Abstract
Over a sufficiently long period of time, or from an appropriate distance, the motion of many swimmers can appear smooth, with their trajectories appearing almost ballistic in nature and slowly varying in character. These long-time behaviours, however, often mask more complex dynamics, such as the side-to-side snakelike motion exhibited by spermatozoa as they swim, propelled by the frequent and periodic beating of their flagellum. Many models of motion neglect these effects in favour of smoother long-term behaviours, which are often of greater practical interest than the small-scale oscillatory motion. Whilst it may be tempting to ignore any yawing motion, simply assuming that any effects of rapid oscillations cancel out over a period, a precise quantification of the impacts of high-frequency yawing is lacking. In this study, we systematically evaluate the long-term effects of general high-frequency oscillations on translational and angular motion, cast in the context of microswimmers but applicable more generally. Via a multiple-scales asymptotic analysis, we show that rapid oscillations can cause a long-term bias in the average direction of progression. We identify sufficient conditions for an unbiased long-term effect of yawing, and we quantify how yawing modifies the speed of propulsion and the effective hydrodynamic shape when in shear flow. Furthermore, we investigate and justify the long-time validity of the derived leading-order solutions and, by direct computational simulation, we evidence the relevance of the presented results to a canonical microswimmer.
Highlights
Swimming in nature is often driven by undulatory deformations, from the snakelike motion of spermatozoa at low Reynolds number to the inertia-dominated movement of fish and aquatic mammals [1]
Though it is tempting to neglect the effects of yawing over long enough timescales, rationalizing this by assuming that the effects of rapid oscillations cancel over a period, a precise investigation of the impacts of high-frequency yawing on the long-term behavior of a swimmer or particle is lacking, even for the simplest models
We have theoretically and numerically explored the effects of high-frequency yawing oscillations on both objects in Stokes flow and the motion of active particles, from the dynamics of a self-propelled particle to the canonical Jeffery’s orbit. In this latter example, employing the asymptotic method of multiple scales as we have done throughout, we have found, perhaps surprisingly, that the slow angular dynamics in the presence of rapid yawing are once again governed by Jeffery’s equation in the plane
Summary
Swimming in nature is often driven by undulatory deformations, from the snakelike motion of spermatozoa at low Reynolds number to the inertia-dominated movement of fish and aquatic mammals [1]. In the study of this swimmer, along with that of many others, a broad variety of methods have been employed of varying degrees of complexity, ranging from the simplest phenomenological models, as we will consider in this work and are common in the large-scale study of active matter [6,7,8], to the most computationally and geometrically intricate [9,10,11,12,13,14] While the latter class of methods inherently includes the effects of rapid yawing in their formulation, it being an emergent property of geometrically faithful study, many simpler models neglect these effects in favour of capturing the smoother, long-term behaviors, which are often of greater interest than the rapid transient motion in applications [6,15,16,17,18,19,20,21,22,23,24]. We will seek to ascertain the effects of a rapidly oscillating φ, modeling the yawing dynamics exhibited by many microswimmers, such as the flagellated spermatozoon or Leishmania mexicana
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