Search

Article

x

留言板

尊敬的读者、作者、审稿人, 关于本刊的投稿、审稿、编辑和出版的任何问题, 您可以本页添加留言。我们将尽快给您答复。谢谢您的支持!

姓名
邮箱
手机号码
标题
留言内容
验证码

Dynamic instability of super-long elastic rod in viscous fluid

Wang Peng Xue Yun Lou Zhi-Mei

Dynamic instability of super-long elastic rod in viscous fluid

Wang Peng, Xue Yun, Lou Zhi-Mei
PDF
Get Citation
  • The external environment affects the structural form of biological system. Many biological systems are surrounded by cell solutions, such as DNA and bacteria. The solution will offer a viscous resistance as the biological system moves in the viscous fluid. How does the viscous resistance affect the stability of biological system and what mode will be selected after instability? In this paper, we establish a super-long elastic rod model which contains the viscous resistance to model this phenomenon. The stability and instability of the super-long elastic rod in the viscous fluid are studied. The dynamic equations of motion of the super-long elastic rod in viscous fluid are given based on the Kirchhoff dynamic analogy. Then a coordinate basis vector perturbation scheme is reviewed. According to the new perturbation method, we obtain the first order perturbation representation of super-long elastic rod dynamic equation in the viscous fluid, which is a group of the second order linear partial differential equations. The stability of the super-long elastic rod can be determined by analyzing the solutions of the second order linear partial differential equations. The results are applied to a twisted planar DNA ring. The stability criterion of the twisted planar DNA ring and its critical region are obtained. The results show that the viscous resistance has no effect on the stability of super-long elastic rod dynamics, but affects its instability. The mode selection and the influence of the viscous resistance on the instability of DNA ring are discussed. The amplitude of the elastic loop becomes smaller under the influence of the viscous resistance, and a bifurcation occurs. The mode number of instability of DNA loop becomes bigger with the increase of viscous resistance.
      Corresponding author: Wang Peng, sdpengwang@163.com
    • Funds: Project supported by the National Nature Science Foundation of China (Grant Nos. 11262019, 11372195, 11472177).
    [1]

    Beham C J 1977 Proc. Natl. Acad. Sci. USA 74 2397

    [2]

    Le Bret M 1978 Biopolymers 17 1939

    [3]

    Travers A A, Thompson J M T 2004 Phil. Trans. R. Soc. Lond. A 362 1265

    [4]

    Benham C J, Mielke S P 2005 Annu. Rev. Biomed. Eng. 7 21

    [5]

    Shi Y M, Hearst J E 1994 J. Chem. Phys. 101 5186

    [6]

    Zhou H J, Ouyang Z C 1999 J. Chem. Phys. 110 1247

    [7]

    Xue Y, Liu Y Z, Chen L Q 2004 Chin. Phys. 13 794

    [8]

    Wang P, Xue Y 2016 Nonlinear Dyn. 83 1815

    [9]

    Liu Y Z 2006 Nonlinear Mechanics of Thin Elastic Rod-Theoretical Basis of Mechanical Model of DNA (Beijing: Tsinghua Press Springer) p85 (in Chinese) [刘延柱 2006 弹性细杆非线性力学-DNA力学模型的理论基础 (北京: 清华大学出版社 Springer) 第85页]

    [10]

    Bustamante C, Bryant Z 2003 Nature 421 423

    [11]

    Tobias I, Swigon D, Coleman B D 2000 Phys. Rev. E 61 747

    [12]

    Manning R S, Bluman G B 2005 Proc. R. Soc. Lond. A 461 2423

    [13]

    Liu Y Z, Zu J W 2004 Acta Mech. 164 29

    [14]

    Liu Y Z, Sheng L W 2007 Acta Phys. Sin. 56 2305 (in Chinese) [刘延柱, 盛立伟 2007 物理学报 56 2305]

    [15]

    Xue Y, Liu Y Z 2009 Acta Phys. Sin. 58 6737 (in Chinese) [薛纭, 刘延柱 2009 物理学报 58 6737]

    [16]

    Xue Y, Chen L Q, Liu Y Z 2004 Acta Phys. Sin. 53 4029 (in Chinese) [薛纭, 陈立群, 刘延柱 2004 物理学报 53 4029]

    [17]

    Goriely A, Tabor M 1997 Physica D 105 20

    [18]

