On the motion of flexible strings and filaments in inertial and viscous regimes

Abstract

Study of the dynamics of strings and filaments has broad applications, for instance, macroscopic coil motion in petroleum engineering and microscopic one-armed swimmers in biological science. In this work, we study the motions of flexible strings and thin filaments in two different regimes, inertial and viscous, theoretically and experimentally. Quantitative experiments on the whirling string show that steady motion exists only when the string whirls at its natural frequencies and that whirling motions for other frequencies exhibit rich dynamics. Furthermore, three kinds of response have been observed experimentally for the planar excitation: planar steady oscillation; two-dimensional (2D) to three-dimensional (3D) transient response; 3D steady whirling motion. These phenomena repeat as the driving frequency is increased. The forced response of a string subjected to planar excitation is analyzed through a perturbation technique and multiple time scale method. The steady-state whirling motion of linear elastic filaments under self-weight with rotary excitation at one end and free at the other has been examined; specifically, the effect of bending stiffness has been investigated both theoretically and experimentally. The theoretical predictions have been compared with the experimental results for thin filaments with different bending stiffness to demonstrate the effect of bending stiffness directly. The dynamic response of thin filaments under planar excitation has also been studied experimentally. The two-dimensional dynamics of an Euler elastica in low-Reynolds number regime has been studied. Tension effects have been shown to be either comparable to or dominant over the bending contributions for the microscopic one-armed swimmers. Hence one may change the tension in situ through the externally or internally generated forces, thus changing the effective bending stiffness, and as a consequence controlling the swimming velocity and the propulsion efficiency. Finally, the low-Reynolds-number dynamics of a micro-string has been studied, in order to understand the physics underpinning eukaryotic sperm flagellar swimming. Both linear analysis of small-amplitude swimming and fully numerical simulations show that time-reversal symmetry is broken, which leads to the propulsion. Numerical studies have been performed for different boundary conditions and different forcing levels. Comparison with previous bending model illustrated that, for the same equivalent bending stiffness, the micro-string has higher propulsion efficiency with similar swimming velocity. Excellent agreement between the simulation predictions and the experimentally observed flagellar wave-forms has been obtained. With this theoretical model, observations of swimming characteristics of the sperm of different species are reconciled into a single scaling relationship, characterized by the so-called \string sperm number". Our results imply that tension plays a crucial role in flagellar elasticity and provides impetus for studying a different model underlying the physics of flagellar swimming. For example, it is possible to postulate alternate hypotheses for active force generation by the dynein motors; it also enables the formulation of a different role to the micro-filaments in general, one based on tension rather than one based on bending.