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Reduced-order model for inertial locomotion of a slender swimmer

Raksha Mahalinkam, Felicity Gong, and Aditya S. Khair^*

Department of Chemical Engineering, Carnegie Mellon University, Pittsburgh, Pennsylvania 15213, USA

^*akhair@andrew.cmu.edu

Phys. Rev. E 97, 043102 – Published 3 April, 2018

DOI: https://doi.org/10.1103/PhysRevE.97.043102

Abstract

The inertial locomotion of an elongated model swimmer in a Newtonian fluid is quantified, wherein self-propulsion is achieved via steady tangential surface treadmilling. The swimmer has a length $2 l$ and a circular cross section of longitudinal profile $a R (z)$ , where $a$ is the characteristic width of the cross section, $R (z)$ is a dimensionless shape function, and $z$ is a dimensionless coordinate, normalized by $l$ , along the centerline of the body. It is assumed that the swimmer is slender, $ε = a / l ≪ 1$ . Hence, we utilize slender-body theory to analyze the Navier-Stokes equations that describe the flow around the swimmer. Therefrom, we compute an asymptotic approximation to the swimming speed, $U$ , as $U / u_{s} = 1 - β [V (Re) - \frac{1}{2} \int_{- 1}^{1} z ln R (z) d z] / ln (1 / ε) + O [1 / {ln}^{2} (1 / ε)]$ , where $u_{s}$ is the characteristic speed of the surface treadmilling, $Re$ is the Reynolds number based on the body length, and $β$ is a dimensionless parameter that differentiates between “pusher” (propelled from the rear, $β < 0$ ) and “puller” (propelled from the front, $β > 0$ ) -type swimmers. The function $V (Re)$ increases monotonically with increasing $Re$ ; hence, fluid inertia causes an increase (decrease) in the swimming speed of a pusher (puller). Next, we demonstrate that the power expenditure of the swimmer increases monotonically with increasing $Re$ . Further, the power expenditures of a puller and pusher with the same value of $| β |$ are equal. Therefore, pushers are superior in inertial locomotion as compared to pullers, in that they achieve a faster swimming speed for the same power expended. Finally, it is demonstrated that the flow structure predicted from our reduced-order model is consistent with that from direct numerical simulation of swimmers at intermediate $Re$ .

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