致时,外界磁场的作用会加速界面波头向前传播,反之则会减慢界面波头的传播.当磁场梯度方向与流速方向垂直时,则会增加界面的侧向不稳定性,使其对称性遭到破坏,而且在侧面会形成更多的泡沫状漩涡.这些结果对如何通过外界磁场的大小和方向来控制铁磁流体R-T不稳定性的发展具有参考价值.
Some special physical characteristics of ferrofluid make it is widely ap-plied in the fields of chemistry, mechanism, materials science, magnetic circuit design, medicine, etc. So, modeling ferrofluid and its numerical computation have become a hot topic in field of computational mechanics. Although some great progresses on the topic have been achieved, some important problems remain unresolved. This thesis focuses on modeling and numerical compu-tation of three classes of problems, chemical mechanical polishing (CMP) of ferrofluid, heat conduction of ferrofluid and R-T instability of ferrofluid. Contribution of this thesis on them can be described as follows.
As for CMP modeling of ferrofluid, a CMP model with centrifugal force and ferrofluid slurry is installed firstly. The simulation results show that external magnetic fields can effectively change pressure distribution and in-fluences of nominal clearance, rolling angle and pitch angle on the dimen-sionless resultant forces and moments. Considering convective effect of or-dinary fluid, a Reynolds lubrication equation is derived by simplifying the 2-D Navier-Stokes equations, which is used to build the corresponding CMP model. Rationality of the model is verified by analyzing influences of nomi-nal clearance, rolling angle, pitch angle on the dimensionless resultant forces and moments, in term of numerical computation. Based on the convective effects of fluid, CMP process model of ferrofluid is studied. Simulation anal-ysis shows that distribution and magnitude of the pressure will be changed by changing external magnetic field. At last, by comparing with numerical results, roles of external magnetic fields to the pressure distribution are stud-ied in different ferrofluid CMP models; and influences of centrifugal force or convective effect to the pressure distribution are also researches in different external magnetic fields. These results can work as reference for improving CMP technologies.
As for two kinds of ferrofluid heat conduction problems, nature con-vection in 2-D closed square cavity and heat conduction between two plates, model equations about them are established. When gradient of external mag-netic field is uniform, the two kinds of equations are solved by the SIMPLE algorithm. When direction of gradient of external magnetic field is the same as that of gradient of temperature, the numerical results show that the ex-ternal magnetic field can increase the convection and the pressure in the 2-D closed square cavity, and the trends of increase of the influence on the con-vection and the pressure is decreased with increase of the magnetic Rayleigh number. For the heat conduction between two plates, a kind of open system, the numerical results show that if the magnetic Rayleigh is increased, trends of increase of the influence on the thermal conductivity is decreased too. And if direction of gradient of external magnetic field is parallel to that of the plates, external magnetic can accelerate the heat conduction. Based on these model equations and the diffusion parabolized theory, the heat conduc-tion equations of ferrofluid are simplified and the grade structure theory of the equations is also derived. As an example, the heat conduction model be-tween two plates is calculated numerically. The numerical results show that the simplified equations are more reasonable, compared with the previous ones.
As for the R-T instability problem of ferrofluid, the simplified model equations are derived firstly. Under uniform mesh, the corresponding discrete systems are installed by using weighted essentially non-oscillatory (WENO) scheme and TVD R-K method. Simulation results show that, when direction of the external magnetic filed is the same as that of gravity, velocities of the interface arc increased by the external magnetic field, vice versa. When the direction of the external magnetic field is perpendicular to that of gravity, the case become that instability of the interface is increased, symmetry of the interface is destructed, and more foam like spiral vortexes emerge on the boundaries. These results are very helpful for controlling R-T instability of ferrofluid by adjusting magnitude and direction of external magnetic field.
引文
[1]S.P. Anjali Devi and P.T. Hemamalini. Nonlinear Rayleigh-Taylor instability of two superposed magnetic fluids under parallel rotation and a normal mag-netic field[J]. Journal of Magnetism and Magnetic Materials,314(2):135-139,2007.
[2]N.N. Anuchina, V.I. Volkov, V.A. Gordeychuk, N.S. Es'kov, O.S. Ilyutina, and O.M. Kozyrev. Numerical simulations of Rayleigh-Taylor and Richtmyer-Meshkov instability using MAH-3 code[J]. Journal of Computa-tional and Applied Mathematics,168(1-2):11-20,2004.
[3]R. Aris. Vectors, Tensors and the Basic Equations of Fluid Mechanics [M]. Dover Publications,1990.
[4]D.S. Balsara and C.W. Shu. Monotonicity preserving weighted essentially non-oscillatory schemes with increasingly high order of accuracy [J]. Journal of Computational Physics,160(2):405-452,2000.
[5]V.G. Bashtovoi, G. Challant, and O.Y. Volkova. Boiling heat transfer in magnetic fluids[J]. Journal of Magnetism and Magnetic Materials,122:305-308,1993.
[6]D. Biskamp. Nonlinear Magnetohydrodynamics[M]. Cambridge University Press, Cambridge,1993.
[7]M. Bhushan, R. Rouse, and J.E. Lukens. Chemical-mechanical polish-ing in semidirect contact mode[J]. Journal of the Electrochemical Society, 142(11):3845-3851,1995.
[8]T.J. Black and W.B. Mraz. On-site fillable ferrofluidic seal[P]. Google Patents. US Patent 5,560,620,1996.
[9]E. Blums. Heat and mass transfer phenomena[J]. Journal of Magnetism and Magnetic Materials,252:189-193,2002.
[10]E. Blums, G. Kronkalns, A. Mezulis and V. Sints. Non-isothermal mass transfer of ferrocolloids through porous membrane [C].2th International Conference on Magnetic Fluids,2010.
[11]A. Bozhko and G. Putin. Thermomagnetic convection as a tool for heat and mass transfer control in nanosize materials under microgravity conditions [J]. Microgravity Science and Technology,21(1):89-93,2009.
[12]C.M. Braams and P.E. Stott. Nuclear Fusion:Half a Century of Magnetic Confinement Fusion Research[M]. Taylor & Francis Publishers,2002.
[13]V. Bychkov, M. Marklund, and M. Modestov. The Rayleigh-Taylor in-stability and internal waves in quantum plasmas[J]. Physics Letters A, 372(17):3042-3045,2008.
[14]C. Canuto, M.Y. Hussaini, A. Quarteroni, T.A. Zang. Spectral Methods for Fluid Dynamics[M]. Spring er-Verlag, Berlin,1988.
[15]陈材侃.计算流体力学[M].重庆：重庆出版社,1992.
[16]陈国谦,陈耀松,高智.简化Navier-Stokes方程的基本形式[J].水动力学研究与进展A辑,5(3)：25-32,1990.
[17]陈国谦,陈耀松,高智.简化Navier-Stokes方程的一般理论[J].中国科学A辑,20(12)：12721281,1990.
[18]C.H. Cho, S.S. Park, and Y. Ahn. Three-dimensional wafer scale hydro-dynamic modeling for chemical mechanical polishing[J]. Thin Solid Films, 389(1-2):254-260,2001.
[19]B. Cockburn. Discontinuous galerkin methods[J]. ZAMM-Journal of Applied Mathematics and Mechanics/Zeitschrift fur Angewandte Mathematik und Mechanik,83(11):731-754,2003.
[20]F.R. Cunha, and Y.D. Sobral Characterization of the physical parameters in a process of magnetic separation and pressure-driven flow of a magnetic fluid [J]. Physica A,343:36-64,2004.
[21]R.M. Darlington, T.L. McAbee, and G. Rodrigue. A study of ALE simula-tions of Rayleigh-Taylor instability [J]. Computer Physics Communications, 135(1):58-73,2001.
[22]R.M. Darlington, T.L. McAbee, and G. Rodrigue. Large eddy simulation and ALE mesh motion in Rayleigh-Taylor instability simulation [J]. Computer Physics Communications,144(3):261-276,2002.
[23]P.A. Davidson. An Introduction to Magnetohydrodynamics[M]. Cambridge University Press, Cambridge,2001.
[24]丁明,张锋,孙虹,韩效钊.磁性液体的制备及应用进展[J].合肥学院学报：自然科学版,12(4)：1-6,2004.
[25]Y.O. El-Dib. Nonlinear hydromagnetic Rayleigh-Taylor instability for strong viscous fluids in porous media[J]. Journal of Magnetism and Magnetic Ma-terials,260(1-2):1-18,2003.
