We theoretically investigate the motions of an object immersed in a background flow at a low Reynolds number, generalizing the Jeffery equation for the angular dynamics to the case of an object with n-fold rotational symmetry (n ≥ 3). We demonstrate that when n ≥ 4, the dynamics are identical to those of a helicoidal object for which two parameters related to the shape of the object, namely, the Bretherton constant and a chirality parameter, determine the dynamics. When n = 3, however, we find that the equations require a new parameter that is related to the shape and represents the strength of triangularity. On the basis of detailed symmetry arguments, we show theoretically that microscopic objects can be categorized into a small number of classes that exhibit different dynamics in a background flow. We perform further analyses of the angular dynamics in a simple shear flow, and we find that the presence of triangularity can lead to chaotic angular dynamics, although the dynamics typically possess stable periodic orbits, as further demonstrated by an example of a triangular object. Our findings provide a comprehensive viewpoint concerning the dynamics of an object in a flow, emphasizing the notable simplification of the dynamics resulting from the symmetry of the object’s shape, and they will be useful in studies of fluid–structure interactions at a low Reynolds number.

中文翻译：

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具有离散旋转对称性的物体的杰弗里轨道

我们从理论上研究了物体在低雷诺数下浸入背景流中的运动，将角动力学的杰弗里方程推广到具有 n 重旋转对称性 (n ≥ 3) 的物体的情况。我们证明，当 n ≥ 4 时，动力学与螺旋形物体的动力学相同，其中与物体形状相关的两个参数，即 Bretherton 常数和手性参数，决定了动力学。然而，当 n = 3 时，我们发现方程需要一个与形状相关并表示三角形强度的新参数。基于详细的对称性论证，我们从理论上表明，微观对象可以分为少数类别，这些类别在背景流中表现出不同的动态。我们对简单剪切流中的角动力学进行了进一步分析，我们发现三角形的存在会导致角动力学混乱，尽管动力学通常具有稳定的周期轨道，正如三角形物体的例子所进一步证明的那样。我们的研究结果提供了关于流体中物体动力学的综合观点，强调了由于物体形状的对称性而显着简化了动力学，它们将有助于研究低雷诺数下的流固耦合。