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应用同伦分析法研究微重力环境下圆管毛细流动解析近似解问题, 给出了级数解的表达公式. 不同于其他解析近似方法, 该方法从根本上克服了摄动理论对小参数的过分依赖, 其有效性与所研究的非线性问题是否含有小参数无关, 适用范围广. 同伦分析法提供了选取基函数的自由, 可以选取较好的基函数, 更有效地逼近问题的解, 通过引入辅助参数和辅助函数来调节和控制级数解的收敛区域和收敛速度, 同伦分析法为圆管毛细流动问题的解析近似求解开辟了一个全新的途径. 通过具体算例, 将同伦分析法与四阶龙格库塔方法数值解做了比较, 结果表明, 该方法具有很高的计算精度.The capillary flow in a circular tube under microgravity environment is investigated by the homotopy analysis method (HAM), and the approximate analytical solution in the form of series solution is obtained. Different from other analytical approximate methods, the HAM is totally independent of small physical parameters, and thus it is suitable for most nonlinear problems. The HAM provides us a great freedom to choose basis functions of solution series, so that a nonlinear problem can be approximated more effectively, and it adjusts and controls the convergence region and the convergence rate of the series solution through introducing auxiliary parameter and the auxiliary function. The HAM hews out a new approach to the analytical approximate solutions of capillary flow in a circular tube. Through the specific example and comparing homotopy approximate analytical solution with the numerical solution which is obtained by the fourth-order Runge-Kutta method, the computed result indicate that this method has the good computational accuracy.
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Keywords:
- circular tubes /
- microgravity /
- capillary flow /
- homotopy analysis method
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[2] Washburn E W 1921 Phys. Rev. 17 273
[3] Bell J M, Cameron F K 1906 J. Phys. Chem. 10 658
[4] Rideal E K 1922 Philos. Mag. 44 1152
[5] LeGrand E J, Rense W A 1945 J. Appl. Phys. 16 843
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[8] Jeje A A 1979 J. Colloid Interf. Sci. 69 420
[9] Ichikawa N, Satoda Y 1994 J. Colloid Interf. Sci. 162 350
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[11] Quéré D 1997 Europhys. Lett. 39 533
[12] Levine S, Reed P, Watson E J, Neale G 1976 In Colloid and Interface Science (New York: Academic) p403
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[14] Wang C X, Xu S H, Sun Z W, Hu W R 2009 AIAA J. 11 2642
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[16] Cheng J, Liao S J 2007 Acta Mech. Sin. 39 715 (in Chinese) [成均, 廖世俊 2007 力学学报 39 715]
[17] Liao S J 2003 J. Fluid Mech. 488 189
[18] Li Y Q, Zhu D W, Li F 2009 Chin. J. Mech. Eng. 45 37 (in Chinese) [李永强, 朱大巍, 李锋 2009 机械工程学报 45 37]
[19] Li Y Q, Li F, Zhu D W 2010 Compos. Struct. 92 1110
[20] Yuan P X, Li Y Q 2010 Appl. Math. Mech. 31 1293
[21] Li Y Q, Li L, He Y L 2011 Compos. Struct. 93 360
[22] Li Y Q, Zhu D W 2011 Compos. Struct. 93 880
[23] Shi Y R, Yang H J 2010 Acta Phys. Sin. 59 67 (in Chinese) [石玉仁, 杨红娟 2010 59 67]
[24] Yang P, Chen Y, Li Z B 2010 Acta Phys. Sin. 59 3668 (in Chinese) [杨沛, 陈勇, 李志斌 2010 59 3668]
[25] Liao S J 2012 Homotopy Analysis Method for Nonlinear Differential Equations (Beijing: Higher Education Press) p285
[26] Dreyer M E 2007 Spring Tracts in Mordern Physics 221 51
[27] Sparrow E M, Lin S H, Lundgren T S 1964 Phys. Fluids 7 338
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[1] Lucas V R 1918 Kolloid-Z. 23 15
[2] Washburn E W 1921 Phys. Rev. 17 273
[3] Bell J M, Cameron F K 1906 J. Phys. Chem. 10 658
[4] Rideal E K 1922 Philos. Mag. 44 1152
[5] LeGrand E J, Rense W A 1945 J. Appl. Phys. 16 843
[6] Siegel R 1961 J. Appl. Mech. 83 165
[7] Petrash D A, Nelson T M, Otto E W 1963 NASA TN D-1582
[8] Jeje A A 1979 J. Colloid Interf. Sci. 69 420
[9] Ichikawa N, Satoda Y 1994 J. Colloid Interf. Sci. 162 350
[10] Joos P, Remoortere P, Bracke M{\it} 1990 J. Colloid Interf. Sci. 136 189
[11] Quéré D 1997 Europhys. Lett. 39 533
[12] Levine S, Reed P, Watson E J, Neale G 1976 In Colloid and Interface Science (New York: Academic) p403
[13] Stange M, Dreyer M E, Rath H J 2003 Phys. Fluids 15 2587
[14] Wang C X, Xu S H, Sun Z W, Hu W R 2009 AIAA J. 11 2642
[15] Liao S J 2006 Beyond Perturbation: Introduction to the Homotopy Analysis Method (Beijing: Science Press) p204 (in Chinese) [廖世俊2006 超越摄动–-同伦分析方法导论(北京:科学出版社) 第204页]
[16] Cheng J, Liao S J 2007 Acta Mech. Sin. 39 715 (in Chinese) [成均, 廖世俊 2007 力学学报 39 715]
[17] Liao S J 2003 J. Fluid Mech. 488 189
[18] Li Y Q, Zhu D W, Li F 2009 Chin. J. Mech. Eng. 45 37 (in Chinese) [李永强, 朱大巍, 李锋 2009 机械工程学报 45 37]
[19] Li Y Q, Li F, Zhu D W 2010 Compos. Struct. 92 1110
[20] Yuan P X, Li Y Q 2010 Appl. Math. Mech. 31 1293
[21] Li Y Q, Li L, He Y L 2011 Compos. Struct. 93 360
[22] Li Y Q, Zhu D W 2011 Compos. Struct. 93 880
[23] Shi Y R, Yang H J 2010 Acta Phys. Sin. 59 67 (in Chinese) [石玉仁, 杨红娟 2010 59 67]
[24] Yang P, Chen Y, Li Z B 2010 Acta Phys. Sin. 59 3668 (in Chinese) [杨沛, 陈勇, 李志斌 2010 59 3668]
[25] Liao S J 2012 Homotopy Analysis Method for Nonlinear Differential Equations (Beijing: Higher Education Press) p285
[26] Dreyer M E 2007 Spring Tracts in Mordern Physics 221 51
[27] Sparrow E M, Lin S H, Lundgren T S 1964 Phys. Fluids 7 338
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