[26]Y.O. El-Dib. Nonlinear Rayleigh-Taylor instability for hydromagnetic Dar-cian flow:effect of free surface currents[J]. Journal of Colloid and Interface Science,259(2):309-321,2003.
[27]费翔麟,胡庆康,景思睿.高等流体力学[M].西安：西安交通大学出版社,1989.
[28]J.H. Ferziger, M. Peric, and KW Morton. Computational Methods for Fluid Dynamics[M]. Springer Berlin,1999.
[29]E. Fournier, S. Gauthier, and F. Renaud.2D pseudo-spectral parallel Navier-Stokes simulations of compressible Rayleigh-Taylor instability [J]. Computers & Fluids,31(4-7):569-587,2002.
[30]M.N. Fu and F.C. Chou. Flow simulation for chemical mechanical planariza-tion[J]. Japanese Journal of Applied Physics,38:4709-4714,1999.
[31]R. Ganguly, S.Sen, and I.K.Puri. Heat transfer augmentation using a mag-netic fluid under the influence of a line dipole[J]. Journal of Magnetism and Magnetic Materials,271(01):63-73,2004.
[32]高太元,李明军,高智.CMP润滑方程的Chebyshev加速松弛法求解[J].自然科学进展,17(12)：1724-1728,2007.
[33]高太元,李明军,胡利民,高智.CMP流场的数值模拟及离心力影响分析[J].力学学报,40(6)：729-734,2008.
[34]高智.简化Navier-Stokes方程组及无粘和粘性边界层联合求解[J].力学学报,14(6)：606-611,1982.
[35]高智.论简化Navier-Stokes方程组(SNSE)[J].中国科学A辑,17(10)：1058-1070,1987.
[36]高智.流体力学基本方程组(BEFM)的层次结构理论和简化Navier-Stokes方程组(SNSE)[J].力学学报,20(2)：107-116,1988.
[37]高智.简化Navier-Stokes方程的层次结构及其力学内涵和应用[J].中国科学A辑,18(6)：1058-1070,1988.
[38]高智.粘流-无粘干扰流动(IF)理论[J].力学学报,22(1)：9-19.1990.
[39]高智.粘性流体力学研究的若干进展[J].中国科学院院刊,13(3)：187-192,1998.
[40]高智.高雷诺数流动的控制方程体系和扩散抛物化Navier-Stokes方程组的意义和用途[J].力学进展,35(3)：427-438,2005.
[41]高智,周光.高雷诺数流动理论、算法和应用的若干研究进展[J].力学进展,31(3)：417-436,2001.
[42]葛劢冲.CMP系统技术与市场[J].电子工业专用设备,32(1)：17-24,2003.
[43]国秋菊,郑少华,苏登成.纳米CoFe2O4颗粒制备及性能研究[J].中国粉体技术,13(1)：16-19,2007.
[44]C. Geindreau and J.L. Auriault. Magnetohydrodynamic flows in porous media[J]. Journal of Fluid Mechanics,466:343-363,2002.
[45]R.R. Gold. Magnetohydrodynamic pipe flow. Part 1[J]. Journal of Fluid Mechanics,13(4):505-512,1962,2006.
[46]K. Gottfried, I. Schubert, SE Schulz, and T. Gessner. Cu/barrier CMP on porous low-k based interconnect schemes[J]. Microelectronic Engineering, 83(11-12):2218-2224,2006.
[47]S. Gottlieb and C.W. Shu. Total variation diminishing Runge-Kutta schemes[J]. Mathematics of Computation,67:73-85,1998.
[48]S. Gottlieb, C.W. Shu, and E. Tadmor. Strong stability preserving high order time discretization methods[J]. SIAM Review,43:89-112,2001.
[49]A. Harten and S. Osher. Uniformly high-order accurate nonoscillatory schemes, Ⅰ[J]. SIAM Journal on Numerical Analysis,24(2):279-309,1987.
[50]A. Harten, B. Engquist, S. Osher, and S.R. Chakravarthy. Uniformly high order accurate essentially non-oscillatory schemes, Ⅲ[J]. Journal of Com-putational Physics,131(1):3-47,1997.
[51]X. He, S. Chen, and R. Zhang. A lattice Boltzmann scheme for incompress-ible multiphase flow and its application in simulation of Rayleigh-Taylor instability[J]. Journal of Computational Physics,152(2):642-663,1999.
[52]B.J. Hooper, G. Byrne, and S. Galligan. Pad conditioning in chemical me-chanical polishing[J]. Journal of Materials Processing Tech.,123(1):107-113, 2002.
[53]C. Hu and C.W. Shu. Weighted essentially non-oscillatory schemes on tri-angular meshes[J]. Journal of Computational Physics,150(1):97-127,1999.
[54]H.D. Jeong, K.H. Park, and K.K. Cho. CMP pad break-in time reduc-tion in silicon wafer polishing[J]. CIRP Annals-Manufacturing Technology, 56(1):357-360,2007.
[55]D.G. Jian, L. Huang, and X.L. Qiu. Assembling stabilization of the Rayleigh-Taylor instability by the effects of Finite Larmor radius and sheared axial flow[J]. Plasma Science and Technology,7(3):2805-2809,2005.
[56]G.S. Jiang and C.W. Shu. Efficient implementation of weighted ENO scheme[J]. Journal of Computational Physics,126:202-228,1996.
[57]X. Jin, L.M. Keer, and Q. Wang. A 3D EHL simulation of CMP[J]. Journal of the Electrochemical Society,152(1):G7-G15,2005.
[58]S. Jing, Y.T. Zhang, and C.W. Shu. Resolution of high order WENO schemes for complicated flow structures[J]. Journal of Computational Physics, 186(2):690-696,2003.
[59]A. Jordan, R. Scholz, K. Maier-Hauff, M. Johannsen, P. Wust, J. Nadobny, H. Schirra, H. Schmidt, S. Deger, S. Loening, et al. Presentation of a new magnetic field therapy system for the treatment of human solid tumors with magnetic fluid hyperthermia[J]. Journal of Magnetism and Magnetic Mate-rials,225(1-2):118-126,2001.
[60]S. Kamiyama and J. Ishimoto. Boiling two-phase flows of magnetic fluid in a non-uniform magnetic field [J]. Journal of Magnetism and Magnetic Materials,149(1-2):125-131,1995.
[61]A. Lange. Thermomagnetic convection of magnetic fluids in a cylindrical geometry [J]. Physics of Fluids,14:2059-2064,2002.
[62]R.J. LeVeque. Finite Volume Methods for Hyperbolic Problems[M]. Cam-bridge University Press,2002.
[63]J.A. Levert, F.M. Mess, R.F. Salant, S. Danyluk, and A.R. Baker. Mech-anisms of chemical-mechanical polishing of SiO2 dielectric on integrated circuits [J]. Tribology Transactions,41(4):593-599,1998.
[64]D.J. Lewis. The instability of liquid surfaces when accelerated in a direction perpendicular to their planes. Ⅱ[J]. Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences,202:81-96,1950.
[65]雷红,雒建斌,马俊杰.化学机械抛光(CMP)技术的发展、应用及存在问题[J].润滑与密封,04：7376,2002.
[66]雷红,雒建斌,张朝辉.化学机械抛光技术的研究进展[J].上海大学学报：自然科学版,9(6)：494-502,2003.
[67]B.Q. Li. Discontinuous Finite Elements in Fluid Dynamics and Heat Trans-fer [M]. Springer-Verlag,2006.
[68]李德才.磁性液体理论及应用[M].北京：科学出版社,2003.
[69]李德才.磁性液体密封理论及应用[M].北京：科学出版社,2010.
[70]李国斌,胡连卡,宋顺成,杨润田,曹学军.磁流体密封对齿聚磁结构特性的有限元分析[J].兵器材料科学与工程,29(6)：18-21,2005.
[71]李明军,高智.二维抛物化稳定性方程的特征和次特征[J].应用力学学报,19(3)：124-126,2002.
[72]李荣华,冯果枕.微分方程数值解法[M].北京：高等教育出版社,1996.
[73]李寿佛,叶文华,张瑗,舒适,肖爱国.高阶FD-WENO格式在数值求解Rayleigh-Taylor不稳定性问题中的应用[J].计算物理,25(4)：379-386,2008.
[74]X.Y. Li, K.L. Yao, and Z.L. Liu. CFD study on the magnetic fluid delivering in the vessel in high-gradient magnetic field[J]. Journal of Magnetism and Magnetic Materials,320(11):1753-1758,2008.
