二阶边界滑移气体薄膜挤压特性研究
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摘要
微/纳科技的高速发展对稀薄气体在过渡流区流动的建模提出了迫切需求。气体-固体界面的二阶滑移边界条件将N-S方程的适用范围扩展到过渡流区。由于该模型形式简洁、计算效率高,近年来被广泛应用。本文以二阶滑移边界条件下修正的雷诺方程研究可压缩气体和不可压缩气体的球面挤压特性。
对于不可压缩气体流动,气体压强和球面反力的解答根据边界条件分为九种,其中前四种已有报道。本文给出二阶滑移边界条件下的其余五种解答。求解中当完全解析解无法得到时,采用数值解与解析解结合的方法,用数学解析技巧最大限度减少数值解的使用,使求解过程和最终解保持解析形式。算例显示,二阶滑移的影响随着克努森数和二阶滑移系数的增加而增大;在过渡流区,二阶滑移的影响显著。
对于符合理想气体状态方程的可压缩气体,本文采用有限差分和直接迭代法求解雷诺方程。数值算例同样显示,二阶滑移的影响在过渡流区不可忽略;而在无滑移边界条件下,气体可压缩性的影响随着最小膜厚与球体半径比值的增加而增大。
Development of micro/nanotechnology requires an effective model to investigate rarefied gas flow in the transition regime. Second-order slip boundary condition is used to extend the application scope of Navier-Stokes equations into the transition regime. Recently, this modeling is extensively used due to its simplicity and high-efficiency compared with molecular based theories. In this thesis, a modified Reynolds equation is employed to study the hydrodynamic behavior of incompressible and compressible squeezed gas film confined between spherical surfaces under second-order slip boundary condition.
There are nine different cases for incompressible gas flow under no-slip, first-order slip and second-order slip boundary conditions. Gas pressure and hydrodynamic force of the first four cases have been reported in literature. This thesis presents solutions for the other cases. An approach of analytical method associated with numerical computation is supplemented when analytical method is ineffective. Illustrated examples show that a considerable effect of the second-order slip exists in the transition regime. This effect is enhanced by accretion of Knudsen number and the second-order slip coefficient.
Finite differential method and iteration procedure are employed to solve compressible gas problem. Numerical examples indicate that a significant effect of the second-order slip occurs in the transition regime. It is also shown that the compressible effect under no-slip condition increases with the ratio of minimum gap height to sphere radius.
对于不可压缩气体流动,气体压强和球面反力的解答根据边界条件分为九种,其中前四种已有报道。本文给出二阶滑移边界条件下的其余五种解答。求解中当完全解析解无法得到时,采用数值解与解析解结合的方法,用数学解析技巧最大限度减少数值解的使用,使求解过程和最终解保持解析形式。算例显示,二阶滑移的影响随着克努森数和二阶滑移系数的增加而增大;在过渡流区,二阶滑移的影响显著。
对于符合理想气体状态方程的可压缩气体,本文采用有限差分和直接迭代法求解雷诺方程。数值算例同样显示,二阶滑移的影响在过渡流区不可忽略;而在无滑移边界条件下,气体可压缩性的影响随着最小膜厚与球体半径比值的增加而增大。
Development of micro/nanotechnology requires an effective model to investigate rarefied gas flow in the transition regime. Second-order slip boundary condition is used to extend the application scope of Navier-Stokes equations into the transition regime. Recently, this modeling is extensively used due to its simplicity and high-efficiency compared with molecular based theories. In this thesis, a modified Reynolds equation is employed to study the hydrodynamic behavior of incompressible and compressible squeezed gas film confined between spherical surfaces under second-order slip boundary condition.
There are nine different cases for incompressible gas flow under no-slip, first-order slip and second-order slip boundary conditions. Gas pressure and hydrodynamic force of the first four cases have been reported in literature. This thesis presents solutions for the other cases. An approach of analytical method associated with numerical computation is supplemented when analytical method is ineffective. Illustrated examples show that a considerable effect of the second-order slip exists in the transition regime. This effect is enhanced by accretion of Knudsen number and the second-order slip coefficient.
