基于水平集方法的毛管上升特征研究
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摘要
毛管上升现象与许多行业密切相关,系统地对此现象进行研究具有重大意义。与传统理论研究方法不同,本文使用N-S方程耦合水平集方法模拟毛管气液上升行为。通过与简化条件的解析解对比,验证了模拟方法的可靠性。此外,详细地研究了毛管振荡现象,并分析了影响毛管振荡行为的主要因素。结果表明,水平集方法能够精确地表征毛管振荡现象,与数值解相比具有更高的精度。毛管长度的增加能够减弱液柱振荡,主要归结于非湿相气体的粘滞力作用;湿相密度和湿相粘度同样对毛管振荡现象影响显著。湿相密度越大,惯性力越大,促进了毛管振荡;而湿相粘度变大,会增大粘滞力作用,因此减弱了毛管振荡现象。毛管振荡是由多种影响因素共同控制的,流体的惯性力是造成毛管振荡的主要原因,而粘滞力是减弱毛管振荡行为的主要因素,使液柱振荡逐渐衰减,并稳定至平衡高度。
The study of dynamic capillary rise is crucial for many industries and applications.Unlike the traditional theoretical work on the penetration of liquids into circular cylindrical capillaries,capillary force,gravity,inertial force,viscous force,etc.,are not taken into account entirely.This paper used time-dependent N-S equation coupling level set method for simulating the liquid rise in capillary tubes under different conditions.To begin with,we validated the model by comparing the simulation results with the simplified analytical solutions.Then,we investigated capillary oscillation phenomena in detail and the main impacts on oscillation behaviors in capillary tubes were discussed.The results show that this coupling method can accurately capture characteristics of rebounds of liquids in cylindrical tubes.With the increase of the capillary tube's length,the magnitude of capillary oscillation seems to be weakened which is attributed to the resistance of air theoretically,the capillary oscillation will vanish if the length of tube is indefinite.In addition,the liquid viscosities also produce a remarkable effect on the oscillating behavior in capillary tubes,increasing the viscosity of liquid results in a greater viscous resistance of the wetting liquid,thus the oscillation is reduced and even disappears.Moreover,the liquid density has a positive correlation with the inertial force.Therefore,a more drastic oscillation behavior will emerge when using a higher density of liquid for simulation.In conclusion,the capillary oscillation is controlled by a variety of factors.The inertial force of the liquid is the principal reason that may give rise to the oscillation of liquid column.
The study of dynamic capillary rise is crucial for many industries and applications.Unlike the traditional theoretical work on the penetration of liquids into circular cylindrical capillaries,capillary force,gravity,inertial force,viscous force,etc.,are not taken into account entirely.This paper used time-dependent N-S equation coupling level set method for simulating the liquid rise in capillary tubes under different conditions.To begin with,we validated the model by comparing the simulation results with the simplified analytical solutions.Then,we investigated capillary oscillation phenomena in detail and the main impacts on oscillation behaviors in capillary tubes were discussed.The results show that this coupling method can accurately capture characteristics of rebounds of liquids in cylindrical tubes.With the increase of the capillary tube's length,the magnitude of capillary oscillation seems to be weakened which is attributed to the resistance of air theoretically,the capillary oscillation will vanish if the length of tube is indefinite.In addition,the liquid viscosities also produce a remarkable effect on the oscillating behavior in capillary tubes,increasing the viscosity of liquid results in a greater viscous resistance of the wetting liquid,thus the oscillation is reduced and even disappears.Moreover,the liquid density has a positive correlation with the inertial force.Therefore,a more drastic oscillation behavior will emerge when using a higher density of liquid for simulation.In conclusion,the capillary oscillation is controlled by a variety of factors.The inertial force of the liquid is the principal reason that may give rise to the oscillation of liquid column.
引文
[1] Wan J W,Zhang W J,Bergstrom D J.Recent advances in modeling the underfill process in flip-chip packaging[J].Microelectronics Journal,2007,38(1):67-75.