[75]廖海东,孙承纬,李永池,杨礼兵.有限厚度流体层界面运动Rayleigh-Taylor不稳定性的数值模拟[J].爆炸与冲击,19(2)：139-145,1999.
[76]D. Lima, W. Van Saarloos, and A. De Wit. Rayleigh-Taylor instabil-ity of pulled versus pushed fronts[J]. Physica D:Nonlinear Phenomena, 218(2):158-166,2006.
[77]J. Liu, J. Gu, Z. Lian, and H. Liu. Experiments and mechanism analysis of pool boiling heat transfer enhancement with water-based magnetic fluid [J]. Heat and Mass Transfer,41(2):170-175,2004.
[78]X.D. Liu, Osher S., and Chan T. Weighted essentially non-oscillatory schemes[J]. Journal of Computational Physics,115:200-212,1994.
[79]M.A. Lopez de Bertodano, S. Leonardi, and P.S. Lykoudis. Nucleate pool boiling of mercury in the presence of a magnetic field [J]. International Jour-nal of Heat and Mass Transfer,41 (22):3491-3500,1998.
[80]S. Lugomer and A. Maksimovi. Laser-induced bubble formation and collapse in a nonlinear Rayleigh-Taylor unstable interface in a thin layer of molten metal[J]. Physics Letters A,251(5):303-310,1999.
[811吕玉山,蔡光起,华雷.单晶硅片的区域载荷法平坦化抛光[J].沈阳工业学院学报,22(3)：1-4,2003.
[82]J. Ma, Y. Chen, B. Gan, and M.Y. Yu. Behavior of the Rayleigh-Taylor mode in a dusty plasma with rotational and shear flows [J]. Planetary and Space Science,54(8):719-725,2006.
[83]Y.D. Mahmoud. Nonlinear Rayleigh-Taylor instability of two fluids with cylindrical interfaces:effect of a periodic radial magnetic field[J]. Journal of Magnetism and Magnetic Materials,195(3):779-792,1999.
[84]G.M. Moatimid. Nonlinear waves on the surface of a magnetic fluid jet in porous media[J]. Physica A:Statistical Mechanics and its Applications, 328(3-4):525-544,2003.
[85]R. Moβner, and U. Miiller. A numerical investigation of three-dimensional magnetoconvection in rectangular cavities[J]. International Journal of Heat and Mass Transfer,42(6):1111-1121,1999.
[86]A.E.M.A. Mohamed, E.F.A. Elshehawey, and Y.O. El-Dib. Electroviscoelas-tic Rayleigh-Taylor instability of Kelvin fluids:Effect of a constant tangen-tial electric field [J]. Fluid Dynamics Research,19(5):327-341,1997.
[87]F. Moukalled and M. Darwish. A unified formulation of the segregated class of algorithms for fluid flow at all speeds[J]. Numerical Heat Transfer, Part B:Fundamentals,37(1):103-139,2000.
[88]J.L. Neuringer and R.E. Rosensweig. Ferrohydrodynamics[J]. Physics of Fluids,7(12):1927,1964.
[89]S.H. Ng. Measurement and Modeling of Fluid Pressures in Chemical Me-chanical Polishing[D]. PhD thesis, Georgia Institute of Technology,2005.
[90]S. Odenbach. Microgravity experiments on thermomagnetic convection in magnetic fluids[J]. Journal of Magnetism and Magnetic Materials, 149(1):155-157,1995.
[91]S. Odenbach and S. Thurm. Magnetoviscous Effects in Ferrofluids[M]. Springer, New York,2002.
[92]S.S. Park, C.H. Cho, and Y. Ahn. Hydrodynamic analysis of chemical me-chanical polishing process[J]. Tribology International,33(10):723-730,2000.
[93]N. Patir and H.S. Cheng. An average flow model for determining effects of three-dimensional roughness on partial hydrodynamic lubrication [J]. ASME Journal of Lubrication Technology,100:12-17,1978.
[94]N. Patir and H.S. Cheng. Application of average flow model to lubrication between rough sliding surfaces[J]. ASME Journal of Lubrication Technology, 101:220-230,1978.
[95]A. Postelnicu. Influence of a magnetic field on heat and mass transfer by natural convection from vertical surfaces in porous media considering Soret and Dufour effects [J]. International Journal of Heat and Mass Transfer, 47(6-7):1467-1472,2004.
[96]秦承森,王裴.可压缩与不可压缩流体中Rayleigh-Taylor不稳定性的比较[J].强激光与粒子束,15(10)：977-980,2003.
[97]J.X. Qiu and C.W. Shu. Hermite WENO schemes for Hamilton-Jacobi equations[J]. Journal of Computational Physics,204(1):82-99,2005.
[98]X.M. Qiu, L. Huang, and D.G. Jian. Comparison between mitigation effects of the finite larmor radius and sheared axial flow on Rayleigh-Taylor insta-bility in Z-pinch implosions [J]. Plasma Science and Technology,4(5):1429-1434,2002.
[99]M. Rabiski and K. Zdunek. Rayleigh-Taylor instability in plasma jet from IPD accelerator[J]. Surface and Coatings Technology,174:964-967,2003.
[100]L. Rayleigh. Investigation of the character of the equilibrium of an in-compressible heavy fluid of variable density [J]. Proceedings of the London Mathematical Society,14(1):170-177,1883.
[101]B. Riviere. Discontinuous Galerkin Methods for Solving Elliptic and Parabolic Equations:Theory and Implementation[M]. SIAM,2008.
[102]R. E. Rosensweig. Ferrohydrodynamics[M]. Dover Publications New York, 1997.
[103]R. Rosensweig. Basic equations for magnetic fluids with internal rotations [J]. Ferrofluids, LNP594,594:61-84,2002.
[104]R.E. Rosensweig. Stress Boundary-Conditions in Ferrohydrodynamics[J]. Industrial & Enineering Chemistry Research,46(19):6113-6117,2007.
[105]S.R. Runnels and L.M. Eyman. Tribology Analysis of Chemical-Mechanical Polishing[J]. Journal of the Electrochemical Society,141(6):1698-1701,1994.
[106]G. Sambandan, V.K. Tripathi, and V.K. Sayal. Rayleigh-Taylor mode-RF coupling in a dusty plasma[J]. Physics Letters A,262(2-3):179-185,1999.
[107]S. Shuchi K. Sakatani and H. Yamaguchi. An application of a binary mixture of magnetic fluid for heat transport devices[J]. Journal of Magnetism and Magnetic Materials,289:257-259,2005.
[108]J. Seok, C.P. Sukam, A.T. Kim, J.A. Tichy, and T.S. Cale. Multiscale material removal modeling of chemical mechanical polishing[J]. Wear,254(3-4):307-320,2003.
[109]K. Shan, S. Wizel, T. Prozorov, and A. Gedanken. The use of ultrasound radiation for the preparation of magnetic fluids [J]. Thin Solid Films,318(1-2):38-41,1998.
[110]J.A. Shercliff. Magnetohydrodynamic pipe flow Part2:High Hartmann num-ber[J]. Journal of Fluid Mechanics,13(4):513-518,1962,2006.
[111]M. Shliomis. Ferrohydrodynamics:Retrospective and issues[J]. Ferrofluids, LNP 594,594:85-111,2002.
[112]C.W.Shu. Total-variation-diminishing time discretizations [J]. SIAM Jour-nal on Scientific Computing,9(6):1073-1084,1988.
[113]C.W. Shu. Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws[R]. NASA/CR-97-206253, ICASE Report No.97-65, Institute for Computer Applications in Science and Engineering NASA Langley Research Center Hampton, VA, 1997.
[114]C.W. Shu. High-order finite difference and finite volume WENO schemes and discontinuous Galerkin methods for CFD[J]. International Journal of Computational Fluid Dynamics,17(2):107-118,2003.
[115]C.W. Shu and S. Osher. Efficient implementation of essentially non-oscillatory shock-capturing schemes [J]. Journal of Computational Physics, 77(2):439-471,1988.
[116]C.W. Shu and S. Osher. Efficient implementation of essentially non-oscillatory shock-capturing schemes, Ⅱ[J]. Journal of Computational Physics,83(1):32-78,1989.
[117]水鸿寿.一维流体力学差分格式[M].北京：国防工业出版社,1998.
[118]J.H. Spurk. Fluid Mechanics[M]. Springer-Verlag Berlin Heidelberg,1997.
[119]S. Sundararajan, D.G. Thakurta, D.W. Schwendeman, S.P. Murarka, and W.N. Gill. Two-dimensional wafer-scale chemical mechanical planarization models based on lubrication theory and mass transport [J]. Journal of the Electrochemical Society,146(2):761-766,1999.