Finite differential method and iteration procedure are employed to solve compressible gas problem. Numerical examples indicate that a significant effect of the second-order slip occurs in the transition regime. It is also shown that the compressible effect under no-slip condition increases with the ratio of minimum gap height to sphere radius.
引文
[1] STECKELMACHER W. Knudsen flow 75 years on: the current state of the art for flow of rarefied gases in tubes and systems [J].Reports on Progress in Physics, 1986, 49:1083-1107.
[2] ARKILIC E B. Measurement of the mass flow and tangential momentum accommodation coefficient in silicon micromachined channels [D]. Massachusetts: Massachusetts Institute of Technology, 1997.
[3] SCHAAF S A, CHAMBRE P L. Flow of rarified gases [M]. New Jersey: Princeton University Press, 1961.
[4] GAD-EL-HAK M. The fluid mechanics of microdevices-the freeman scholar lecture [J]. Transactions of the ASME-Journal of Fluids Engineering, 1999,121:5-33.
[5] BARBER R W, EMERSON D R. Challenges in modeling gas-phase flow in microchannels: from slip to transition [J].Heat Transfer Engineering,2006,27(4):3-12.
[6] COLIN S. Rarefaction and compressibility effects on steady and transient gas flows in microchannels [J].Microfluidics and Nanofluidics, 2005, 1:268-279.
[7] KARNIADAKIS G E, BESKOK A, ALURU N. Microflows and nanoflows: fundamentals and simulation [M]. New York: Springer-Verlag, 2005.
[8] BESKOK A, KARNIADAKIS G E. A model for flows in channels, pipes, and ducts at micro and nano scales [J]. Microscale Thermophysical Engineering, 1999, 3:43-77.
[9] CERCIGNANI C. Rarefied gas dynamics from basic concepts to actual calculations [M].Cambridge: Cambridge University Press, 2000.
[10] 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 [J]. Physical Review,1954, 94(3): 511-525.
[11] BIRD G A. Molecular gas dynamics and the direct simulation of gas flows [M]. Oxford:Oxford University Press, 1994.
[12] CHEN S Y, GARY D D. Lattice Boltzmann method for fluid flows [J]. Annual Review of Fluid Mechanics, 1998, 30:329-364.
[13] GAD-EL-HAK M. Comments on "critical view on new results in micro-fluid mechanics"[J]. International Journal of Heat and Mass Transfer, 2003, 46:3941-3945.
[14] KENNARD E H. Kinetic theory of gases: with an introduction to statistical mechanics [M]. New York: McGraw-Hill Book Company, 1938.
[15] MAXWELL J C. On stress in rarified gases arising from inequalities of temperature [J].The Philosophical Transaction of the Royal Society of London, 1879, 170:231-256.
[16] MILLIKAN R A. Coefficients of slip in gases and the law of reflection of molecules from the surfaces of solids and liquids [J].Physical Review, 1923, 21:217-238.
[17] SREEKANTH A K. Slip flow through long circular tubes [C]// Proceedings of the 6th International Symposium on Rarefied Gas Dynamics. New York: Wachman Academic Press,1969,1:667-680.
[18] PIEKOS E S, BREUER K S. Numerical modeling of micromechanical devices using the direct simulation Monte Carlo method [J]. Transactions of the ASME-Journal of Fluids Engineering, 1996, 118:464-469.
[19] SCHAMBERG R. The fundamental differential equations and the boundary conditions for high speed slip-flow, and their application to several specific problems [D]. Pasadena:California Institute of Technology, 1947.
[20] TANG G H, HE Y L, TAO W Q. Comparison of gas slip models with solutions of linearized Boltzmann equation and direct simulation of Monte Carlo method [J]. International Journal of Modern Physics C, 2007,18(2):203-216.
