Institute of Fundamental Technological Research, Polish Academy of Sciences, Paw-ińskiego 5B, 02-106, Warsaw, Poland.
Soft Matter. 2019 Sep 18;15(36):7262-7274. doi: 10.1039/c9sm00598f.
We consider elastic microfilaments which form closed loops. We investigate how the loops change shape and orientation while settling under gravity in a viscous fluid. Loops are circular at the equilibrium. Their dynamics are investigated numerically based on the Stokes equations for the fluid motion and the bead-spring model of the microfilament. The Rotne-Prager approximation for the bead mobility is used. We demonstrate that the relevant dimensionless parameter is the ratio of the bending resistance of the filament to the gravitation force corrected for buoyancy. The inverse of this ratio, called the elasto-gravitation number B, is widely used in the literature for sedimenting elastic linear filaments. We assume that B is of the order of 104-106, which corresponds to easily deformable loops. We find out that initially tilted circles evolve towards different sedimentation modes, depending on B. Very stiff or stiff rings attain almost planar, oval shapes, which are vertical or tilted, respectively. More flexible loops deform significantly and converge towards one of several characteristic periodic motions. These sedimentation modes are also detected when starting from various shapes, and for different loop lengths. In general, multi-stability is observed: an elastic ring converges to one of several sedimentation modes, depending on the initial conditions. This effect is pronounced for very elastic loops. The surprising diversity of long-lasting periodic motions and shapes of elastic rings found in this work gives a new perspective for the dynamics of more complex deformable objects at micrometer and nanometer scales, sedimenting under gravity or rotating in a centrifuge, such as red blood cells, ring polymers or circular DNA.
我们研究了形成封闭环的弹性微丝。我们研究了在粘性流体中受重力影响时，环如何改变形状和方向。环在平衡时是圆形的。我们基于流体运动的 Stokes 方程和微丝的珠-簧模型对其动力学进行了数值研究。珠的迁移率采用 Rotne-Prager 近似。我们证明，相关的无量纲参数是纤维的弯曲阻力与浮力修正后的重力之比。该比值的倒数称为弹性-重力数 B，在沉降弹性线性丝的文献中被广泛使用。我们假设 B 的量级为 10^4-10^6，这对应于容易变形的环。我们发现，最初倾斜的圆会根据 B 演变成不同的沉降模式。非常硬或硬的环会达到几乎平面的椭圆形，分别是垂直或倾斜的。更灵活的环会显著变形，并收敛到几种特征周期性运动之一。当从各种形状开始或对于不同的环长度时，也会检测到这些沉降模式。一般来说，会观察到多稳定性：弹性环会根据初始条件收敛到几种沉降模式之一。对于非常弹性的环，这种效果很明显。在这项工作中发现的弹性环持久的周期性运动和形状的多样性为在微米和纳米尺度下受重力沉降或在离心机中旋转的更复杂变形物体的动力学提供了新的视角，例如红细胞、环状聚合物或圆形 DNA。
