Dynamics of a Closed Rod with Twist and Bend in Fluid

We investigate the instability and subsequent dynamics of a closed rod with twist and bend in a viscous, incompressible fluid. A new version of the immersed boundary (IB) method is used in which the immersed boundary applies torque as well as force to the surrounding fluid and in which the equations of motion of the immersed boundary involve the local angular velocity as well as the local linear velocity of the fluid. An important feature of the IB method in this context is that self-crossing of the rod is automatically avoided because the rod moves in a continuous (interpolated) velocity field. A rod with a uniformly distributed twist that has been slightly perturbed away from its circular equilibrium configuration is used as an initial condition, with the fluid initially at rest. If the twist in the rod is sufficiently small, the rod simply returns to its circular equilibrium configuration, but for larger twists that equilibrium configuration becomes unstable, and the rod undergoes large excursions before relaxing to a stable coiled configuration.

[1]  S. Antman.  Nonlinear problems of elasticity , 1994 .

[2]  T Schlick,et al.  Modeling superhelical DNA: recent analytical and dynamic approaches. , 1995, Current opinion in structural biology.

[3]  Ricardo Cortez,et al.  Investigation of bend and shear waves in a geometrically exact elastic rod model , 2004 .

[4]  T. Strick,et al.  Behavior of supercoiled DNA. , 1998, Biophysical journal.

[5]  C. Peskin,et al.  Fluid Dynamics of the Heart and its Valves , 1996 .

[6]  C. Peskin.  Flow patterns around heart valves: A numerical method , 1972 .

[7]  W. Olson,et al.  Finite element analysis of DNA supercoiling , 1993 .

[8]  N H Mendelson,et al.  Helical growth of Bacillus subtilis: a new model of cell growth. , 1976, Proceedings of the National Academy of Sciences of the United States of America.

[9]  Charles S. Peskin,et al.  Two-Dimensional Simulations of Valveless Pumping Using the Immersed Boundary Method , 2001, SIAM J. Sci. Comput..

[10]  D. Gaver,et al.  A microscale model of bacterial swimming, chemotaxis and substrate transport. , 1995, Journal of theoretical biology.

[11]  E. Givelberg.  Modeling elastic shells immersed in fluid , 2003 .

[12]  Kathleen A. Hoffman,et al.  Link, twist, energy, and the stability of DNA minicircles. , 2003, Biopolymers.

[13]  N H Mendelson,et al.  Bacterial growth and division: genes, structures, forces, and clocks. , 1982, Microbiological reviews.

[14]  I. Klapper.  Biological Applications of the Dynamics of Twisted Elastic Rods , 1996 .

[15]  B. D. Coleman,et al.  Elastic stability of DNA configurations. I. General theory. , 2000, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[16]  C. Peskin.  Numerical analysis of blood flow in the heart , 1977 .

[17]  A. Love.  A treatise on the mathematical theory of elasticity , 1892 .

[18]  Tamar Schlick,et al.  Trefoil Knotting Revealed by Molecular Dynamics Simulations of Supercoiled DNA , 1992, Science.

[19]  C. Peskin.  The immersed boundary method , 2002, Acta Numerica.

[20]  N H Mendelson,et al.  Mechanics of bacterial macrofiber initiation , 1995, Journal of bacteriology.

[21]  David Swigon,et al.  Theory of self–contact in Kirchhoff rods with applications to supercoiling of knotted and unknotted DNA plasmids , 2004, Philosophical Transactions of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[22]  H. Berg.  The rotary motor of bacterial flagella. , 2003, Annual review of biochemistry.

[23]  Charles S. Peskin,et al.  Simulations of the Whirling Instability by the Immersed Boundary Method , 2004, SIAM J. Sci. Comput..

[24]  P. Strevens.  Iii , 1985 .

[25]  E. Dill,et al.  Kirchhoff's theory of rods , 1992 .

[26]  Alain Goriely,et al.  Nonlinear dynamics of filaments II. nonlinear analysis , 1997 .

[27]  C. Peskin,et al.  Simulation of a Flapping Flexible Filament in a Flowing Soap Film by the Immersed Boundary Method , 2002 .

[28]  Alain Goriely,et al.  Nonlinear dynamics of filaments I. Dynamical instabilities , 1997 .

[29]  B. D. Coleman,et al.  Elastic stability of DNA configurations. II. Supercoiled plasmids with self-contact. , 2000, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[30]  L. Fauci,et al.  A computational model of aquatic animal locomotion , 1988 .

[31]  Raymond E. Goldstein,et al.  Dynamic Supercoiling Bifurcations of Growing Elastic Filaments , 2003, cond-mat/0312563.

[32]  Alain Goriely,et al.  The Nonlinear Dynamics of Filaments , 1997 .

[33]  Raymond E. Goldstein,et al.  VISCOUS NONLINEAR DYNAMICS OF TWIST AND WRITHE , 1998 .

[34]  R. Stephenson.  A and V , 1962, The British journal of ophthalmology.

[35]  A. Fogelson.  A MATHEMATICAL MODEL AND NUMERICAL METHOD FOR STUDYING PLATELET ADHESION AND AGGREGATION DURING BLOOD CLOTTING , 1984 .