title: Nonlinear dynamics of a rotating elastic rod in a viscous fluid.
abstract: The dynamics of an elastic rod in a viscous fluid at zero Reynolds number is investigated when the bottom end of the rod is tethered at a point in space and rotates at a prescribed angular frequency, while the other part of the rod freely moves through the fluid. A rotating elastic rod, which is intrinsically straight, exhibits three dynamical motions: twirling, overwhirling, and whirling. The first two motions are stable, whereas the last motion is unstable. The stability of dynamical motions is determined by material and geometrical properties of the rod, fluid properties, and the angular frequency of the rod. We employ the regularized Stokes flow to describe the fluid motion and the Kirchhoff rod model to describe the elastic rod. Our simulation results display subcritical Hopf bifurcation diagrams indicating the bistability region. We also investigate the whirling motion generated by the rotation of an intrinsically bent rod. It is observed that the angular frequency determines the handedness of the whirling rod and thus the flow direction and that there is a critical frequency which separates the positive (upward) flow at frequencies above it from the negative (downward) flow at frequencies below it.
authors.affiliation: National Institute for Mathematical Sciences, KT Daeduk 2 Research Center, 70, Yuseong-daero 1689-gil, Yuseong-gu, Daejeon 305-811, Republic of Korea.
authors.forename: Wanho
authors.initials: W
authors.lastname: Lee
authors.affiliation: Department of Mathematics, Chung-Ang University, Dongjakgu, Heukseokdong, Seoul 156-756, Republic of Korea.
authors.forename: Yongsam
authors.initials: Y
authors.lastname: Kim
authors.affiliation: Department of Mathematical Sciences, Worcester Polytechnic Institute, 100 Institute Road Worcester, Massachusetts 01609, USA.
authors.forename: Sarah D
authors.initials: SD
authors.lastname: Olson
authors.affiliation: Department of Mathematical Sciences, University of Cincinnati, 4199 French Hall West, Cincinnati, Ohio 45221, USA.
authors.forename: Sookkyung
authors.initials: S
authors.lastname: Lim
mesh_terms: D004548:Elasticity; D017711:Nonlinear Dynamics; D012399:Rotation; D014783:Viscosity