Lee Wanho, Kim Yongsam, Olson Sarah D, Lim Sookkyung
National Institute for Mathematical Sciences, KT Daeduk 2 Research Center, 70, Yuseong-daero 1689-gil, Yuseong-gu, Daejeon 305-811, Republic of Korea.
Department of Mathematics, Chung-Ang University, Dongjakgu, Heukseokdong, Seoul 156-756, Republic of Korea.
Phys Rev E Stat Nonlin Soft Matter Phys. 2014 Sep;90(3):033012. doi: 10.1103/PhysRevE.90.033012. Epub 2014 Sep 22.
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.
当弹性杆的底端固定在空间中的某一点并以规定的角频率旋转，而杆的其他部分在粘性流体中自由移动时，研究了零雷诺数下粘性流体中弹性杆的动力学。一根本质上是直的旋转弹性杆呈现出三种动力学运动：扭转、过度扭转和回旋。前两种运动是稳定的，而最后一种运动是不稳定的。动力学运动的稳定性由杆的材料和几何特性、流体特性以及杆的角频率决定。我们采用正则化斯托克斯流来描述流体运动，采用基尔霍夫杆模型来描述弹性杆。我们的模拟结果显示了亚临界霍普夫分岔图，表明了双稳区域。我们还研究了本质上弯曲的杆的旋转产生的回旋运动。据观察，角频率决定了回旋杆的旋向，从而决定了流动方向，并且存在一个临界频率，该频率将其上方频率下的正向（向上）流动与下方频率下的负向（向下）流动分开。