[120]K.S. Suslick, M. Fang, and T. Hyeon. Sonochemical synthesis of iron col-loids[J]. Journal of the American Chemical Society,118(47):11960-11961, 1996.
[121]M. Takahashi, K. Shinbo, R. Ohkawa, M. Matsuzaki, and A. Inoue. Nucleate pool boiling heat transfer of magnetic fluid in a magnetic field [J]. Journal of Magnetism and Magnetic Materials,122(1-3):301-304,1993.
[122]H.S. Takhar and O.A. Beg. Effects of transverse magnetic field, Prandtl number and Reynolds number on non-Darcy mixed convective flow of an incompressible viscous fluid past a porous vertical flat plate in a saturated porous medium[J]. International Journal of Energy Research,21(1):87-100, 1997.
[123]J.C. Tannehill, D.A. Anderson, and R.H. Pletcher. Computational Fluid Mechanics and Heat Transfer[M]. Taylor (?) Francis Pubishers,1997.
[124]汤文辉,毛益明.Rayleigh-Taylor不稳定性的SPH模拟[J].国防科技大学学报,25(1)：21-23,2004.
[125]陶文铨.数值传热学(第2版)M].西安交通大学出版社,2001.
[126]V.E. Tarasov. Magnetohydrodynamics of fractal media[J]. Physics of plas-mas,13:052107,2006.
[127]G. Taylor. The instability of liquid surfaces when accelerated in a direc-tion perpendicular to their planes. Ⅰ[J]. Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences,201:192-196,1950.
[128]D.G. Thakurta, C.L. Borst, D.W. Schwendeman, R.J. Gutmann, and W.N. Gill. Pad porosity, compressibility and slurry delivery effects in chemical-mechanical planarization:modeling and experiments [J]. Thin Solid Films, 366(1-2):181-190,2000.
[129]D.G. Thakurta, D.W. Schwendeman, R.J. Gutmann, S. Shankar, L. Jiang, and W.N. Gill. Three-dimensional wafer-scale copper chemical-mechanical planarization model[J]. Thin Solid Films,414(1):78-90,2002.
[130]J. Tichy, J.A. Levert, L. Shan, and S. Danyluk. Contact mechanics and lubrication hydrodynamics of chemical mechanical polishing[J]. Journal of the Electrochemical Society,146(4):1523-1528,1999.
[131]童志义.CMP设备市场及技术现状[J].电子元器件应用,4(1-2)：53-58,2002.
[132]童志义.化学机械抛光技术现状与发展趋势[J]. 电子工业专用设备,33(6)：1-6,2004.
[133]W.T. Tseng and Y.L. Wang. Re-examination of pressure and speed depen-dences of removal rate during chemical-mechanical polishing processes [J]. Journal of the Electrochemical Society,144(2):L15,1997.
[134]T. Volker, F. Blums, and S. Odenbach. Thermodiffusion in magnetic flu-ids [J]. Journal of Magnetism and Magnetic Materials,252:218-220,2002.
[135]T. Volker and S. Odenbach. Thermodiffusion in ferrofluids in the presence of a magnetic field [J]. Physics of Fluids,17:037104,2005.
[136]T. Volker and S. Odenbach. Thermodiffusion in magnetic fluids[J]. Journal of Magnetism and Magnetic Materials,289:289-291,2005.
[137]T. Volker, E. Blums, and S. Odenbach. Heat and mass transfer phenomena in magnetic fluids [J]. GAMM-Mitteilungen,30(1):185-194,2007.
[138]L.L. Wang and J.C. Li. Fluid mixing due to Rayleigh-Taylor instability in a time-dependent acceleration field [J]. Communications in Nonlinear Science and Numerical Simulation,10(5):571-577,2005.
[139]王金瑞.氮化铁磁流体的制备与稳定性[J].科技通报,21(3)：342-346,2005.
[140]王汝权,焦履琼.简化Navier-Stokes方程组推进解法的数值分析[J].数值计算与计算机科学,1(1)：53-61,1980.
[141]王汝权,焦履琼,刘学宗.简化Navier-Stokes方程的数值解法[J].空气动力学学报,2(1)：85-95,1984.
[142]王汝权,申义庆.扩散抛物化Navier—Stokes方程数值解法评述[J].力学进展,35(4)：481-497,2005.
[143]魏昕,熊伟,黄蕊慰,袁慧.化学机械抛光中抛光垫的研究[J].金刚石与磨料磨具工程,143(5)：40-43,2004.
[144]C.Y. Wen, C.Y. Chen, and S.F. Yang. Flow visualization of natural con-vection of magnetic fluid in a rectangular Hele-Shaw cell[J]. Journal of Magnetism and Magnetic Materials,252:206-208,2002.
[145]C.Y. Wen and W.P. Su. Natural convection of magnetic fluid in a rectangular Hele-Shaw cell[J]. Journal of Magnetism and Magnetic Materials,289:299-302,2005.
[146]吴俊峰,叶文华,张维岩.柱几何Rayleigh-Taylor不稳定性的数值模拟[J].强激光与粒子束,15(1)：64-68,2003.
[147]吴望一.流体力学：上册[M].北京：北京大学出版社,1982,2006.
[148]X. Xie. Existence and uniqueness of analytic solution for Rayleigh-Taylor problem[J]. Journal of Differential Equations,237(1):116-132,2007.
[149]Y. Xing and C.W. Shu. High order finite difference WENO schemes with the exact conservation property for the shallow water equations[J]. Journal of Computational Physics,208(1):206-227,2005.
[150]徐立.R-T不稳定性非线性发展和SPH模拟[D].博士学位论文,中国工程物理研究院北京研究生部,2002.
[151]Y. Xuan, Q. Li, and M. Ye. Investigations of convective heat transfer in fer-rofluid microflows using lattice-Boltzmann approach[J]. International Jour-nal of Thermal Sciences,46(2):105-111,2007.
[152]H. Yamaguchi, I. Kobori, and N. Kobayashi. Numerical study of flow state for a magnetic fluid heat transport device[J]. Journal of Magnetism and Magnetic Materials,201:260-263,1999.
[153]H. Yamaguchi, Z. Zhang, S. Shuchi, and K. Shimada. Heat transfer char-acteristics of magnetic fluid in a partitioned rectangular box[J]. Journal of Magnetism and Magnetic Materials,252:203-205,2002.
[154]H. Yamaguchi, Z. Zhang, S. Shuchi, and K. Shimada. Gravity simulation of natural convection in magnetic fluid[J]. JSME International Journal Series B,45(1):61-65,2002.
[155]H. Yamaguchi. Engineering Fluid Mechanics[M]. Springer Verlag,2008.
[156]H. Yamuguchi, X.R. Zhang, S. Higashi, and M.J. Li. Study on power genera-tion using electro-conductive polymer and its mixture with magnetic fluid [J]. Journal of Magnetism and Magnetic Materials,320(7):1406-1411,2008.
[157]杨玉月.修正系数格式及其在Rayleigh-Taylor不稳定性数值模拟中的应用[D].硕士学位论文,湘潭大学,2005.
[158]叶荣昌,刘书进,高宏,and杨志伊.磁流体制备技术的研究现状及其存在问题[J].机械工程材料,27(3)：33-34,50,2003.
[159]蔚喜军.数值模拟Rayleigh-Taylor不稳定性中不同介质界面的处理方法[R].中国核物理报告.中国核物理中心,原子能出版社,2000.
[160]L.Q. Yu, L.J. Zheng, and J.X. Yang. Study of preparation and properties on magnetization and stability for ferromagnetic fluids[J]. Materials Chemistry and Physics,66(1):6-9,2000.
[161]T.K. Yu, C.C. Yu, and M. Orlowski. A statistical polishing pad model for chemical-mechanical polishing[C], In Electron Devices Meeting,1993. IEDM'93. Technical Digest., International, pp.865-868,1993.
[162]D. Zablockis, V. Frishfelds, and E. Blums. Numerical investigation of ther-momagnetic convection in a heated cylinder under the magnetic field of a solenoid[J]. Journal of Physics:Condensed Matter,20:204134,2008.
[163]P.B. Zantye, A. Kumar, and A.K. Sikder. Chemical mechanical pianarization for microelectronics applications [J]. Materials Science and Engineering:R: Reports,45(3-6):89-220,2004.
[164]O.C. Zienkiewicz and R. L. Taylor. The Finite Element Method, Vol.3: Fluid Dynamics,5th Edition[M]. Elsevier Butterworth Heinemann, Oxford, 2000.