[21] DEISSLER R G. An analysis of second-order slip flow and temperature-jump boundary conditions for rarefied gases [J]. International Journal of Heat and Mass Transfer, 1964, 7:681-694.
[22] HADJICONSTANTINOU N G. Comment on Cercignani' s second-order slip coefficient [J].Physics of Fluids, 2003, 15(8):2352-2354.
[23] HSIA Y T, DOMOTO G A. An experimental investigation of molecular rarefaction effects in gas lubricated bearings at ultra-low clearances [J]. Transactions of the ASME-Journal of Lubrication Technology, 1983, 105:120-130.
[24] MITSUYA Y. Modified Reynolds equation for ultra-thin film gas lubrication using 1. 5-order slip-flow model and considering surface accommodation coefficient [J]. Transactions of the ASME-Journal of Tribology, 1993, 115:289-294.
[25] SUN Y H, CHAN W K, LIU N Y. A slip model with molecular dynamics [J].Journal of Micromechanics and Microengineering, 2002, 12:316-322.
[26] CERCIGNANI C, DANERI A. Flow of a rarefied gas between two parallel plates [J]. Journal of Applied Physics,1963, 34(12):3509-3513.
[27] MAURER J, TABELING P, JOSEPH P, et al. Second-order slip laws in microchannels for helium and nitrogen [J].Physics of Fluids,2003, 15(9):2613-2621.
[28] EWART T, PERRIER P, GRAUR I, et al. Tangential momemtum accommodation in microtube [J].Microfluidics and Nanofluidics,2007,3:689-695.
[29] EWART T, PERRIER P, GRAUR I, et al. Mass flow rate measurements in a microchannel,from hydrodynamic to near free molecular regimes [J].Journal of Fluid Mechanics,2007,584:337-356.
[30]AUBERT C,COLIN S.High-order boundary conditions for gaseous flows in rectangular microducts[J].Microscale Thermophysical Engineering,2001,5:41-54.
[31]COLIN S,LALONDE P,CAEN R.Validation of a second-order slip flow model in rectangular microchannels[J].Heat Transfer Engineering,2004,25(3):23-30.
[32]HADJICONSTANTINOU N G.Validation of a second-order slip model for dilute gas flows [J].Microscale Thermophysical Engineering,2005,9(2):137-153.
[33]DONGARI N,AGRAWAL A,AGRAWAL A.Analytical solution of gaseous slip flow in long microchannels[J].International Journal of Heat and Mass Transfer,2007,50:3411-3421.
[34]LOCKERBY D A,REESE J M.On the modelling of isothermal gas flows at the microscale [J].Journal of Fluid Mechanics,2008,604:235-261.
[35]XUE H,FAN Q.A new analytic solution of the Navier-Stokes equations for microchannel flows[J].Microscale Thermophysical Engineering,2000,4:125-143.
[36]JIE D,DIAO X,CHEONG K B,et al.Navier-Stokes simulations of gas flow in micro devices [J].Journal of Micromechanics and gicroengineering,2000,10:372-379.
[37]NETO C,EVANS D R,BONACCURSO E,et al.Boundary slip in Newtonian liquids:a review of experimental studies[J].Reports on Progress in Physics,2005,68:2859-2897.
[38]MAALI A,BHUSHANB.Slip-length measurement of confined air flow using dynamic atomic force microscopy[J].Physical Review E,2008,78:027302.
[39]TAGAWA N.State of the art for nanospacing flying head slider mechanisms in magnetic recording disk storage[J].Wear,1993,168:43-47.
[40]VINOGRADOVAO I.Drainage of a thin liquid film confined between hydrophobic surfaces [J].Langmuir,1995,11:2213-2220.
[41]CAMERON A,ETTLES C M.Basic lubrication theory[M].3rd ed.Chichester UK:E.Horwood Ltd,1981.
[42]斯米尔诺夫.高等数学教程卷一第二分册[M].上海:商务印书馆,1953.