[2] Wan J W,Zhang W J,Bergstrom D J.Experimental verification of models for underfill flow driven by capillary forces in flip-chip packaging[J].Microelectronics Reliability,2008,48(3):425-430.
[3] Bouaidat S,Hansen O,Bruus H,et al.Surface -directed capillary system;theory,experiments and applications[J].Lab on a Chip,2005,5(8):827.
[4] Chen J M,Huang P C,Lin M G.Analysis and experiment of capillary valves for microfluidics on a rotating disk[J].Microfluidics and Nanofluidics,2008,4(5):427-437.
[5] Lichner L,Hallett P D,Drongová Z,et al.Algae influence the hydrophysical parameters of a sandy soil[J].Catena,2013,108:58-68.
[6] Liu Z,Yu X,Wan L.Capillary rise method for the measurement of the contact angle of soils[J].Acta Geotechnica,2016,11(1):21-35.
[7] Nia S F,Jessen K.Theoretical analysis of capillary rise in porous media[J].Transport in Porous Media,2015,110(1):141-155.
[8] Mason G,Morrow N R.Developments in spontaneous imbibition and possibilities for future work[J].Journal of Petroleum Science and Engineering,2013,110:268-293.
[9] Carnali J O,Kotkin C A.Determination of the capi-llary nature of simple woven textiles[J].Journal of Colloid and Interface Science,1993,159(2):319-323.
[10] Marmur A,Cohen R D.Characterization of porous media by the kinetics of liquid penetration:The vertical capillaries model [J].Journal of Colloid and Interface Science,1997,189(2):299-304.
[11] 蔡建超,郁伯铭.多孔介质自发渗吸研究进展[J].力学进展,2012,42(6):735-754.(CAI Jian-chao,YU Bo -ming.Advances in studies of spontaneous imbibition in porous media[J].Advances in Mechanics,2012,42(6):735-754.(in Chinese))
[12] Washburn E W.The dynamics of capillary flow[J].Physical Review,1921,17(3):273-283.
[13] Quéré D.Wetting and roughness[J].Annual Review of Materials Research,2008,38:71-99.
[14] Zhmud B V,Tiberg F,Hallstensson K.Dynamics of capillary rise[J].Journal of Colloid and Interface Science,2000,228(2):263-269.
[15] Hamraoui A,Nylander T.Analytical approach for the Lucas-Washburn equation[J].Journal of Colloid and Interface Science,2002,250(2):415-421.
[16] Fries N,Dreyer M.An analytic solution of capillary rise restrained by gravity[J].Journal of Colloid and Interface Science,2008,320(1):259-263.
[17] Szekely J,Neumann A W,Chuang Y K.The rate of capillary penetration and the applicability of the washburn equation[J].Journal of Colloid and Interface Science,1971,35(2):273-278.
[18] Hamraoui A,Thuresson K,Nylander T,et al.Can a dynamic contact angle be understood in terms of a friction coefficient?[J].Journal of Colloid and Interface Science,2000,226(2):199-204.
[19] Quéré D,Rapha?l é,Ollitrault J Y.Rebounds in a capillary tube[J].Langmuir,1999,15(10):3679-3682.
[20] Quéré D.Inertial capillarity[J].Europhysics Letters(EPL),1997,39(5):533-538.
[21] Xiao Y,Yang F Z,Pitchumani R.A generalized analysis of capillary flows in channels[J].Journal of Colloid and Interface Science,2006,298(2):880-888.
[22] Cai J C,Yu B M,Mei M F,et al.Capillary rise in a single tortuous capillary[J].Chinese Physics Letters,2010,27(5):054701.
[23] Siebold A,Nardin M,Schultz J,et al.Effect of dynamic contact angle on capillary rise phenomena[J].Colloids and Surfaces A:Physicochemical and Engineering Aspects,2000,161(1):81-87.
[24] Popescu M N,Ralston J,Sedev R.Capillary rise with velocity-dependent dynamic contact angle[J].Langmuir the Acs Journal of Surfaces & Colloids,2008,24(21):12710-12716.
