Simulations of the Whirling Instability by the Immersed Boundary Method

When an elastic filament spins in a viscous incompressible fluid it may undergo a whirling instability, as studied asymptotically by Wolgemuth, Powers, and Goldstein [Phys. Rev. Lett., 84 (2000), pp. 16--23]. We use the immersed boundary (IB) method to study the interaction between the elastic filament and the surrounding viscous fluid as governed by the incompressible Navier--Stokes equations. This allows the study of the whirling motion when the shape of the filament is very different from the unperturbed straight state.

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

[2]  William H. Press,et al.  Numerical recipes , 1990 .

[3]  C. Jones,et al.  The bacterial flagellum and flagellar motor: structure, assembly and function. , 1991, Advances in microbial physiology.

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

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

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

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

[8]  S. Childress.  Mechanics of swimming and flying: Frontmatter , 1977 .

[9]  C. R. Calldine.  Change of waveform in bacterial flagella : the role of mechanics at the molecular level , 1978 .

[10]  J. Marsden,et al.  A mathematical introduction to fluid mechanics , 1979 .

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

[12]  B. Kendall.  Nonlinear Dynamics and Chaos , 2001 .

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

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

[15]  M. Lai,et al.  An Immersed Boundary Method with Formal Second-Order Accuracy and Reduced Numerical Viscosity , 2000 .

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

[17]  Joseph B. Keller,et al.  Slender-body theory for slow viscous flow , 1976, Journal of Fluid Mechanics.

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

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

[20]  Charles S. Peskin,et al.  Modeling prosthetic heart valves for numerical analysis of blood flow in the heart , 1980 .

[21]  M. Yanagida,et al.  Structure of straight flagellar filaments from a mutant of Escherichia coli. , 1975, Journal of molecular biology.

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

[23]  C. Peskin,et al.  Heart Simulation by an Immersed Boundary Method with Formal Second-order Accuracy and Reduced Numerical Viscosity , 2001 .

[24]  T. Powers,et al.  Twirling and whirling: viscous dynamics of rotating elastic filaments. , 1999, Physical review letters.

[25]  W. Press,et al.  Numerical Recipes: The Art of Scientific Computing , 1987 .

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