### Abstract

The dynamics of slender filaments or fibers suspended in Stokesian fluids are fundamental to understanding many flows arising in physics, biology and engineering. Such filaments can have aspect ratios of length to radius ranging from a few tens to several thousands. Full discretizations of such 3D flows are very costly. Instead, we employ a non-local slender body theory that yields an integral equation, along the filament centerline, relating the force exerted on the body to the filament velocity. This hydrodynamical description takes into account the effect of the filament on the fluid, and is extended to capture the interaction of multiple filaments as mediated by the intervening fluid. We consider filaments that are inextensible and elastic. Replacing the force in the slender body integral equation by an explicit expression that uses Euler-Bernoulli theory to model bending and tensile forces yields an integral expression for the velocity of the filament centerlines, coupled to auxiliary integro-differential equations for the filament tensions. Based on a regularized version of these slender body equations that is asymptotically equivalent to the original formulation, we construct a numerical method which uses a combination of finite differences, implicit time-stepping to avoid severe stability constraints, and special quadrature methods for nearly singular integrals. We present simulations of single flexible filaments, as well as multiple interacting filaments, evolving in a background shear flow. These simulations show shear induced buckling and relaxation of the filaments, leading to the storage and release of elastic energy. These dynamics are responsible for the development of positive first normal stress differences, commonly associated with visco-elastic fluids that are suspensions of microscopic elastic fibers.

Original language | English (US) |
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Pages (from-to) | 8-40 |

Number of pages | 33 |

Journal | Journal of Computational Physics |

Volume | 196 |

Issue number | 1 |

DOIs | |

State | Published - May 1 2004 |

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### ASJC Scopus subject areas

- Computer Science Applications
- Physics and Astronomy(all)

### Cite this

*Journal of Computational Physics*,

*196*(1), 8-40. https://doi.org/10.1016/j.jcp.2003.10.017

**Simulating the dynamics and interactions of flexible fibers in Stokes flows.** / Tornberg, Anna Karin; Shelley, Michael.

Research output: Contribution to journal › Article

*Journal of Computational Physics*, vol. 196, no. 1, pp. 8-40. https://doi.org/10.1016/j.jcp.2003.10.017

}

TY - JOUR

T1 - Simulating the dynamics and interactions of flexible fibers in Stokes flows

AU - Tornberg, Anna Karin

AU - Shelley, Michael

PY - 2004/5/1

Y1 - 2004/5/1

N2 - The dynamics of slender filaments or fibers suspended in Stokesian fluids are fundamental to understanding many flows arising in physics, biology and engineering. Such filaments can have aspect ratios of length to radius ranging from a few tens to several thousands. Full discretizations of such 3D flows are very costly. Instead, we employ a non-local slender body theory that yields an integral equation, along the filament centerline, relating the force exerted on the body to the filament velocity. This hydrodynamical description takes into account the effect of the filament on the fluid, and is extended to capture the interaction of multiple filaments as mediated by the intervening fluid. We consider filaments that are inextensible and elastic. Replacing the force in the slender body integral equation by an explicit expression that uses Euler-Bernoulli theory to model bending and tensile forces yields an integral expression for the velocity of the filament centerlines, coupled to auxiliary integro-differential equations for the filament tensions. Based on a regularized version of these slender body equations that is asymptotically equivalent to the original formulation, we construct a numerical method which uses a combination of finite differences, implicit time-stepping to avoid severe stability constraints, and special quadrature methods for nearly singular integrals. We present simulations of single flexible filaments, as well as multiple interacting filaments, evolving in a background shear flow. These simulations show shear induced buckling and relaxation of the filaments, leading to the storage and release of elastic energy. These dynamics are responsible for the development of positive first normal stress differences, commonly associated with visco-elastic fluids that are suspensions of microscopic elastic fibers.

AB - The dynamics of slender filaments or fibers suspended in Stokesian fluids are fundamental to understanding many flows arising in physics, biology and engineering. Such filaments can have aspect ratios of length to radius ranging from a few tens to several thousands. Full discretizations of such 3D flows are very costly. Instead, we employ a non-local slender body theory that yields an integral equation, along the filament centerline, relating the force exerted on the body to the filament velocity. This hydrodynamical description takes into account the effect of the filament on the fluid, and is extended to capture the interaction of multiple filaments as mediated by the intervening fluid. We consider filaments that are inextensible and elastic. Replacing the force in the slender body integral equation by an explicit expression that uses Euler-Bernoulli theory to model bending and tensile forces yields an integral expression for the velocity of the filament centerlines, coupled to auxiliary integro-differential equations for the filament tensions. Based on a regularized version of these slender body equations that is asymptotically equivalent to the original formulation, we construct a numerical method which uses a combination of finite differences, implicit time-stepping to avoid severe stability constraints, and special quadrature methods for nearly singular integrals. We present simulations of single flexible filaments, as well as multiple interacting filaments, evolving in a background shear flow. These simulations show shear induced buckling and relaxation of the filaments, leading to the storage and release of elastic energy. These dynamics are responsible for the development of positive first normal stress differences, commonly associated with visco-elastic fluids that are suspensions of microscopic elastic fibers.

UR - http://www.scopus.com/inward/record.url?scp=1942475755&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=1942475755&partnerID=8YFLogxK

U2 - 10.1016/j.jcp.2003.10.017

DO - 10.1016/j.jcp.2003.10.017

M3 - Article

AN - SCOPUS:1942475755

VL - 196

SP - 8

EP - 40

JO - Journal of Computational Physics

JF - Journal of Computational Physics

SN - 0021-9991

IS - 1

ER -