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.
Highlights
Ever since Stokes1 first derived an expression for the hydrodynamic drag on a moving sphere in a viscous fluid, the so-called Stokes law of drag, microscopic particles in fluids have usually been modeled as spheres
II, we introduce hydrodynamics at a low Reynolds number to describe the dynamics of an object in a background flow, together with the symmetry arguments regarding resistance tensors under n-fold rotational symmetry, demonstrating that when n ≥ 4, the dynamics coincide with those obtained for a helicoidal object, which is a C4-object
We have theoretically investigated the dynamics of a microscopic object, which can be either a passive particle or an active swimmer, with a n-fold rotational symmetry (n ≥ 3), which we have referred to in this paper as a Cn-object, using the Schoenflies notation
Summary
Ever since Stokes first derived an expression for the hydrodynamic drag on a moving sphere in a viscous fluid, the so-called Stokes law of drag, microscopic particles in fluids have usually been modeled as spheres. The primary aim of our paper is to generalize the Jeffery equations to an arbitrary microscopic object, which can be either a passive particle or an active swimmer, with n-fold rotational symmetry but without a mirror plane, where n ≥ 3. Our secondary aim is to classify the shape of an object based on its dynamics in a flow, showing that the number of parameters in the generalized Jeffery equations is reduced from the original degrees of freedom in the resistance tensors Another important type of discrete symmetry can be found in a particle with triaxially mirror-symmetric planes, such as an ellipsoid.
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