PHYSICAL REVIEW E 97, 043102 (2018) 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 (Received 22 January 2018; published 3 April 2018) 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 2l and a circular cross section of longitudinal profile aR(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, ǫ = 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) − 1 2  1 −1 z ln R(z) dz]/ ln(1/ǫ ) + O[1/ ln 2 (1/ǫ )], 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. DOI: 10.1103/PhysRevE.97.043102 I. INTRODUCTION The application of reduced-order models to mathematically describe locomotion of swimming organisms was initiated by Taylor [1], who considered transverse oscillations of a two- dimensional infinite sheet as a model for flagellar propulsion at zero Reynolds number Re. Taylor’s pioneering work has been generalized in many problems involving microscale propulsion, such as hydrodynamic interactions between or- ganisms [1–4], swimming in porous media [5], transient propulsion [6], and swimming in non-Newtonian fluids [7–10]. Taylor’s model was also studied at finite Re [11] and in inviscid flow (for a finite length sheet) [12,13]. In this regard, note that the fluid mechanical regimes of small and large organisms are completely different [14]. That is, the movement of microor- ganisms is characterized by the dominance of viscous forces over inertial forces (Re ≪ 1), whereas the reverse is true for large swimmers such as most fish. In contrast, both viscous and inertial forces play a role in the (intermediate Re) locomotion of organisms with linear dimension on the order of millimeters, including crustaceans (e.g., copepods and euphausiids), large ciliates, and small jellyfish. The spherical squirmer, introduced by Lighthill [15] and Blake [16], is another reduced-order model for self-propulsion, where a spherical body achieves locomotion through small axisymmetric deformations of its surface. A further simplified squirmer model that swims through steady surface tread- milling (that is, steady tangential motion of its surface) has * akhair@andrew.cmu.edu been employed to examine various facets of locomotion in Stokes (Re = 0) flow, including enhanced diffusion of passive scalars [17,18], nutrient transport [19,20], hydrodynamic inter- actions of swimmers [21,22], and swimming in non-Newtonian fluids [23,24]. Furthermore, the squirmer model has recently been used to study the impact of fluid inertia on self-propulsion, using matched asymptotic expansions at small Re [25,26] and numerical computation across a wide range of Re [27]. It was found that spherical squirmers that generate thrust from their rear (“pushers”) tend to swim faster than those that generate thrust from their front (“pullers”) at nonzero Re. Of course, most intermediate Re swimmers are not spheri- cal; in fact, they tend to be elongated. The ciliate Paramecium roughly resembles a prolate spheroid with an approximate length and width of 100 μm and 40 μm, respectively, and has an escape swimming speed of 10 mm/s, corresponding to Re ≈ 2[28]. Zooplankton such as copepods and krill, with length of 1 mm to 1 cm, are associated with a sus- tained swimming speed of around 5 cm/s and Re ≈ 10–100 [29–31]. Ngo and McHenry [32] studied the locomotion of a water boatman (Corixidae), which has length and width of around 5 mm and 1.5 mm, respectively. The water boatman generates thrust via paddling appendages and covers the regime 10  Re  200. There has been relatively little modeling work on swimming at intermediate Re, compared to the extensive literature on locomotion in Stokes flow and at large Re. Therefore, the central purpose of this article is to analyze a reduced-order model for the locomotion of a slender swimmer at nonzero Re. Naturally, such a model cannot capture in detail the locomotion of any particular organism; however, it can provide results and intuition relevant to the qualitative 2470-0045/2018/97(4)/043102(8) 043102-1 ©2018 American Physical Society