[25] Chibbaro S.Capillary filling with pseudo-potential binary Lattice -Boltzmann model[J].The European Physical Journal E,2008,27(1):99-106.
[26] Zacharoudiou I,Boek E S.Capillary filling and haines jump dynamics using free energy lattice boltzmann simulations[J].Advances in Water Resources,2016,92:43-56.
[27] 陈海楠,孙东科,戴挺,等.凝固前沿和气泡相互作用的大密度比格子玻尔兹曼方法模拟[J].物理学报,2013,62(12):120502.(CHEN Hai-nan,SUN Dong-ke,DAI Ting,et al.Modeling of the interaction be -tween solidification interface and bubble using the Lattice Boltzmann method wih large density ratio [J].Acta Physica Sinica,2013,62(12):120502.(in Chinese))
[28] Meakin P,Tartakovsky A M.Modeling and simulation of pore -scale multiphase fluid flow and reactive transport in fractured and porous media[J].Reviews of Geophysics,2009,47(3):RG3002.
[29] Hirt C W,Nichols B D.Volume of fluid (VOF) method for the dynamics of free boundaries[J].Journal of Computational Physics,1981,39(1):201-225.
[30] Ubbink O,Issa R I.A method for capturing sharp fluid interfaces on arbitrary meshes[J].Journal of Computational Physics,1999,153(1):26-50.
[31] Sussman M,Smereka P,Osher S.A level set approach for computing solutions to incompressible two-phase flow[J].Journal of Computational Physics,1994,114(1):146-159.
[32] Prodanovi M,Bryant S L.A level set method for determining critical curvatures for drainage and imbibition[J].Journal of Colloid and Interface Science,2006,304(2):442-458.
[33] Nadiga B T,Zaleski S.Investigations of a two-phase fluid model[J].European Journal of Mechanics-B/Fluids,1995,15(6):885-896.
[34] Jacqmin D.Calculation of two -phase Navier-Stokes flows using phase -field modeling[J].Journal of Computational Physics,1999,155(1):96-127.
[35] Olsson E,Kreiss G,Zahedi S.A conservative level set method for two phase flow [J].Journal of Computational Physics,2005,210(1):225-246.
[36] Xu J J,Ren W Q.A level-set method for two -phase flows with moving contact line and insoluble surfactant[J].Journal of Computational Physics,2014,263:71-90.
[37] Lafaurie B,Nardone C,Scardovelli R,et al.Modelling merging and fragmentation in multiphase flows with SURFER[J].Journal of Computational Physics,1994,113(1):134-147.
[38] Stange M,Dreyer M E,Rath H J.Capillary driven flow in circular cylindrical tubes[J].Physics of Fluids,2003,15(9):2587-2601.
[39] Martic G,Gentner F,Seveno D,et al.A molecular dynamics simulation of capillary imbibition[J].Langmuir,2002,18(21):7971-7976.
[40] Lavi B,Marmur A,Bachmann J.Porous media characterization by the two -liquid method:Effect of dynamic contact angle and inertia[J].Langmuir,2008,24(5):1918-1923.
[41] Weisbrod N,Mcginnis T,Rockhold M L,et al.Effective Darcy-scale contact angles in porous media imbibing solutions of various surface tensions[J].Water Resources Research,2009,45:W00D39.
[42] Masoodi R,Languri E,Ostadhossein A.Dynamics of liquid rise in a vertical capillary tube[J].Journal of Colloid and Interface Science,2013,389(1):268-272.
[43] Liou W W,Peng Y Q,Parker P E.Analytical mode -ling of capillary flow in tubes of nonuniform cross section[J].Journal of Colloid and Interface Science,2009,333(1):389-399.
[44] Hilpert M.Explicit analytical solutions for liquid infiltration into capillary tubes:Dynamic and constant contact angle[J].Journal of Colloid and Interface Science,2010,344(1):198-208.
[45] Fries N,Dreyer M.Dimensionless scaling methods for capillary rise[J].Journal of Colloid and Interface Science,2009,338(2):514-518.