    Goriely A, Tabor M 1996 Phys. Rev. Lett. 77 3537

    [19]

    Moulton D E, Lessinnes T, Goriely A 2013 J. Mech. Phys. Solids 61 398

    [20]

    Klapper I 1996 J. Comput. Phys. 125 325

    [21]

    Goldstein R E, Powers T R, Wiggins C H 1998 Phys. Rev. Lett. 80 5232

    [22]

    Wolgemuth C W, Powers T R, Goldstein R E 2000 Phys. Rev. Lett. 84 1623

    [23]

    Liu Y Z, Sheng L W 2007 Chin. Phys. 16 0891

    [24]

    Keller J B, Rubinow S I 1976 J. Fluid Mech. 75 705

    [25]

    Manning R S, Maddocks J H, Kahn J D 1996 J. Chem. Phys. 105 5626

    [26]

    Kehrbaum S 1997 Ph. D. Dissertation (Maryland: University of Maryland, College Park, USA)

    [27]

    Hagerman P 1988 Rev. Biophys. Chem. 17 265

    [28]

    Schlick T 1995 Curr. Opinion Struct. Biol. 5 245

    [29]

    Zajac E E 1962 Trans. ASME. J. Appl. Mech. 29 136

  • [1]

    Beham C J 1977 Proc. Natl. Acad. Sci. USA 74 2397

    [2]

    Le Bret M 1978 Biopolymers 17 1939

    [3]

    Travers A A, Thompson J M T 2004 Phil. Trans. R. Soc. Lond. A 362 1265

    [4]

    Benham C J, Mielke S P 2005 Annu. Rev. Biomed. Eng. 7 21

    [5]

    Shi Y M, Hearst J E 1994 J. Chem. Phys. 101 5186

    [6]

    Zhou H J, Ouyang Z C 1999 J. Chem. Phys. 110 1247

    [7]

    Xue Y, Liu Y Z, Chen L Q 2004 Chin. Phys. 13 794

    [8]

    Wang P, Xue Y 2016 Nonlinear Dyn. 83 1815

    [9]

    Liu Y Z 2006 Nonlinear Mechanics of Thin Elastic Rod-Theoretical Basis of Mechanical Model of DNA (Beijing: Tsinghua Press Springer) p85 (in Chinese) [刘延柱 2006 弹性细杆非线性力学-DNA力学模型的理论基础 (北京: 清华大学出版社 Springer) 第85页]

    [10]

    Bustamante C, Bryant Z 2003 Nature 421 423

    [11]

    Tobias I, Swigon D, Coleman B D 2000 Phys. Rev. E 61 747

    [12]

    Manning R S, Bluman G B 2005 Proc. R. Soc. Lond. A 461 2423

    [13]

    Liu Y Z, Zu J W 2004 Acta Mech. 164 29

    [14]

    Liu Y Z, Sheng L W 2007 Acta Phys. Sin. 56 2305 (in Chinese) [刘延柱, 盛立伟 2007 物理学报 56 2305]

    [15]

    Xue Y, Liu Y Z 2009 Acta Phys. Sin. 58 6737 (in Chinese) [薛纭, 刘延柱 2009 物理学报 58 6737]

    [16]

    Xue Y, Chen L Q, Liu Y Z 2004 Acta Phys. Sin. 53 4029 (in Chinese) [薛纭, 陈立群, 刘延柱 2004 物理学报 53 4029]

    [17]

    Goriely A, Tabor M 1997 Physica D 105 20

    [18]

    Goriely A, Tabor M 1996 Phys. Rev. Lett. 77 3537

    [19]

    Moulton D E, Lessinnes T, Goriely A 2013 J. Mech. Phys. Solids 61 398

    [20]

    Klapper I 1996 J. Comput. Phys. 125 325

    [21]

    Goldstein R E, Powers T R, Wiggins C H 1998 Phys. Rev. Lett. 80 5232

    [22]

    Wolgemuth C W, Powers T R, Goldstein R E 2000 Phys. Rev. Lett. 84 1623

    [23]

    Liu Y Z, Sheng L W 2007 Chin. Phys. 16 0891

    [24]

    Keller J B, Rubinow S I 1976 J. Fluid Mech. 75 705

    [25]

    Manning R S, Maddocks J H, Kahn J D 1996 J. Chem. Phys. 105 5626

    [26]

    Kehrbaum S 1997 Ph. D. Dissertation (Maryland: University of Maryland, College Park, USA)

