Fluid elasticity increases the locomotion of flexible swimmers

We conduct experiments with flexible swimmers to address the impact of fluid viscoelasticity on their locomotion. The swimmers are composed of a magnetic head actuated in rotation by a frequency-controlled magnetic field and a flexible tail whose deformation leads to forward propulsion. We consider both viscous Newtonian and glucose-based Boger fluids with similar viscosities. We find that the elasticity of the fluid systematically enhances the locomotion speed of the swimmer, and that this enhancement increases with Deborah number. Using Particle Image Velocimetry to visualize the flow field, we find a significant difference in the amount of shear between the rear and leading parts of the swimmer head. We conjecture that viscoelastic normal stresses lead to a net elastic forces in the swimming direction and thus a faster swimming speed.

A field originally started by Sir G. I. Taylor [1], the fluid dynamics of swimming microorganisms was very active in the 1970's [2 -6]. Renewed interest was recently prompted by new series of questions arising on nonlinear and nonlocal swimming behavior, including cell-cell hydrodynamic interactions, collective locomotion, instabilities of active suspensions, and synthetic swimming systems [7]. One of such questions, and the focus of this paper, concerns locomotion in complex fluids.
Many situations of biological importance involve locomotion in, and transport of, polymeric fluids with large relaxation times in viscous dominated regimes. Well-studied examples include mucus transport by lung cilia [8] and locomotion of mammalian spermatozoa in cervical mucus [9]. Except for a handful of investigations [10][11][12] all past experimental and theoretical work in the field focused solely on Newtonian fluids.
Recently, a series of studies has addressed locomotion in complex fluids [13][14][15][16][17][18][19][20][21], with somewhat contradictory results. Analytical studies in the small-deformation limit showed that in a Oldroyd viscoelastic fluid, locomotion under a given gait (two-or three-dimensional waving) always lead to a decrease of the swimming speed with the Deborah number, De, to ratio of the fluid relaxation time scale to the typical time scale of the swimming motion [13,14,16,17]. In contrast, numerical simulations for a two-dimensional swimmer in an Oldroyd fluid showed that, for large-amplitude motion, an increase in swimming speed could be obtained, which the authors attributed to high-strain regions behind the swimmer [18].
Similar to the modeling approaches, experimental investigations have lead to two different results. Shen and Arratia studied the locomotion of C. elegans nematodes in shear-thinning polymeric fluids and found, in agreement with the small-deformation analysis, that viscoelasticity hinders locomotion [19]. Liu et al. [20] considered the force-free translation of externally-rotated thin rigid helices, as a model for the dynamic of bacterial flagella in complex fluids. They found that, for large-amplitude motion in a constant-viscosity Boger fluid [22], the velocity leading to the force-free condition (equivalent to a swimming speed) could be increased by the presence of fluid elasticity.
In this paper we address the effect of fluid elasticity on flexible swimmers. We measure the swimming speed of magnetic bodies equipped with flexible tails actuated externally by a time-periodic magnetic field. The combination of filament flexibility, periodic actuation, and drag forces from the fluid break the time-reversibility for the filament shape and leads to forward propulsion [23]. In a Boger fluid with constant viscosity we show that fluid elasticity always enhance the locomotion speed of the swimmers. In addition, the ratio between the non-Newtonian velocity and that measured in a Newtonian flow systematically increases with the Deborah number.
A sketch of the swimmer is shown in Fig. 1  viscosity and finite relaxation times are obtained [22]. In Table I we show the properties of the all fluids used; note that fluids N1 and B1 are more viscous than N2 and B2. The density, ρ, is measured using a pycnometer (Simax, 50 ml). The rheological properties are   study are therefore attributable only to the elasticity of the fluids and not to their sheardependance. The mean relaxation times are calculated considering the scheme proposed in Ref. [20]. The experimental values of G ′ (ω) and G ′′ (ω) are fitted to a generalized Maxwell model [27], , with N = 4 leading to an excellent fit. The values of the fitting parameters (g i and λ i ) are then used to estimate the mean relaxation time as Note that the value of the solvent viscosity was 3.4 and 2.6 Pa s for the fluids B1 and B2, respectively.
