Gait and speed selection in slender inertial swimmers

Abstract

Inertial swimmers use flexural movements to push water and generate thrust. We quantify this dynamical process for a slender body in a fluid by accounting for passive elasticity and hydrodynamics and active muscular force generation and proprioception. Our coupled elastohydrodynamic model takes the form of a nonlinear eigenvalue problem for the swimming speed and locomotion gait. The solution of this problem shows that swimmers use quantized resonant interactions with the fluid environment to enhance speed and efficiency. Thus, a fish is like an optimized diode that converts a prescribed alternating transverse motion to forward motion. Our results also allow for a broad comparative view of swimming locomotion and provide a mechanistic basis for the empirical relation linking the swimmer's speed U, length L, and tail beat frequency f, given by U=L ∼ f [Bainbridge R (1958) J Exp Biol 35:109-133]. Furthermore, we show that a simple form of proprioceptive sensory feedback, wherein local muscle activation is function of body curvature, suffices to drive elastic instabilities associated with thrust production and leads to a spontaneous swimming gait without the need for a central pattern generator. Taken together, our results provide a simple mechanistic view of swimming consistent with natural observations and suggest ways to engineer artificial swimmers for optimal performance.inertial swimming | gait selection | proprioception U nderstanding locomotory behavior requires that we integrate the neural control of muscular dynamics with the body mechanics of the organism as it interacts with the environment. Simultaneously, we must also account for sensory feedback from the environment and from the organism's sense of its own shape. However, translating these concepts to quantitative theories is challenging because of the variety of organism sizes and shapes and the complexity of their physical and biological environment. Therefore, investigations typically focus on specific organisms and try to glean principles that might be of broader relevance. In the context of terrestrial locomotion, recent experimental and theoretical work on the undulating worm Caenorhabditis elegans (1) shows that proprioceptive feedback suffices to coordinate undulatory motions. Complementing these studies, general theoretical models have explored the conditions under which coordinated crawling occurs in a coupled brain-body-environment system (2). These models explain observations in larvae of Drosophila melanogaster, showing how a sensorimotor coupling that links brain, body, and environment can robustly lead to crawling in a range of conditions.Translating these ideas to macroscopic aquatic locomotion, when inertia dominates viscous forces, is a formidable challenge owing to the presence of complex hydrodynamic nonlinearities, which must be coupled to body mechanics and neural dynamics of proprioceptive and sensory feedback. The hydrodynamics of swimming has been the subject of a variety of studies for more than half a century from experimen...

Study of the thrust–drag balance with a swimming robotic fish

“…For elastic plates forced by an oscillating torque distribution, the highest velocities are also observed near the resonant frequency forcing. 21 Data collected from natural swimmers show that the swimming characteristics in the turbulent regime, i.e., a Reynolds number Re larger than 3000, 22 are governed by the relative uniformity of two dimensionless numbers, the Strouhal number St = Af /U 0 ∼ 0.3 [22][23][24][25][26][27] and the ratio A/L ∼ 0.2, [26][27][28][29] with swimming velocity U 0 , fish length L, tailbeat amplitude A, and frequency f. To understand these data in terms of swimming efficiency, numerous Refs. 4-7, 10, and 30-34 have assumed that the mean thrust F T scales with the dynamic pressure and that the thrust coefficient should be defined as F T / 1 2 ρS T U 2 0 , where ρ is the fluid density and a) Author to whom correspondence should be addressed: mederic.argentina@ unice.fr S T is the effective propulsive area of the fish.…”

“…They also reported that in the laminar regime (Reynolds number smaller than 3000), the relevant dimensionless parameter changes due to a change in the drag expression to account for the viscous boundary layer. Actually a subtle hypothesis is hidden behind this simple balance: 21 to the leading order, thrust and drag can be treated separately, with thrust mostly independent of the swimming velocity 11,21,35 and drag mostly independent of the fin frequency or amplitude. 21 It is remarkable that this idea was already in the mind of Lighthill 8 and that his definition of the thrust coefficient C T seems to be the natural one, 9,36…”

Modelling of an actuated elastic swimmer

“…Borazjani & Sotiropoulos 2010;van Rees et al 2013van Rees et al , 2015Li et al 2016). It should be noted that the nonlinear character of this resistive term sets it apart from the viscous dissipation that has been included in other models of similar problems (Argentina & Mahadevan 2005;Gazzola et al 2015). Concerning artificial elastic swimmers with localised actuation, Ramananarivo, Godoy-Diana & Thiria (2014a) showed that the resistive term is determinant for the establishment of the propagative wave that mimics the kinematics of animal swimmers, and Paraz et al (2016) have incorporated it in a successful model predicting thrust generation of a heaving foil.…”

Phototactic guidance of a tissue-engineered soft-robotic ray

“…Inspired by histological analysis (fig. S1) as well as theoretical considerations (26), we channeled muscular work and elastic energy into forward motion by breaking fore-aft symmetry through a varying body rigidity along the anterior-posterior axis. This was achieved via a thicker body, and a denser and radially further reaching skeleton pattern in the front (Fig.…”