[43]卡姆克.常微分方程手册[M].北京:科学出版社,1977.
[44]南京大学数学系计算数学专业.常微分方程数值解法[M].北京:科学出版社,1979.
[45]WARFIELD C N.Tentative tables for the properties of the upper atmosphere[R].Washington:NACA Technical Note NO.1200,1947.
[2] ARKILIC E B. Measurement of the mass flow and tangential momentum accommodation coefficient in silicon micromachined channels [D]. Massachusetts: Massachusetts Institute of Technology, 1997.
[3] SCHAAF S A, CHAMBRE P L. Flow of rarified gases [M]. New Jersey: Princeton University Press, 1961.
[4] GAD-EL-HAK M. The fluid mechanics of microdevices-the freeman scholar lecture [J]. Transactions of the ASME-Journal of Fluids Engineering, 1999,121:5-33.
[5] BARBER R W, EMERSON D R. Challenges in modeling gas-phase flow in microchannels: from slip to transition [J].Heat Transfer Engineering,2006,27(4):3-12.
[6] COLIN S. Rarefaction and compressibility effects on steady and transient gas flows in microchannels [J].Microfluidics and Nanofluidics, 2005, 1:268-279.
[7] KARNIADAKIS G E, BESKOK A, ALURU N. Microflows and nanoflows: fundamentals and simulation [M]. New York: Springer-Verlag, 2005.
[8] BESKOK A, KARNIADAKIS G E. A model for flows in channels, pipes, and ducts at micro and nano scales [J]. Microscale Thermophysical Engineering, 1999, 3:43-77.
[9] CERCIGNANI C. Rarefied gas dynamics from basic concepts to actual calculations [M].Cambridge: Cambridge University Press, 2000.
[10] 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 [J]. Physical Review,1954, 94(3): 511-525.
[11] BIRD G A. Molecular gas dynamics and the direct simulation of gas flows [M]. Oxford:Oxford University Press, 1994.
[12] CHEN S Y, GARY D D. Lattice Boltzmann method for fluid flows [J]. Annual Review of Fluid Mechanics, 1998, 30:329-364.
[13] GAD-EL-HAK M. Comments on "critical view on new results in micro-fluid mechanics"[J]. International Journal of Heat and Mass Transfer, 2003, 46:3941-3945.
[14] KENNARD E H. Kinetic theory of gases: with an introduction to statistical mechanics [M]. New York: McGraw-Hill Book Company, 1938.
[15] MAXWELL J C. On stress in rarified gases arising from inequalities of temperature [J].The Philosophical Transaction of the Royal Society of London, 1879, 170:231-256.
[16] MILLIKAN R A. Coefficients of slip in gases and the law of reflection of molecules from the surfaces of solids and liquids [J].Physical Review, 1923, 21:217-238.
[17] SREEKANTH A K. Slip flow through long circular tubes [C]// Proceedings of the 6th International Symposium on Rarefied Gas Dynamics. New York: Wachman Academic Press,1969,1:667-680.
[18] PIEKOS E S, BREUER K S. Numerical modeling of micromechanical devices using the direct simulation Monte Carlo method [J]. Transactions of the ASME-Journal of Fluids Engineering, 1996, 118:464-469.
[19] SCHAMBERG R. The fundamental differential equations and the boundary conditions for high speed slip-flow, and their application to several specific problems [D]. Pasadena:California Institute of Technology, 1947.
[20] TANG G H, HE Y L, TAO W Q. Comparison of gas slip models with solutions of linearized Boltzmann equation and direct simulation of Monte Carlo method [J]. International Journal of Modern Physics C, 2007,18(2):203-216.
[21] DEISSLER R G. An analysis of second-order slip flow and temperature-jump boundary conditions for rarefied gases [J]. International Journal of Heat and Mass Transfer, 1964, 7:681-694.
[22] HADJICONSTANTINOU N G. Comment on Cercignani' s second-order slip coefficient [J].Physics of Fluids, 2003, 15(8):2352-2354.