In Fig. 3 we show a typical measurement of the swimmer kinematics in our study: position of the head along (•) and across ( ) the swimming direction as well as head angle (⋄). Filled symbols (blue online) correspond to the Boger (B2) fluid while empty symbols (red online) are for the Newtonian (N2) fluid. As the swimmer head moves its head sideways, a net forward motion is produced. Even though the exact same swimmer was used in the two fluids under the same magnetic driving, the motion in the viscoelastic case appears to be larger, both for sideways and forward displacements. The angle amplitude of the head appears to also be larger in the Boger fluid.
In Fig. 4  This is similar to past measurements for Newtonian fluids [28]. In all cases, the swimming speed is seen to be larger for the viscoelastic fluids than that reached in the Newtonian fluid. Our results are therefore in qualitative agreement with the numerical results of [18] and the experiments of [20] showing enhanced swimming velocity in viscoelastic fluids.
To help rationalize our results we plot our measurements in dimensionless terms. The relative importance of the bending forces in the tail and viscous drag forces are quantified by the so-called sperm number [7] Sp = L ωε ⊥ EI where ω is the oscillation frequency, ε ⊥ is a viscous resistance coefficient for fluid motion perpendicular to the tail, E is the material's Young's modulus, and I is the second moment of inertia of the tail cross-sectional area (I = πa 4 /4 for a circular cross section). With the approximate value ε ⊥ ≈ 4πµ/ln(L/a), the sperm number can be written as where L * = L/a is the dimensionless tail length. The quantity in the first parenthesis on the right-hand side of Eq.
(2), µω/E, is a pseudo-Weissenberg number that compares the which showed an increase of swimming in complex fluids [18,20] have found that there is a critical Deborah number at which the velocity increase is maximum. In contrast here, and for the range of parameters tested, we observe a continuous increase of the swimming velocity for De numbers as large as 5.
Why does this swimmer move systematically faster in a viscoelastic media? To help rationalize our results, we conducted a visualization study of the velocity fields around the swimmer [29]. We employed two-dimensional PIV to characterize the velocity field around the swimmer with a flat tail. In Fig. 7a we show the typical velocity and vorticity fields around the swimmer at a given instant in the beating period. A large vortical structure develops around the swimmer; during each cycle of oscillation, a vortex forms, dissipates quickly and a new one forms in the opposite direction.
With our measurement of the velocity field, we can compute all four in-plane components of the velocity gradient tensor. The rate-of-strain tensor, D, can therefore be measured around the swimmer, D = 1 2 [∇v + (∇v) T ], where v is the velocity in the fluid. We show in Fig. 7b the field of the phase-averaged magnitude of D, defined as |D| = √ D : D , where · denotes averaging in time. This plot quantifies therefore regions where shear deformations are large throughout the periodic beating of the swimmer tail. Clearly, on average, the rear part of the head of the swimmer is subject to a larger shear than its front.
As is well known, viscoelastic fluids subject to shear deformations result in additional normal stresses, due to the stretching of polymeric molecules along flow streamlines [30].
Based on our observation of the difference in the amount of shear between the leading and trailing sides of the head, we conjecture that it is the presence of these non-Newtonian normal stresses which induce a nonzero elastic force in the swimming direction, leading to an increase in the swimming speed.
One important aspect that needs to be addressed is the potential difference in tail kinematics between different fluids. As was pointed out by Teran et. al [18], the details of the waveform do affect the propulsion speed, and swimming enhancement occurs only for specific tail shapes. To address this effect in our experiments, we tracked the shape of the tail at different instants throughout the oscillation cycle. We show in Fig. 8