Initially straight slender elastic filaments and rods with geometrically constrained ends buckle and form stable two-dimensional shapes when compressed by bringing the ends together. It is known that beyond a critical value of this pre-stress, clamped rods transition to bent, twisted three-dimensional equilibrium shapes. Here, we analyze the three-dimensional instabilities and dynamics of such pre- stressed, initially twisted filaments subject to active follower forces and dissipative fluid drag. We find that degree of boundary constraint and the directionality of active forces determines if oscillatory instabilities can arise. When filaments are clamped at one end and pinned at the other with follower forces directed towards the clamped end, stable planar flapping oscillations result; reversing the directionality of the active forces quenches the instability. When both ends are clamped however, computations reveal a novel secondary instability wherein planar oscillations are destabilized by off-planar perturbations resulting in three-dimensional swirling patterns with periodic flips. These swirl-flip transitions are characterized by two distinct and time-scales. The first corresponds to unidirectional swirling rotation around the end-to-end axis. The second captures the time between flipping events when the direction of swirling reverses. We find that this spatiotemporal dance resembles relaxation oscillations with each cycle initiated by a sudden jump in torsional deformation and then followed by a period of gradual decrease in net torsion until the next cycle of variations. Our work reveals the rich tapestry of spatiotemporal patterns when weakly inertial strongly damped rods are deformed by non-conservative active forces. Practically, our results suggest avenues by which pre-stress, elasticity and activity may be used to design synthetic fluidic elements to pump or mix fluid at macroscopic length scales.

中文翻译：

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拍打，涡流和翻转：预应力有源细丝的非线性动力学

最初，细长的弹性长丝和带有受几何约束的端部的棒弯曲并通过将端部放在一起压缩而形成稳定的二维形状。已知的是，超过该预应力的临界值，夹紧的杆过渡到弯曲的，扭曲的三维平衡形状。在这里，我们分析了这种预应力的，最初加捻的细丝在主动从动力和耗散流体阻力作用下的三维不稳定性和动力学。我们发现边界约束的程度和主动力的方向性决定了是否会出现振荡不稳定性。当细丝的一端被夹紧而另一端被钉住，而从动力则指向被夹紧的一端，因此会产生稳定的平面拍打振荡。反向作用力的方向性可以消除不稳定性。但是，当两端都被夹紧时，计算会发现一种新的二次不稳定性，其中平面振荡会因非平面扰动而不稳定，从而导致带有周期性翻转的三维涡旋模式。这些旋涡翻转转变的特征是两个不同的时间尺度。第一个对应于围绕端到端轴的单向回旋旋转。第二个捕捉旋转方向相反时翻转事件之间的时间。我们发现，这种时空舞动类似于放松振荡，每个周期都是由扭转变形的突然跳跃引发的，然后是逐渐减小的净扭转直到下一个变化周期。我们的工作揭示了当弱惯性强阻尼杆由于非保守作用力而变形时，时空模式的丰富挂毯。实际上，我们的结果提出了预应力，弹性和活动性可用于设计合成流体元件的途径，从而可以在宏观长度尺度上泵送或混合流体。