[23] HSIA Y T, DOMOTO G A. An experimental investigation of molecular rarefaction effects in gas lubricated bearings at ultra-low clearances [J]. Transactions of the ASME-Journal of Lubrication Technology, 1983, 105:120-130.
[24] MITSUYA Y. Modified Reynolds equation for ultra-thin film gas lubrication using 1. 5-order slip-flow model and considering surface accommodation coefficient [J]. Transactions of the ASME-Journal of Tribology, 1993, 115:289-294.
[25] SUN Y H, CHAN W K, LIU N Y. A slip model with molecular dynamics [J].Journal of Micromechanics and Microengineering, 2002, 12:316-322.
[26] CERCIGNANI C, DANERI A. Flow of a rarefied gas between two parallel plates [J]. Journal of Applied Physics,1963, 34(12):3509-3513.
[27] MAURER J, TABELING P, JOSEPH P, et al. Second-order slip laws in microchannels for helium and nitrogen [J].Physics of Fluids,2003, 15(9):2613-2621.
[28] EWART T, PERRIER P, GRAUR I, et al. Tangential momemtum accommodation in microtube [J].Microfluidics and Nanofluidics,2007,3:689-695.
[29] EWART T, PERRIER P, GRAUR I, et al. Mass flow rate measurements in a microchannel,from hydrodynamic to near free molecular regimes [J].Journal of Fluid Mechanics,2007,584:337-356.
[30]AUBERT C,COLIN S.High-order boundary conditions for gaseous flows in rectangular microducts[J].Microscale Thermophysical Engineering,2001,5:41-54.
[31]COLIN S,LALONDE P,CAEN R.Validation of a second-order slip flow model in rectangular microchannels[J].Heat Transfer Engineering,2004,25(3):23-30.
[32]HADJICONSTANTINOU N G.Validation of a second-order slip model for dilute gas flows [J].Microscale Thermophysical Engineering,2005,9(2):137-153.
[33]DONGARI N,AGRAWAL A,AGRAWAL A.Analytical solution of gaseous slip flow in long microchannels[J].International Journal of Heat and Mass Transfer,2007,50:3411-3421.
[34]LOCKERBY D A,REESE J M.On the modelling of isothermal gas flows at the microscale [J].Journal of Fluid Mechanics,2008,604:235-261.
[35]XUE H,FAN Q.A new analytic solution of the Navier-Stokes equations for microchannel flows[J].Microscale Thermophysical Engineering,2000,4:125-143.
[36]JIE D,DIAO X,CHEONG K B,et al.Navier-Stokes simulations of gas flow in micro devices [J].Journal of Micromechanics and gicroengineering,2000,10:372-379.
[37]NETO C,EVANS D R,BONACCURSO E,et al.Boundary slip in Newtonian liquids:a review of experimental studies[J].Reports on Progress in Physics,2005,68:2859-2897.
[38]MAALI A,BHUSHANB.Slip-length measurement of confined air flow using dynamic atomic force microscopy[J].Physical Review E,2008,78:027302.
[39]TAGAWA N.State of the art for nanospacing flying head slider mechanisms in magnetic recording disk storage[J].Wear,1993,168:43-47.
[40]VINOGRADOVAO I.Drainage of a thin liquid film confined between hydrophobic surfaces [J].Langmuir,1995,11:2213-2220.
[41]CAMERON A,ETTLES C M.Basic lubrication theory[M].3rd ed.Chichester UK:E.Horwood Ltd,1981.
[42]斯米尔诺夫.高等数学教程卷一第二分册[M].上海:商务印书馆,1953.
[43]卡姆克.常微分方程手册[M].北京:科学出版社,1977.
[44]南京大学数学系计算数学专业.常微分方程数值解法[M].北京:科学出版社,1979.
[45]WARFIELD C N.Tentative tables for the properties of the upper atmosphere[R].Washington:NACA Technical Note NO.1200,1947.