[165]张健,史宝军,杜运东,杨廷毅.化学机械抛光技术研究现状与展望[J].山东建筑大学学报,24(2)：168-174,2009.
[166]张朝辉,雒建斌,温诗铸.CMP润滑方程的多重网格法求解[J].自然科学进展,13(11)：1224-1227,2003.
[167]C.H. Zhang, J. B. Luo, and S.Z. Wen. Multigrid technique incorporated algorithm for CMP lubrication equations[J]. Progress in Natural Science, 14(4):369-372,2004.
[168]C.H. Zhang, J.B. Luo, and S.Z. Wen. Modeling CMP with couple stress fluids[J]. Tsinghua science and technology,9(3):270-273,2005.
[169]张朝辉,雒建斌,温诗铸.化学机械抛光中纳米颗粒的作用分析[J].物理学报,54(5)：2123-2127,2005.
[170]张朝辉,雒建斌,温诗铸.考虑抛光垫特性的CMP流动性能[J].机械工程学报,42(4)：13-17,2006.
[171]张朝辉,杜永平,雒建斌.CMP中接触与流动关系的分析[J].科学通报,51(16)：1691-1695,08 2006.
[172]张朝辉,杜永平,常秋英,雒建斌.化学机械抛光中抛光垫作用分析[J].北京交通大学学报：自然科学版,31(1)：18-21,2007.
[173]张朝辉,叶巍.抛光垫特性对抛光中流体运动的影响分析[J].润滑与密封,32(11)：59-61,64,2007.
[174]Y. Zhao and L. Chang. A micro-contact and wear model for chemical-mechanical polishing of silicon wafers [J]. Wear,252(3-4):220-226,2002.
[175]庄逢甘,张德良.扩散抛物化(DP) NS方程组的意义及其在计算流体力学中的应用[J].空气动力学学报,21(1)：1-9,2003.
[176]庄礼贤,尹协远,马晖扬.流体力学[M].合肥：中国科学技术大学出版社,1991.
[2]N.N. Anuchina, V.I. Volkov, V.A. Gordeychuk, N.S. Es'kov, O.S. Ilyutina, and O.M. Kozyrev. Numerical simulations of Rayleigh-Taylor and Richtmyer-Meshkov instability using MAH-3 code[J]. Journal of Computa-tional and Applied Mathematics,168(1-2):11-20,2004.
[3]R. Aris. Vectors, Tensors and the Basic Equations of Fluid Mechanics [M]. Dover Publications,1990.
[4]D.S. Balsara and C.W. Shu. Monotonicity preserving weighted essentially non-oscillatory schemes with increasingly high order of accuracy [J]. Journal of Computational Physics,160(2):405-452,2000.
[5]V.G. Bashtovoi, G. Challant, and O.Y. Volkova. Boiling heat transfer in magnetic fluids[J]. Journal of Magnetism and Magnetic Materials,122:305-308,1993.
[6]D. Biskamp. Nonlinear Magnetohydrodynamics[M]. Cambridge University Press, Cambridge,1993.
[7]M. Bhushan, R. Rouse, and J.E. Lukens. Chemical-mechanical polish-ing in semidirect contact mode[J]. Journal of the Electrochemical Society, 142(11):3845-3851,1995.
[8]T.J. Black and W.B. Mraz. On-site fillable ferrofluidic seal[P]. Google Patents. US Patent 5,560,620,1996.
[9]E. Blums. Heat and mass transfer phenomena[J]. Journal of Magnetism and Magnetic Materials,252:189-193,2002.
[10]E. Blums, G. Kronkalns, A. Mezulis and V. Sints. Non-isothermal mass transfer of ferrocolloids through porous membrane [C].2th International Conference on Magnetic Fluids,2010.
[11]A. Bozhko and G. Putin. Thermomagnetic convection as a tool for heat and mass transfer control in nanosize materials under microgravity conditions [J]. Microgravity Science and Technology,21(1):89-93,2009.
[12]C.M. Braams and P.E. Stott. Nuclear Fusion:Half a Century of Magnetic Confinement Fusion Research[M]. Taylor & Francis Publishers,2002.
[13]V. Bychkov, M. Marklund, and M. Modestov. The Rayleigh-Taylor in-stability and internal waves in quantum plasmas[J]. Physics Letters A, 372(17):3042-3045,2008.
[14]C. Canuto, M.Y. Hussaini, A. Quarteroni, T.A. Zang. Spectral Methods for Fluid Dynamics[M]. Spring er-Verlag, Berlin,1988.
[15]陈材侃.计算流体力学[M].重庆：重庆出版社,1992.
[16]陈国谦,陈耀松,高智.简化Navier-Stokes方程的基本形式[J].水动力学研究与进展A辑,5(3)：25-32,1990.
[17]陈国谦,陈耀松,高智.简化Navier-Stokes方程的一般理论[J].中国科学A辑,20(12)：12721281,1990.
[18]C.H. Cho, S.S. Park, and Y. Ahn. Three-dimensional wafer scale hydro-dynamic modeling for chemical mechanical polishing[J]. Thin Solid Films, 389(1-2):254-260,2001.
[19]B. Cockburn. Discontinuous galerkin methods[J]. ZAMM-Journal of Applied Mathematics and Mechanics/Zeitschrift fur Angewandte Mathematik und Mechanik,83(11):731-754,2003.
[20]F.R. Cunha, and Y.D. Sobral Characterization of the physical parameters in a process of magnetic separation and pressure-driven flow of a magnetic fluid [J]. Physica A,343:36-64,2004.
[21]R.M. Darlington, T.L. McAbee, and G. Rodrigue. A study of ALE simula-tions of Rayleigh-Taylor instability [J]. Computer Physics Communications, 135(1):58-73,2001.
[22]R.M. Darlington, T.L. McAbee, and G. Rodrigue. Large eddy simulation and ALE mesh motion in Rayleigh-Taylor instability simulation [J]. Computer Physics Communications,144(3):261-276,2002.
[23]P.A. Davidson. An Introduction to Magnetohydrodynamics[M]. Cambridge University Press, Cambridge,2001.
[24]丁明,张锋,孙虹,韩效钊.磁性液体的制备及应用进展[J].合肥学院学报：自然科学版,12(4)：1-6,2004.
[25]Y.O. El-Dib. Nonlinear hydromagnetic Rayleigh-Taylor instability for strong viscous fluids in porous media[J]. Journal of Magnetism and Magnetic Ma-terials,260(1-2):1-18,2003.
[26]Y.O. El-Dib. Nonlinear Rayleigh-Taylor instability for hydromagnetic Dar-cian flow:effect of free surface currents[J]. Journal of Colloid and Interface Science,259(2):309-321,2003.
[27]费翔麟,胡庆康,景思睿.高等流体力学[M].西安：西安交通大学出版社,1989.
[28]J.H. Ferziger, M. Peric, and KW Morton. Computational Methods for Fluid Dynamics[M]. Springer Berlin,1999.
[29]E. Fournier, S. Gauthier, and F. Renaud.2D pseudo-spectral parallel Navier-Stokes simulations of compressible Rayleigh-Taylor instability [J]. Computers & Fluids,31(4-7):569-587,2002.
[30]M.N. Fu and F.C. Chou. Flow simulation for chemical mechanical planariza-tion[J]. Japanese Journal of Applied Physics,38:4709-4714,1999.
[31]R. Ganguly, S.Sen, and I.K.Puri. Heat transfer augmentation using a mag-netic fluid under the influence of a line dipole[J]. Journal of Magnetism and Magnetic Materials,271(01):63-73,2004.
[32]高太元,李明军,高智.CMP润滑方程的Chebyshev加速松弛法求解[J].自然科学进展,17(12)：1724-1728,2007.
[33]高太元,李明军,胡利民,高智.CMP流场的数值模拟及离心力影响分析[J].力学学报,40(6)：729-734,2008.
[34]高智.简化Navier-Stokes方程组及无粘和粘性边界层联合求解[J].力学学报,14(6)：606-611,1982.
[35]高智.论简化Navier-Stokes方程组(SNSE)[J].中国科学A辑,17(10)：1058-1070,1987.
[36]高智.流体力学基本方程组(BEFM)的层次结构理论和简化Navier-Stokes方程组(SNSE)[J].力学学报,20(2)：107-116,1988.
[37]高智.简化Navier-Stokes方程的层次结构及其力学内涵和应用[J].中国科学A辑,18(6)：1058-1070,1988.
