Hydrodynamic interactions are important in biophysics research because they influence both the collective and the individual behavior of micro-organisms and self-propelled particles. For instance, hydrodynamic interactions at the microswimmer level determine the attraction or repulsion between individuals, and hence the rise of collective behavior. Meanwhile, hydrodynamic interactions between swimming appendages (e.g., cilia and flagella) influence the emergence of swimming gaits, synchronized bundles, and metachronal waves, and hence the propulsive capacity of the individual swimmer. In this study we address the issue of hydrodynamic interactions between slender filaments separated by a distance larger than their contour length $(d>L)$ by means of asymptotic calculations and numerical simulations. We first derive analytical expressions for the extended resistance matrix of two arbitrarily shaped rigid filaments, as a series expansion in inverse powers of $d/L>1$. The coefficients in our asymptotic series expansion are then evaluated using two well-established methods for slender filaments, resistive-force theory (RFT) and slender-body theory (SBT), and our asymptotic theory is verified using numerical simulations based on SBT for the case of two parallel helical filaments. The theory is able to capture the qualitative features of the interactions in the regime $d/L>1$, which opens the path to a deeper physical understanding of hydrodynamically governed phenomena such as interfilament synchronization and multiflagellar propulsion. To demonstrate the usefulness of our results, we next apply our theory to the case of two helical filaments rotating side-by-side, where we quantify the dependence of all forces and torques on the distance and phase difference between the helices. Using our understanding of pairwise interactions, we then provide physical intuition for the case of a circular array of rotating helices. Our theoretical results will be useful for the study of hydrodynamic effects that emerge between interacting bacterial flagella, nodal cilia, and slender microswimmers, both artificial and biological.

中文翻译：

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细长丝之间流体动力学相互作用的渐近理论

流体动力学相互作用在生物物理学研究中很重要，因为它们影响微生物和自推进粒子的集体和个体行为。例如，微型游泳者级别的流体动力学相互作用决定了个体之间的吸引力或排斥力，从而导致集体行为的兴起。同时，游泳附属物（例如纤毛和鞭毛）之间的水动力相互作用影响游泳步态、同步束和异时波的出现，从而影响个体游泳者的推进能力。在这项研究中，我们解决了间距大于轮廓长度的细长丝之间的流体动力学相互作用问题$(d>升)$通过渐近计算和数值模拟。我们首先推导出两个任意形状的刚性细丝的扩展电阻矩阵的解析表达式，作为$d/升>1$. 我们的渐近级数展开中的系数然后使用两种成熟的细长细丝方法，电阻力理论 (RFT) 和细长体理论 (SBT) 进行评估，并且我们的渐近理论使用基于 SBT 的数值模拟来验证两条平行螺旋丝的情况。该理论能够捕捉制度中相互作用的定性特征$d/升>1$，这为更深入地了解流体动力学控制的现象（例如丝间同步和多鞭毛推进）开辟了道路。为了证明我们的结果的有用性，我们接下来将我们的理论应用于两个并排旋转的螺旋丝的情况，在那里我们量化所有力和扭矩对螺旋之间的距离和相位差的依赖性。利用我们对成对相互作用的理解，我们为旋转螺旋的圆形阵列的情况提供了物理直觉。我们的理论结果将有助于研究相互作用的细菌鞭毛、结节纤毛和细长的微型游泳者（人工和生物）之间出现的流体动力学效应。