Jeffery orbits for an object with discrete rotational symmetry
Kenta Ishimoto
Abstract
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.