    [27]

    Hagerman P 1988 Rev. Biophys. Chem. 17 265

    [28]

    Schlick T 1995 Curr. Opinion Struct. Biol. 5 245

    [29]

    Zajac E E 1962 Trans. ASME. J. Appl. Mech. 29 136

  • [1] Jiang Tao, Ren Jin-Lian, Xu Lei, Lu Lin-Guang. A corrected smoothed particle hydrodynamics approach to solve the non-isothermal non-Newtonian viscous fluid flow problems. Acta Physica Sinica, 2014, 63(21): 210203. doi: 10.7498/aps.63.210203
    [2] Chen Li-Qun, Liu Yan-Zhu, Xue Yun. Methods of analytical mechanics for dynamics of the Kirchhoff elastic rod. Acta Physica Sinica, 2006, 55(8): 3845-3851. doi: 10.7498/aps.55.3845
    [3] Cui Jian-Xin, Gao Hai-Bo, Hong Wen-Xue. Mei symmetries and the Noether conserved quantities of super-thin elastic rod. Acta Physica Sinica, 2009, 58(11): 7426-7430. doi: 10.7498/aps.58.7426
    [4] Zhou Guo-Cheng, Cao Jin-Bin, Wei Xin-Hua, Li Liu-Yuan. Low-frequency electromagnetic instabilities in a collisionless current sheet:magnetohydrodynamic model. Acta Physica Sinica, 2005, 54(7): 3228-3235. doi: 10.7498/aps.54.3228
    [5] Wei Qi, E Wen-Ji. Thermodynamic analyses of dewetting instability in thin films. Acta Physica Sinica, 2012, 61(16): 160508. doi: 10.7498/aps.61.160508
    [6] Chen Li-Qun, Xue Yun, Liu Yan-Zhu. Special solutions of Kirchhoff equations and their Lyapunov stability. Acta Physica Sinica, 2004, 53(12): 4029-4036. doi: 10.7498/aps.53.4029
    [7] Zhang Heng, Duan Wen-Shan. The modulational instability of the solition wave in two-dimensional Bose-Einstein condensates. Acta Physica Sinica, 2013, 62(4): 044703. doi: 10.7498/aps.62.044703
    [8] Chen Tao, Zhang Ting, Wang Guang-Chang, Zheng Zhi-Jian, Gu Yu-Qiu. Study of transport of hot electrons in solid targets using transition radiation. Acta Physica Sinica, 2007, 56(2): 982-987. doi: 10.7498/aps.56.982
    [9] Du Hui, Wei Gang, Zhang Yuan-Ming, Xu Xiao-Hui. Experimental investigations on the propagation characteristics of internal solitary waves over a gentle slope. Acta Physica Sinica, 2013, 62(6): 064704. doi: 10.7498/aps.62.064704
    [10] Zhang Peng-Fei, Qiao Chun-Hong, Feng Xiao-Xing, Huang Tong, Li Nan, Fan Cheng-Yu, Wang Ying-Jian. Linearization theory of small scale thermal blooming effect in non-Kolmogorov turbulent atmosphere. Acta Physica Sinica, 2017, 66(24): 244210. doi: 10.7498/aps.66.244210
    [11] Yuan Yong-Teng, Hao Yi-Dan, Hou Li-Fei, Tu Shao-Yong, Deng Bo, Hu Xin, Yi Rong-Qing, Cao Zhu-Rong, Jiang Shao-En, Liu Shen-Ye, Ding Yong-Kun, Miao Wen-Yong. The study of hydrodynamic instability growth measurement. Acta Physica Sinica, 2012, 61(11): 115203. doi: 10.7498/aps.61.115203
    [12] Liu Ying, Chen Zhi-Hua, Zheng Chun. Kelvin-Helmholtz instability in anisotropic viscous magnetized fluid. Acta Physica Sinica, 2019, 68(3): 035201. doi: 10.7498/aps.68.20181747
    [13] Fan Zheng-Feng, Ye Wen-Hua, Sun Yan-Qian, Zheng Bing-Song, Li Ying-Jun, Wang Li-Feng. Kelvin-Helmholtz instability in compressible fluids. Acta Physica Sinica, 2009, 58(9): 6381-6386. doi: 10.7498/aps.58.6381
    [14] MA TENG-CAI, GONG YE. RESISTIVE FLUID INSTABILITY AT THE NONMONOTONOUS CURRENT PROFILE. Acta Physica Sinica, 1984, 33(8): 1112-1119. doi: 10.7498/aps.33.1112
    [15] WANG XIN-YI, LIN LEI. ELECTROHYDRODYNAMIC INSTABILITIES OF NEMATIC LIQUID CRYSTALS——EFFECT OF AN INCLINED ELECTRIC FIELD. Acta Physica Sinica, 1983, 32(12): 1565-1573. doi: 10.7498/aps.32.1565
    [16] Li De-Mei, Lai Hui-Lin, Xu Ai-Guo, Zhang Guang-Cai, Lin Chuan-Dong, Gan Yan-Biao. Discrete Boltzmann simulation of Rayleigh-Taylor instability in compressible flows. Acta Physica Sinica, 2018, 67(8): 080501. doi: 10.7498/aps.67.20171952
    [17] Ye Wen-Hua, Fan Zheng-Feng, Teng Ai-Ping, Tao Ye-Sheng, Lin Chuan-Dong, Li Ying-Jun, Wang Li-Feng. Velocity gradient in Kelvin-Helmholtz instability for supersonic fluid. Acta Physica Sinica, 2009, 58(12): 8426-8431. doi: 10.7498/aps.58.8426
    [18] Fan Zheng-Feng, Ye Wen-Hua, Li Ying-Jun, Wang Li-Feng. Study on the Kelvin-Helmholtz instability in two-dimensional incompressible fluid. Acta Physica Sinica, 2009, 58(7): 4787-4792. doi: 10.7498/aps.58.4787
    [19] Zhang Wei-Yan, ]Ye Wen-Hua, He Xian-Tu, Wu Jun-Feng. Nonlinear threshold of two-dimensional Rayleigh-Taylor growth for incompressible liquid. Acta Physica Sinica, 2003, 52(7): 1688-1693. doi: 10.7498/aps.52.1688
    [20] Yang Xiu-Feng, Liu Mou-Bin. Improvement on stress instability in smoothed particle hydrodynamics. Acta Physica Sinica, 2012, 61(22): 224701. doi: 10.7498/aps.61.224701
  • Citation:
Metrics
  • Abstract views:  179
  • PDF Downloads:  143
  • Cited By: 0
Publishing process
  • Received Date:  14 November 2016
  • Accepted Date:  18 December 2016
  • Published Online:  05 May 2017

