Paul M R, Clark M T, Cross M C
Department of Mechanical Engineering, Virginia Tech, Blacksburg, Virginia 24061, USA.
Phys Rev E Stat Nonlin Soft Matter Phys. 2013 Oct;88(4):043012. doi: 10.1103/PhysRevE.88.043012. Epub 2013 Oct 24.
We study the coupled dynamics of two closely spaced micron or nanoscale elastic objects immersed in a viscous fluid. The dynamics of the elastic objects are coupled through the motion of the surrounding viscous fluid. We consider two cases: (i) one object is driven externally by an imposed harmonic actuation force and the second object is passive and (ii) both objects are driven by a Brownian force to yield stochastic dynamics. Using a harmonic oscillator approximation for the elastic objects and the unsteady Stokes equations to describe the fluid dynamics, we develop analytical expressions for the amplitude and phase of the displacement of the oscillating objects. For the case of an imposed actuation we use an impulse in force to determine the resulting dynamics over all frequencies. For the Brownian-driven objects the stochastic dynamics are found using the fluctuation-dissipation theorem. We validate our theoretical expressions by comparison with results from finite-element numerical simulations of the complete fluid-solid interaction problem. Our results yield interesting features in the amplitude and phase of the displacement of the elastic objects due to the fluid motion. We find that the dynamics depend on the separation of the objects, a measure of the mass loading due to the fluid, and the frequency parameter which acts as a frequency-based Reynolds number. Our results are valid over the range of parameters typical of micron and nanoscale elastic objects in fluid. The range of dynamics found can be understood in terms of the interplay between the viscous and potential components of the fluid flow field described by the unsteady Stokes equation for an oscillating cylinder. For small values of the frequency parameter, typical of nanoscale elastic objects, the dynamics are overdamped due to the dominance of viscous forces over inertial forces. For moderate and large values of the frequency parameter, typical of micron-scale elastic objects, we find that the dynamics of the fluid-coupled objects exhibits an interesting mode splitting to yield a bimodal signature in the amplitude-frequency plots. We find that the mode splitting can be described using a normal mode analysis containing only potential fluid interactions between the cylinders.
我们研究了浸没在粘性流体中的两个紧密间隔的微米或纳米级弹性物体的耦合动力学。弹性物体的动力学通过周围粘性流体的运动相互耦合。我们考虑两种情况：（i）一个物体由施加的谐波驱动力外部驱动，第二个物体是被动的；（ii）两个物体都由布朗力驱动以产生随机动力学。使用弹性物体的谐振子近似和非定常斯托克斯方程来描述流体动力学，我们推导出了振荡物体位移的幅度和相位的解析表达式。对于施加驱动力的情况，我们使用力脉冲来确定所有频率下的结果动力学。对于布朗驱动的物体，使用涨落耗散定理来发现随机动力学。我们通过与完整流固相互作用问题的有限元数值模拟结果进行比较，验证了我们的理论表达式。我们的结果在弹性物体由于流体运动而产生的位移幅度和相位方面产生了有趣的特征。我们发现动力学取决于物体之间的间距、由于流体引起的质量负载的度量以及作为基于频率的雷诺数的频率参数。我们的结果在流体中微米和纳米级弹性物体的典型参数范围内有效。所发现的动力学范围可以根据振荡圆柱体的非定常斯托克斯方程所描述的流体流场的粘性和势分量之间的相互作用来理解。对于纳米级弹性物体典型的小频率参数值，由于粘性力对惯性力的主导，动力学是过阻尼的。对于微米级弹性物体典型的中等和大频率参数值，我们发现流体耦合物体的动力学表现出有趣的模式分裂，在幅度 - 频率图中产生双峰特征。我们发现模式分裂可以使用仅包含圆柱体之间势流体相互作用的正常模式分析来描述。
