Extensibility enables locomotion under isotropic drag

Abstract

Anisotropic viscous drag is usually believed to be a requirement for the low Reynolds number locomotion of slender bodies such as flagella and cilia. Here we show that locomotion under isotropic drag is possible for extensible slender bodies. After general considerations, a two-ring swimmer and a model dinoflagellate flagellum are studied analytically to illustrate how extensibility can be exploited for self-propulsion without drag anisotropy. This new degree of freedom could be useful for some complex swimmer geometries and locomotion in complex fluid environments where drag anisotropy is weak or even absent.Due to the absence of inertial forces, low-Reynolds number locomotion is subject to interesting mathematical and physical constraints [1][2][3]. In particular, locomotion by time-reversible strokes is ruled out by Purcell's scallop theorem [4,5]. To escape these constraints, microorganisms swim by either propagating deformation waves along slender appendages, termed flagella, or rotating them. Anisotropic viscous drag is believed to be the fundamental property enabling drag-based propulsion of slender filaments [3,6,7]. It is a classical result that for a slender filament moving in an unbounded Newtonian fluid, the Stokes drag is almost twice when moving perpendicular than parallel to its axis [3,8]. This drag anisotropy allows propulsive forces to be created perpendicularly to the deformation of the filament. Under isotropic drag, it is generally accepted that locomotion of this kind would be impossible [6,7,[9][10][11].Unlike in Newtonian flows, drag laws in more complex media, and their consequences on locomotion, remain largely unexplored. Theoretical studies, via Brinkman models, suggest that porosity enhances drag anisotropy [12,13], explaining, e.g., the increase in propulsion speed of C. elegans in a granular medium. Recent experiments also measured and characterized granular drag in beds of glass beads and granular media [14][15][16][17], which have been applied to study locomotion in sand [18]. Besides the fluid medium, the geometry of a swimming body also plays a role in the drag law. Some flagella, such as those of Ochromonas, possess rigid projections termed mastigonemes, protruding into the fluid [1]. In these geometries the viscous drag in the longitudinal direction of the flagellum is increased, possibly resulting in a more isotropic drag. In situations where drag anisotropy is weak or even absent, what are the alternative mechanisms, if any, offered by physics to achieve locomotion? In this letter, we point out a new degree of freedom enabling inertialess swimming, namely extensibility. Using a general derivation and two simple geometrical models, we demonstrate that the periodic stretching and contraction of a filament allow self-propulsion even under isotropic drag. * Corresponding author. Email: elauga@ucsd.eduWe start by considering the general calculation of Becker et al. [7] showing that drag anisotropy is required for the propulsion of inextensible swimmers. Here we revisit their der...

Hydrodynamics of the double-wave structure of insect spermatozoa flagella

“…A similar example is a toroidal helix (a helix built upon a circle), which is an idealized model studied recently for dinoflagellates [42]. We expect that the propagation of a wave along a toroidal helix should also require extensibility, and that propulsion is still possible even without drag anisotropy [41]. …”

“…The superhelical kinematics described (equations (2.12) -(2.14)) are possible only when extensibility is allowed; the minor helix is built upon another curved structure (the major helix) and local extension and contraction is implied in the wave kinematics. When extensibility is permitted, the relaxation of the drag anisotropy requirement has been recently shown [41]. That the swimming speed is non-zero is owing to intrinsic variations in length (and hence drag) embedded in the curved geometry of the superhelices.…”

Rotation of slender swimmers in isotropic-drag media

“…However, drag anisotropy is the critical physical ingredient to allow motion of the geometric center of a swimmer, and without it a net translation is impossible [27]. Indeed, consider an inextensible [28] filament of length L described by the centerline location r(s, t) and deforming its shape with the instantaneous velocity U(s, t) in the laboratory frame of reference ( Fig. 1).…”

“…since the swimmer is force-free at all times. Allowing the body to be extensible can break this condition, prompting the creation of many popular theoretical models, like extensible filament swimming [28] and three sphere swimmers [29,30]. However for many microswimmers, which are inextensible, drag anisotropy is a fundamental constraint on whether an organism can translate at low Reynolds number.…”

Microscale locomotion in a nematic liquid crystal

“…The anisotropy is also important when the tumbling parameter l approaches unity, marks the transition from tumbling to flow-aligning nematic fluids 45 . Equations (6,7,15,16) are all singular in the limit l ! 1 .The thickness of the boundary layer is seen to be 1-11 | 7 5 (Color online) Dimensionless swimming speed U vs Er = tw for a transverse-wave swimmer with µ = µ 1 = µ 2 = 1, Kr = 1.2, and 0.6, with anchoring strengths w = 0 (blue), w = 0.1 (red), w = 1 (green), and w = 5 (brown).…”