Dynamic instability of super-long elastic rod in viscous fluid

    Corresponding author: Wang Peng, sdpengwang@163.com
  • 1. School of Civil Engineering and Architecture, University of Jinan, Jinan 250022, China;
  • 2. School of Mechanical Engineering, Shanghai Institute of Technology, Shanghai 201418, China;
  • 3. Department of Physics, Shaoxing University, Shaoxing 312000, China
Fund Project:  Project supported by the National Nature Science Foundation of China (Grant Nos. 11262019, 11372195, 11472177).

Abstract: The external environment affects the structural form of biological system. Many biological systems are surrounded by cell solutions, such as DNA and bacteria. The solution will offer a viscous resistance as the biological system moves in the viscous fluid. How does the viscous resistance affect the stability of biological system and what mode will be selected after instability? In this paper, we establish a super-long elastic rod model which contains the viscous resistance to model this phenomenon. The stability and instability of the super-long elastic rod in the viscous fluid are studied. The dynamic equations of motion of the super-long elastic rod in viscous fluid are given based on the Kirchhoff dynamic analogy. Then a coordinate basis vector perturbation scheme is reviewed. According to the new perturbation method, we obtain the first order perturbation representation of super-long elastic rod dynamic equation in the viscous fluid, which is a group of the second order linear partial differential equations. The stability of the super-long elastic rod can be determined by analyzing the solutions of the second order linear partial differential equations. The results are applied to a twisted planar DNA ring. The stability criterion of the twisted planar DNA ring and its critical region are obtained. The results show that the viscous resistance has no effect on the stability of super-long elastic rod dynamics, but affects its instability. The mode selection and the influence of the viscous resistance on the instability of DNA ring are discussed. The amplitude of the elastic loop becomes smaller under the influence of the viscous resistance, and a bifurcation occurs. The mode number of instability of DNA loop becomes bigger with the increase of viscous resistance.

Reference (29)

Catalog

    /

    返回文章
    返回