A growing body of work on the dynamics of eukaryotic flagella has noted that their oscillation frequencies are sufficiently high that the viscous penetration depth of unsteady Stokes flow is comparable to the scales over which flagella synchronize. Incorporating these effects into theories of synchronization requires an understanding of the global unsteady flows around oscillating bodies. Yet, there has been no precise experimental test on the microscale of the most basic aspects of such unsteady Stokes flow: the orbits of passive tracers and the position-dependent phase lag between the oscillating response of the fluid at a distant point and that of the driving particle. Here, we report the first such direct Lagrangian measurement of this unsteady flow. The method uses an array of 30 submicron tracer particles positioned by a time-shared optical trap at a range of distances and angular positions with respect to a larger, central particle, which is then driven by an oscillating optical trap at frequencies up to 400 Hz. In this microscale regime, the tracer dynamics is considerably simplified by the smallness of both inertial effects on particle motion and finite-frequency corrections to the Stokes drag law. The tracers are found to display elliptical Lissajous figures whose orientation and geometry are in agreement with a low-frequency expansion of the underlying dynamics, and the experimental phase shift between motion parallel and orthogonal to the oscillation axis exhibits a predicted scaling form in distance and angle. Possible implications of these results for synchronization dynamics are discussed.

中文翻译：

---

使用光学驱动微球直接测量不稳定的微尺度斯托克斯流量

真核鞭毛动力学研究的不断发展，已经注意到它们的振荡频率足够高，以致不稳定的斯托克斯血流的粘性渗透深度可与鞭毛同步的尺度相媲美。将这些影响整合到同步理论中需要了解围绕振荡体的全局不稳定流。然而，对于这种不稳定的斯托克斯流的最基本方面，还没有关于微观尺度的精确实验测试：无源示踪剂的轨道和远处流体的振动响应与远处流体的振动响应之间的位置相关的相位滞后。驱动粒子。在这里，我们报告了这种非恒定流的首次此类直接拉格朗日测量。该方法使用了一个30个亚微米级示踪粒子的阵列，这些粒子由一个分时的光阱相对于一个较大的中心粒子在一定距离和角位置范围内定位，然后由一个振荡的光阱以高达400 Hz的频率驱动。在这种微尺度状态下，示踪剂动力学被粒子运动的惯性效应和斯托克斯阻力定律的有限频率校正的微小程度大大简化了。发现示踪剂显示椭圆的李沙育图形，其方向和几何形状与基础动力学的低频扩展一致，并且平行于振动轴和垂直于振动轴的运动之间的实验相移在距离和角度上显示出预期的缩放形式。