[2] Wan J W,Zhang W J,Bergstrom D J.Experimental verification of models for underfill flow driven by capillary forces in flip-chip packaging[J].Microelectronics Reliability,2008,48(3):425-430.
[3] Bouaidat S,Hansen O,Bruus H,et al.Surface -directed capillary system;theory,experiments and applications[J].Lab on a Chip,2005,5(8):827.
[4] Chen J M,Huang P C,Lin M G.Analysis and experiment of capillary valves for microfluidics on a rotating disk[J].Microfluidics and Nanofluidics,2008,4(5):427-437.
[5] Lichner L,Hallett P D,Drongová Z,et al.Algae influence the hydrophysical parameters of a sandy soil[J].Catena,2013,108:58-68.
[6] Liu Z,Yu X,Wan L.Capillary rise method for the measurement of the contact angle of soils[J].Acta Geotechnica,2016,11(1):21-35.
[7] Nia S F,Jessen K.Theoretical analysis of capillary rise in porous media[J].Transport in Porous Media,2015,110(1):141-155.
[8] Mason G,Morrow N R.Developments in spontaneous imbibition and possibilities for future work[J].Journal of Petroleum Science and Engineering,2013,110:268-293.
[9] Carnali J O,Kotkin C A.Determination of the capi-llary nature of simple woven textiles[J].Journal of Colloid and Interface Science,1993,159(2):319-323.
[10] Marmur A,Cohen R D.Characterization of porous media by the kinetics of liquid penetration:The vertical capillaries model [J].Journal of Colloid and Interface Science,1997,189(2):299-304.
[11] 蔡建超,郁伯铭.多孔介质自发渗吸研究进展[J].力学进展,2012,42(6):735-754.(CAI Jian-chao,YU Bo -ming.Advances in studies of spontaneous imbibition in porous media[J].Advances in Mechanics,2012,42(6):735-754.(in Chinese))
[12] Washburn E W.The dynamics of capillary flow[J].Physical Review,1921,17(3):273-283.
[13] Quéré D.Wetting and roughness[J].Annual Review of Materials Research,2008,38:71-99.
[14] Zhmud B V,Tiberg F,Hallstensson K.Dynamics of capillary rise[J].Journal of Colloid and Interface Science,2000,228(2):263-269.
[15] Hamraoui A,Nylander T.Analytical approach for the Lucas-Washburn equation[J].Journal of Colloid and Interface Science,2002,250(2):415-421.
[16] Fries N,Dreyer M.An analytic solution of capillary rise restrained by gravity[J].Journal of Colloid and Interface Science,2008,320(1):259-263.
[17] Szekely J,Neumann A W,Chuang Y K.The rate of capillary penetration and the applicability of the washburn equation[J].Journal of Colloid and Interface Science,1971,35(2):273-278.
[18] Hamraoui A,Thuresson K,Nylander T,et al.Can a dynamic contact angle be understood in terms of a friction coefficient?[J].Journal of Colloid and Interface Science,2000,226(2):199-204.
[19] Quéré D,Rapha?l é,Ollitrault J Y.Rebounds in a capillary tube[J].Langmuir,1999,15(10):3679-3682.
[20] Quéré D.Inertial capillarity[J].Europhysics Letters(EPL),1997,39(5):533-538.
[21] Xiao Y,Yang F Z,Pitchumani R.A generalized analysis of capillary flows in channels[J].Journal of Colloid and Interface Science,2006,298(2):880-888.
[22] Cai J C,Yu B M,Mei M F,et al.Capillary rise in a single tortuous capillary[J].Chinese Physics Letters,2010,27(5):054701.
[23] Siebold A,Nardin M,Schultz J,et al.Effect of dynamic contact angle on capillary rise phenomena[J].Colloids and Surfaces A:Physicochemical and Engineering Aspects,2000,161(1):81-87.
[24] Popescu M N,Ralston J,Sedev R.Capillary rise with velocity-dependent dynamic contact angle[J].Langmuir the Acs Journal of Surfaces & Colloids,2008,24(21):12710-12716.