[38]高智.粘流-无粘干扰流动(IF)理论[J].力学学报,22(1)：9-19.1990.
[39]高智.粘性流体力学研究的若干进展[J].中国科学院院刊,13(3)：187-192,1998.
[40]高智.高雷诺数流动的控制方程体系和扩散抛物化Navier-Stokes方程组的意义和用途[J].力学进展,35(3)：427-438,2005.
[41]高智,周光.高雷诺数流动理论、算法和应用的若干研究进展[J].力学进展,31(3)：417-436,2001.
[42]葛劢冲.CMP系统技术与市场[J].电子工业专用设备,32(1)：17-24,2003.
[43]国秋菊,郑少华,苏登成.纳米CoFe2O4颗粒制备及性能研究[J].中国粉体技术,13(1)：16-19,2007.
[44]C. Geindreau and J.L. Auriault. Magnetohydrodynamic flows in porous media[J]. Journal of Fluid Mechanics,466:343-363,2002.
[45]R.R. Gold. Magnetohydrodynamic pipe flow. Part 1[J]. Journal of Fluid Mechanics,13(4):505-512,1962,2006.
[46]K. Gottfried, I. Schubert, SE Schulz, and T. Gessner. Cu/barrier CMP on porous low-k based interconnect schemes[J]. Microelectronic Engineering, 83(11-12):2218-2224,2006.
[47]S. Gottlieb and C.W. Shu. Total variation diminishing Runge-Kutta schemes[J]. Mathematics of Computation,67:73-85,1998.
[48]S. Gottlieb, C.W. Shu, and E. Tadmor. Strong stability preserving high order time discretization methods[J]. SIAM Review,43:89-112,2001.
[49]A. Harten and S. Osher. Uniformly high-order accurate nonoscillatory schemes, Ⅰ[J]. SIAM Journal on Numerical Analysis,24(2):279-309,1987.
[50]A. Harten, B. Engquist, S. Osher, and S.R. Chakravarthy. Uniformly high order accurate essentially non-oscillatory schemes, Ⅲ[J]. Journal of Com-putational Physics,131(1):3-47,1997.
[51]X. He, S. Chen, and R. Zhang. A lattice Boltzmann scheme for incompress-ible multiphase flow and its application in simulation of Rayleigh-Taylor instability[J]. Journal of Computational Physics,152(2):642-663,1999.
[52]B.J. Hooper, G. Byrne, and S. Galligan. Pad conditioning in chemical me-chanical polishing[J]. Journal of Materials Processing Tech.,123(1):107-113, 2002.
[53]C. Hu and C.W. Shu. Weighted essentially non-oscillatory schemes on tri-angular meshes[J]. Journal of Computational Physics,150(1):97-127,1999.
[54]H.D. Jeong, K.H. Park, and K.K. Cho. CMP pad break-in time reduc-tion in silicon wafer polishing[J]. CIRP Annals-Manufacturing Technology, 56(1):357-360,2007.
[55]D.G. Jian, L. Huang, and X.L. Qiu. Assembling stabilization of the Rayleigh-Taylor instability by the effects of Finite Larmor radius and sheared axial flow[J]. Plasma Science and Technology,7(3):2805-2809,2005.
[56]G.S. Jiang and C.W. Shu. Efficient implementation of weighted ENO scheme[J]. Journal of Computational Physics,126:202-228,1996.
[57]X. Jin, L.M. Keer, and Q. Wang. A 3D EHL simulation of CMP[J]. Journal of the Electrochemical Society,152(1):G7-G15,2005.
[58]S. Jing, Y.T. Zhang, and C.W. Shu. Resolution of high order WENO schemes for complicated flow structures[J]. Journal of Computational Physics, 186(2):690-696,2003.
[59]A. Jordan, R. Scholz, K. Maier-Hauff, M. Johannsen, P. Wust, J. Nadobny, H. Schirra, H. Schmidt, S. Deger, S. Loening, et al. Presentation of a new magnetic field therapy system for the treatment of human solid tumors with magnetic fluid hyperthermia[J]. Journal of Magnetism and Magnetic Mate-rials,225(1-2):118-126,2001.
[60]S. Kamiyama and J. Ishimoto. Boiling two-phase flows of magnetic fluid in a non-uniform magnetic field [J]. Journal of Magnetism and Magnetic Materials,149(1-2):125-131,1995.
[61]A. Lange. Thermomagnetic convection of magnetic fluids in a cylindrical geometry [J]. Physics of Fluids,14:2059-2064,2002.
[62]R.J. LeVeque. Finite Volume Methods for Hyperbolic Problems[M]. Cam-bridge University Press,2002.
[63]J.A. Levert, F.M. Mess, R.F. Salant, S. Danyluk, and A.R. Baker. Mech-anisms of chemical-mechanical polishing of SiO2 dielectric on integrated circuits [J]. Tribology Transactions,41(4):593-599,1998.
[64]D.J. Lewis. The instability of liquid surfaces when accelerated in a direction perpendicular to their planes. Ⅱ[J]. Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences,202:81-96,1950.
[65]雷红,雒建斌,马俊杰.化学机械抛光(CMP)技术的发展、应用及存在问题[J].润滑与密封,04：7376,2002.
[66]雷红,雒建斌,张朝辉.化学机械抛光技术的研究进展[J].上海大学学报：自然科学版,9(6)：494-502,2003.
[67]B.Q. Li. Discontinuous Finite Elements in Fluid Dynamics and Heat Trans-fer [M]. Springer-Verlag,2006.
[68]李德才.磁性液体理论及应用[M].北京：科学出版社,2003.
[69]李德才.磁性液体密封理论及应用[M].北京：科学出版社,2010.
[70]李国斌,胡连卡,宋顺成,杨润田,曹学军.磁流体密封对齿聚磁结构特性的有限元分析[J].兵器材料科学与工程,29(6)：18-21,2005.
[71]李明军,高智.二维抛物化稳定性方程的特征和次特征[J].应用力学学报,19(3)：124-126,2002.
[72]李荣华,冯果枕.微分方程数值解法[M].北京：高等教育出版社,1996.
[73]李寿佛,叶文华,张瑗,舒适,肖爱国.高阶FD-WENO格式在数值求解Rayleigh-Taylor不稳定性问题中的应用[J].计算物理,25(4)：379-386,2008.
[74]X.Y. Li, K.L. Yao, and Z.L. Liu. CFD study on the magnetic fluid delivering in the vessel in high-gradient magnetic field[J]. Journal of Magnetism and Magnetic Materials,320(11):1753-1758,2008.
[75]廖海东,孙承纬,李永池,杨礼兵.有限厚度流体层界面运动Rayleigh-Taylor不稳定性的数值模拟[J].爆炸与冲击,19(2)：139-145,1999.
[76]D. Lima, W. Van Saarloos, and A. De Wit. Rayleigh-Taylor instabil-ity of pulled versus pushed fronts[J]. Physica D:Nonlinear Phenomena, 218(2):158-166,2006.
[77]J. Liu, J. Gu, Z. Lian, and H. Liu. Experiments and mechanism analysis of pool boiling heat transfer enhancement with water-based magnetic fluid [J]. Heat and Mass Transfer,41(2):170-175,2004.
[78]X.D. Liu, Osher S., and Chan T. Weighted essentially non-oscillatory schemes[J]. Journal of Computational Physics,115:200-212,1994.
[79]M.A. Lopez de Bertodano, S. Leonardi, and P.S. Lykoudis. Nucleate pool boiling of mercury in the presence of a magnetic field [J]. International Jour-nal of Heat and Mass Transfer,41 (22):3491-3500,1998.
[80]S. Lugomer and A. Maksimovi. Laser-induced bubble formation and collapse in a nonlinear Rayleigh-Taylor unstable interface in a thin layer of molten metal[J]. Physics Letters A,251(5):303-310,1999.
[811吕玉山,蔡光起,华雷.单晶硅片的区域载荷法平坦化抛光[J].沈阳工业学院学报,22(3)：1-4,2003.
[82]J. Ma, Y. Chen, B. Gan, and M.Y. Yu. Behavior of the Rayleigh-Taylor mode in a dusty plasma with rotational and shear flows [J]. Planetary and Space Science,54(8):719-725,2006.
[83]Y.D. Mahmoud. Nonlinear Rayleigh-Taylor instability of two fluids with cylindrical interfaces:effect of a periodic radial magnetic field[J]. Journal of Magnetism and Magnetic Materials,195(3):779-792,1999.
[84]G.M. Moatimid. Nonlinear waves on the surface of a magnetic fluid jet in porous media[J]. Physica A:Statistical Mechanics and its Applications, 328(3-4):525-544,2003.
