Slender-ribbon theory
Slender-ribbon theory
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细带理论
DOI：
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发表时间：
2015
期刊：
影响因子：
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通讯作者：
E. Lauga
中科院分区：
文献类型：
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作者：
Lyndon Koens;E. Lauga
带状物是具有三种不同材料长度尺度（厚度、宽度和长度）的狭长条，这使得它们能够产生线材或长丝无法获得的独特形状。例如，当一个丝带有一半的扭曲和弯曲成一个圆圈，它产生一个M“奥比斯带。在确定带状物的结构形状方面已经做了大量的工作，但对它们在粘性流体中的行为却知之甚少。在本文中，我们确定，渐近，领先阶的流体动力学行为的细长带在斯托克斯流。这个推导让人联想到细丝的细长体理论，它假设带状物的长度远大于其宽度，而宽度本身又远大于其厚度。最后的结果是一个积分方程的力密度的数学直纹面，称为带状平面，位于带内。我们的推导的数值实现显示出良好的协议与已知的流体动力学的长扁椭球体，并成功地捕捉到最近实验探索的人工微观游泳者的游泳行为。我们还研究了弯曲成螺旋线的带状物的渐近行为，即扭曲的椭球体，并且我们研究了带状物的流体动力学如何准确地被细长的细丝有效地捕获。我们的渐近结果提供了必要的基本框架，以预测在各种生物和工程问题的细长带在低雷诺数的行为。
Ribbons are long narrow strips possessing three distinct material length scales (thickness, width, and length) which allow them to produce unique shapes unobtainable by wires or filaments. For example when a ribbon has half a twist and is bent into a circle it produces a M"obius strip. Significant effort has gone into determining the structural shapes of ribbons but less is know about their behavior in viscous fluids. In this paper we determine, asymptotically, the leading-order hydrodynamic behavior of a slender ribbon in Stokes flows. The derivation, reminiscent of slender-body theory for filaments, assumes that the length of the ribbon is much larger than its width, which itself is much larger than its thickness. The final result is an integral equation for the force density on a mathematical ruled surface, termed the ribbon plane, located inside the ribbon. A numerical implementation of our derivation shows good agreement with the known hydrodynamics of long flat ellipsoids, and successfully captures the swimming behavior of artificial microscopic swimmers recently explored experimentally. We also study the asymptotic behavior of a ribbon bent into a helix, that of a twisted ellipsoid, and we investigate how accurately the hydrodynamics of a ribbon can be effectively captured by that of a slender filament. Our asymptotic results provide the fundamental framework necessary to predict the behavior of slender ribbons at low Reynolds numbers in a variety of biological and engineering problems.