[25] Chibbaro S.Capillary filling with pseudo-potential binary Lattice -Boltzmann model[J].The European Physical Journal E,2008,27(1):99-106.
[26] Zacharoudiou I,Boek E S.Capillary filling and haines jump dynamics using free energy lattice boltzmann simulations[J].Advances in Water Resources,2016,92:43-56.
[27] 陈海楠,孙东科,戴挺,等.凝固前沿和气泡相互作用的大密度比格子玻尔兹曼方法模拟[J].物理学报,2013,62(12):120502.(CHEN Hai-nan,SUN Dong-ke,DAI Ting,et al.Modeling of the interaction be -tween solidification interface and bubble using the Lattice Boltzmann method wih large density ratio [J].Acta Physica Sinica,2013,62(12):120502.(in Chinese))
[28] Meakin P,Tartakovsky A M.Modeling and simulation of pore -scale multiphase fluid flow and reactive transport in fractured and porous media[J].Reviews of Geophysics,2009,47(3):RG3002.
[29] Hirt C W,Nichols B D.Volume of fluid (VOF) method for the dynamics of free boundaries[J].Journal of Computational Physics,1981,39(1):201-225.
[30] Ubbink O,Issa R I.A method for capturing sharp fluid interfaces on arbitrary meshes[J].Journal of Computational Physics,1999,153(1):26-50.
[31] Sussman M,Smereka P,Osher S.A level set approach for computing solutions to incompressible two-phase flow[J].Journal of Computational Physics,1994,114(1):146-159.
[32] Prodanovi M,Bryant S L.A level set method for determining critical curvatures for drainage and imbibition[J].Journal of Colloid and Interface Science,2006,304(2):442-458.
[33] Nadiga B T,Zaleski S.Investigations of a two-phase fluid model[J].European Journal of Mechanics-B/Fluids,1995,15(6):885-896.
[34] Jacqmin D.Calculation of two -phase Navier-Stokes flows using phase -field modeling[J].Journal of Computational Physics,1999,155(1):96-127.
[35] Olsson E,Kreiss G,Zahedi S.A conservative level set method for two phase flow [J].Journal of Computational Physics,2005,210(1):225-246.
[36] Xu J J,Ren W Q.A level-set method for two -phase flows with moving contact line and insoluble surfactant[J].Journal of Computational Physics,2014,263:71-90.
[37] Lafaurie B,Nardone C,Scardovelli R,et al.Modelling merging and fragmentation in multiphase flows with SURFER[J].Journal of Computational Physics,1994,113(1):134-147.
[38] Stange M,Dreyer M E,Rath H J.Capillary driven flow in circular cylindrical tubes[J].Physics of Fluids,2003,15(9):2587-2601.
[39] Martic G,Gentner F,Seveno D,et al.A molecular dynamics simulation of capillary imbibition[J].Langmuir,2002,18(21):7971-7976.
[40] Lavi B,Marmur A,Bachmann J.Porous media characterization by the two -liquid method:Effect of dynamic contact angle and inertia[J].Langmuir,2008,24(5):1918-1923.
[41] Weisbrod N,Mcginnis T,Rockhold M L,et al.Effective Darcy-scale contact angles in porous media imbibing solutions of various surface tensions[J].Water Resources Research,2009,45:W00D39.
[42] Masoodi R,Languri E,Ostadhossein A.Dynamics of liquid rise in a vertical capillary tube[J].Journal of Colloid and Interface Science,2013,389(1):268-272.
[43] Liou W W,Peng Y Q,Parker P E.Analytical mode -ling of capillary flow in tubes of nonuniform cross section[J].Journal of Colloid and Interface Science,2009,333(1):389-399.
[44] Hilpert M.Explicit analytical solutions for liquid infiltration into capillary tubes:Dynamic and constant contact angle[J].Journal of Colloid and Interface Science,2010,344(1):198-208.
[45] Fries N,Dreyer M.Dimensionless scaling methods for capillary rise[J].Journal of Colloid and Interface Science,2009,338(2):514-518.