[85]R. Moβner, and U. Miiller. A numerical investigation of three-dimensional magnetoconvection in rectangular cavities[J]. International Journal of Heat and Mass Transfer,42(6):1111-1121,1999.
[86]A.E.M.A. Mohamed, E.F.A. Elshehawey, and Y.O. El-Dib. Electroviscoelas-tic Rayleigh-Taylor instability of Kelvin fluids:Effect of a constant tangen-tial electric field [J]. Fluid Dynamics Research,19(5):327-341,1997.
[87]F. Moukalled and M. Darwish. A unified formulation of the segregated class of algorithms for fluid flow at all speeds[J]. Numerical Heat Transfer, Part B:Fundamentals,37(1):103-139,2000.
[88]J.L. Neuringer and R.E. Rosensweig. Ferrohydrodynamics[J]. Physics of Fluids,7(12):1927,1964.
[89]S.H. Ng. Measurement and Modeling of Fluid Pressures in Chemical Me-chanical Polishing[D]. PhD thesis, Georgia Institute of Technology,2005.
[90]S. Odenbach. Microgravity experiments on thermomagnetic convection in magnetic fluids[J]. Journal of Magnetism and Magnetic Materials, 149(1):155-157,1995.
[91]S. Odenbach and S. Thurm. Magnetoviscous Effects in Ferrofluids[M]. Springer, New York,2002.
[92]S.S. Park, C.H. Cho, and Y. Ahn. Hydrodynamic analysis of chemical me-chanical polishing process[J]. Tribology International,33(10):723-730,2000.
[93]N. Patir and H.S. Cheng. An average flow model for determining effects of three-dimensional roughness on partial hydrodynamic lubrication [J]. ASME Journal of Lubrication Technology,100:12-17,1978.
[94]N. Patir and H.S. Cheng. Application of average flow model to lubrication between rough sliding surfaces[J]. ASME Journal of Lubrication Technology, 101:220-230,1978.
[95]A. Postelnicu. Influence of a magnetic field on heat and mass transfer by natural convection from vertical surfaces in porous media considering Soret and Dufour effects [J]. International Journal of Heat and Mass Transfer, 47(6-7):1467-1472,2004.
[96]秦承森,王裴.可压缩与不可压缩流体中Rayleigh-Taylor不稳定性的比较[J].强激光与粒子束,15(10)：977-980,2003.
[97]J.X. Qiu and C.W. Shu. Hermite WENO schemes for Hamilton-Jacobi equations[J]. Journal of Computational Physics,204(1):82-99,2005.
[98]X.M. Qiu, L. Huang, and D.G. Jian. Comparison between mitigation effects of the finite larmor radius and sheared axial flow on Rayleigh-Taylor insta-bility in Z-pinch implosions [J]. Plasma Science and Technology,4(5):1429-1434,2002.
[99]M. Rabiski and K. Zdunek. Rayleigh-Taylor instability in plasma jet from IPD accelerator[J]. Surface and Coatings Technology,174:964-967,2003.
[100]L. Rayleigh. Investigation of the character of the equilibrium of an in-compressible heavy fluid of variable density [J]. Proceedings of the London Mathematical Society,14(1):170-177,1883.
[101]B. Riviere. Discontinuous Galerkin Methods for Solving Elliptic and Parabolic Equations:Theory and Implementation[M]. SIAM,2008.
[102]R. E. Rosensweig. Ferrohydrodynamics[M]. Dover Publications New York, 1997.
[103]R. Rosensweig. Basic equations for magnetic fluids with internal rotations [J]. Ferrofluids, LNP594,594:61-84,2002.
[104]R.E. Rosensweig. Stress Boundary-Conditions in Ferrohydrodynamics[J]. Industrial & Enineering Chemistry Research,46(19):6113-6117,2007.
[105]S.R. Runnels and L.M. Eyman. Tribology Analysis of Chemical-Mechanical Polishing[J]. Journal of the Electrochemical Society,141(6):1698-1701,1994.
[106]G. Sambandan, V.K. Tripathi, and V.K. Sayal. Rayleigh-Taylor mode-RF coupling in a dusty plasma[J]. Physics Letters A,262(2-3):179-185,1999.
[107]S. Shuchi K. Sakatani and H. Yamaguchi. An application of a binary mixture of magnetic fluid for heat transport devices[J]. Journal of Magnetism and Magnetic Materials,289:257-259,2005.
[108]J. Seok, C.P. Sukam, A.T. Kim, J.A. Tichy, and T.S. Cale. Multiscale material removal modeling of chemical mechanical polishing[J]. Wear,254(3-4):307-320,2003.
[109]K. Shan, S. Wizel, T. Prozorov, and A. Gedanken. The use of ultrasound radiation for the preparation of magnetic fluids [J]. Thin Solid Films,318(1-2):38-41,1998.
[110]J.A. Shercliff. Magnetohydrodynamic pipe flow Part2:High Hartmann num-ber[J]. Journal of Fluid Mechanics,13(4):513-518,1962,2006.
[111]M. Shliomis. Ferrohydrodynamics:Retrospective and issues[J]. Ferrofluids, LNP 594,594:85-111,2002.
[112]C.W.Shu. Total-variation-diminishing time discretizations [J]. SIAM Jour-nal on Scientific Computing,9(6):1073-1084,1988.
[113]C.W. Shu. Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws[R]. NASA/CR-97-206253, ICASE Report No.97-65, Institute for Computer Applications in Science and Engineering NASA Langley Research Center Hampton, VA, 1997.
[114]C.W. Shu. High-order finite difference and finite volume WENO schemes and discontinuous Galerkin methods for CFD[J]. International Journal of Computational Fluid Dynamics,17(2):107-118,2003.
[115]C.W. Shu and S. Osher. Efficient implementation of essentially non-oscillatory shock-capturing schemes [J]. Journal of Computational Physics, 77(2):439-471,1988.
[116]C.W. Shu and S. Osher. Efficient implementation of essentially non-oscillatory shock-capturing schemes, Ⅱ[J]. Journal of Computational Physics,83(1):32-78,1989.
[117]水鸿寿.一维流体力学差分格式[M].北京：国防工业出版社,1998.
[118]J.H. Spurk. Fluid Mechanics[M]. Springer-Verlag Berlin Heidelberg,1997.
[119]S. Sundararajan, D.G. Thakurta, D.W. Schwendeman, S.P. Murarka, and W.N. Gill. Two-dimensional wafer-scale chemical mechanical planarization models based on lubrication theory and mass transport [J]. Journal of the Electrochemical Society,146(2):761-766,1999.
[120]K.S. Suslick, M. Fang, and T. Hyeon. Sonochemical synthesis of iron col-loids[J]. Journal of the American Chemical Society,118(47):11960-11961, 1996.
[121]M. Takahashi, K. Shinbo, R. Ohkawa, M. Matsuzaki, and A. Inoue. Nucleate pool boiling heat transfer of magnetic fluid in a magnetic field [J]. Journal of Magnetism and Magnetic Materials,122(1-3):301-304,1993.
[122]H.S. Takhar and O.A. Beg. Effects of transverse magnetic field, Prandtl number and Reynolds number on non-Darcy mixed convective flow of an incompressible viscous fluid past a porous vertical flat plate in a saturated porous medium[J]. International Journal of Energy Research,21(1):87-100, 1997.
[123]J.C. Tannehill, D.A. Anderson, and R.H. Pletcher. Computational Fluid Mechanics and Heat Transfer[M]. Taylor (?) Francis Pubishers,1997.
[124]汤文辉,毛益明.Rayleigh-Taylor不稳定性的SPH模拟[J].国防科技大学学报,25(1)：21-23,2004.
[125]陶文铨.数值传热学(第2版)M].西安交通大学出版社,2001.
[126]V.E. Tarasov. Magnetohydrodynamics of fractal media[J]. Physics of plas-mas,13:052107,2006.
[127]G. Taylor. The instability of liquid surfaces when accelerated in a direc-tion perpendicular to their planes. Ⅰ[J]. Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences,201:192-196,1950.
[128]D.G. Thakurta, C.L. Borst, D.W. Schwendeman, R.J. Gutmann, and W.N. Gill. Pad porosity, compressibility and slurry delivery effects in chemical-mechanical planarization:modeling and experiments [J]. Thin Solid Films, 366(1-2):181-190,2000.
[129]D.G. Thakurta, D.W. Schwendeman, R.J. Gutmann, S. Shankar, L. Jiang, and W.N. Gill. Three-dimensional wafer-scale copper chemical-mechanical planarization model[J]. Thin Solid Films,414(1):78-90,2002.
[130]J. Tichy, J.A. Levert, L. Shan, and S. Danyluk. Contact mechanics and lubrication hydrodynamics of chemical mechanical polishing[J]. Journal of the Electrochemical Society,146(4):1523-1528,1999.
[131]童志义.CMP设备市场及技术现状[J].电子元器件应用,4(1-2)：53-58,2002.
[132]童志义.化学机械抛光技术现状与发展趋势[J]. 电子工业专用设备,33(6)：1-6,2004.
[133]W.T. Tseng and Y.L. Wang. Re-examination of pressure and speed depen-dences of removal rate during chemical-mechanical polishing processes [J]. Journal of the Electrochemical Society,144(2):L15,1997.
[134]T. Volker, F. Blums, and S. Odenbach. Thermodiffusion in magnetic flu-ids [J]. Journal of Magnetism and Magnetic Materials,252:218-220,2002.
[135]T. Volker and S. Odenbach. Thermodiffusion in ferrofluids in the presence of a magnetic field [J]. Physics of Fluids,17:037104,2005.
[136]T. Volker and S. Odenbach. Thermodiffusion in magnetic fluids[J]. Journal of Magnetism and Magnetic Materials,289:289-291,2005.
[137]T. Volker, E. Blums, and S. Odenbach. Heat and mass transfer phenomena in magnetic fluids [J]. GAMM-Mitteilungen,30(1):185-194,2007.
[138]L.L. Wang and J.C. Li. Fluid mixing due to Rayleigh-Taylor instability in a time-dependent acceleration field [J]. Communications in Nonlinear Science and Numerical Simulation,10(5):571-577,2005.
[139]王金瑞.氮化铁磁流体的制备与稳定性[J].科技通报,21(3)：342-346,2005.
[140]王汝权,焦履琼.简化Navier-Stokes方程组推进解法的数值分析[J].数值计算与计算机科学,1(1)：53-61,1980.
[141]王汝权,焦履琼,刘学宗.简化Navier-Stokes方程的数值解法[J].空气动力学学报,2(1)：85-95,1984.
[142]王汝权,申义庆.扩散抛物化Navier—Stokes方程数值解法评述[J].力学进展,35(4)：481-497,2005.
[143]魏昕,熊伟,黄蕊慰,袁慧.化学机械抛光中抛光垫的研究[J].金刚石与磨料磨具工程,143(5)：40-43,2004.
[144]C.Y. Wen, C.Y. Chen, and S.F. Yang. Flow visualization of natural con-vection of magnetic fluid in a rectangular Hele-Shaw cell[J]. Journal of Magnetism and Magnetic Materials,252:206-208,2002.
[145]C.Y. Wen and W.P. Su. Natural convection of magnetic fluid in a rectangular Hele-Shaw cell[J]. Journal of Magnetism and Magnetic Materials,289:299-302,2005.
[146]吴俊峰,叶文华,张维岩.柱几何Rayleigh-Taylor不稳定性的数值模拟[J].强激光与粒子束,15(1)：64-68,2003.
[147]吴望一.流体力学：上册[M].北京：北京大学出版社,1982,2006.
[148]X. Xie. Existence and uniqueness of analytic solution for Rayleigh-Taylor problem[J]. Journal of Differential Equations,237(1):116-132,2007.
[149]Y. Xing and C.W. Shu. High order finite difference WENO schemes with the exact conservation property for the shallow water equations[J]. Journal of Computational Physics,208(1):206-227,2005.
[150]徐立.R-T不稳定性非线性发展和SPH模拟[D].博士学位论文,中国工程物理研究院北京研究生部,2002.
[151]Y. Xuan, Q. Li, and M. Ye. Investigations of convective heat transfer in fer-rofluid microflows using lattice-Boltzmann approach[J]. International Jour-nal of Thermal Sciences,46(2):105-111,2007.
[152]H. Yamaguchi, I. Kobori, and N. Kobayashi. Numerical study of flow state for a magnetic fluid heat transport device[J]. Journal of Magnetism and Magnetic Materials,201:260-263,1999.
[153]H. Yamaguchi, Z. Zhang, S. Shuchi, and K. Shimada. Heat transfer char-acteristics of magnetic fluid in a partitioned rectangular box[J]. Journal of Magnetism and Magnetic Materials,252:203-205,2002.
[154]H. Yamaguchi, Z. Zhang, S. Shuchi, and K. Shimada. Gravity simulation of natural convection in magnetic fluid[J]. JSME International Journal Series B,45(1):61-65,2002.
[155]H. Yamaguchi. Engineering Fluid Mechanics[M]. Springer Verlag,2008.
[156]H. Yamuguchi, X.R. Zhang, S. Higashi, and M.J. Li. Study on power genera-tion using electro-conductive polymer and its mixture with magnetic fluid [J]. Journal of Magnetism and Magnetic Materials,320(7):1406-1411,2008.
[157]杨玉月.修正系数格式及其在Rayleigh-Taylor不稳定性数值模拟中的应用[D].硕士学位论文,湘潭大学,2005.
[158]叶荣昌,刘书进,高宏,and杨志伊.磁流体制备技术的研究现状及其存在问题[J].机械工程材料,27(3)：33-34,50,2003.
[159]蔚喜军.数值模拟Rayleigh-Taylor不稳定性中不同介质界面的处理方法[R].中国核物理报告.中国核物理中心,原子能出版社,2000.
[160]L.Q. Yu, L.J. Zheng, and J.X. Yang. Study of preparation and properties on magnetization and stability for ferromagnetic fluids[J]. Materials Chemistry and Physics,66(1):6-9,2000.
[161]T.K. Yu, C.C. Yu, and M. Orlowski. A statistical polishing pad model for chemical-mechanical polishing[C], In Electron Devices Meeting,1993. IEDM'93. Technical Digest., International, pp.865-868,1993.
[162]D. Zablockis, V. Frishfelds, and E. Blums. Numerical investigation of ther-momagnetic convection in a heated cylinder under the magnetic field of a solenoid[J]. Journal of Physics:Condensed Matter,20:204134,2008.
[163]P.B. Zantye, A. Kumar, and A.K. Sikder. Chemical mechanical pianarization for microelectronics applications [J]. Materials Science and Engineering:R: Reports,45(3-6):89-220,2004.
[164]O.C. Zienkiewicz and R. L. Taylor. The Finite Element Method, Vol.3: Fluid Dynamics,5th Edition[M]. Elsevier Butterworth Heinemann, Oxford, 2000.
[165]张健,史宝军,杜运东,杨廷毅.化学机械抛光技术研究现状与展望[J].山东建筑大学学报,24(2)：168-174,2009.
[166]张朝辉,雒建斌,温诗铸.CMP润滑方程的多重网格法求解[J].自然科学进展,13(11)：1224-1227,2003.
[167]C.H. Zhang, J. B. Luo, and S.Z. Wen. Multigrid technique incorporated algorithm for CMP lubrication equations[J]. Progress in Natural Science, 14(4):369-372,2004.
[168]C.H. Zhang, J.B. Luo, and S.Z. Wen. Modeling CMP with couple stress fluids[J]. Tsinghua science and technology,9(3):270-273,2005.
[169]张朝辉,雒建斌,温诗铸.化学机械抛光中纳米颗粒的作用分析[J].物理学报,54(5)：2123-2127,2005.
[170]张朝辉,雒建斌,温诗铸.考虑抛光垫特性的CMP流动性能[J].机械工程学报,42(4)：13-17,2006.
[171]张朝辉,杜永平,雒建斌.CMP中接触与流动关系的分析[J].科学通报,51(16)：1691-1695,08 2006.
[172]张朝辉,杜永平,常秋英,雒建斌.化学机械抛光中抛光垫作用分析[J].北京交通大学学报：自然科学版,31(1)：18-21,2007.
[173]张朝辉,叶巍.抛光垫特性对抛光中流体运动的影响分析[J].润滑与密封,32(11)：59-61,64,2007.
[174]Y. Zhao and L. Chang. A micro-contact and wear model for chemical-mechanical polishing of silicon wafers [J]. Wear,252(3-4):220-226,2002.
[175]庄逢甘,张德良.扩散抛物化(DP) NS方程组的意义及其在计算流体力学中的应用[J].空气动力学学报,21(1)：1-9,2003.
[176]庄礼贤,尹协远,马晖扬.流体力学[M].合肥：中国科学技术大学出版社,1991.