2. Overview

Gagnon et al. (Reference Gagnon, Keim and Arratia2014) consider the locomotion of the millimetre-sized nematode C. elegans in a variety of shear-thinning fluids (figure 1 a). They measure the locomotion kinematics and the fluid mechanisms around the organism using tracer particles for two kinds of fluids: (a) solutions of the rod-like polymer xanthan gum, which show rheological data consistent with a Carreau fluid with power indices $n$ ranging from 0.3 to 0.9 (i.e. at large shear rates $\dot{{\it\gamma}}$ , the viscosity ${\it\mu}$ varies as ${\it\mu}\sim \dot{{\it\gamma}}^{n-1}$ , with $n<1$ indicating shear-thinning behaviour); and (b) Newtonian fluids with a large range of viscosities (from that of water to a few hundred times above). In all cases, the Reynolds numbers characterizing the flow were well below unity.

Figure 1. (a) Sketch of the experiment studied by Gagnon et al. (Reference Gagnon, Keim and Arratia2014). (b) Streamlines around the swimming worm in a Newtonian fluid. (c) Streamlines in a shear-thinning fluid. (d) Difference between the normalized velocities in the shear-thinning and Newtonian flows, showing flow decrease near the head and increase near the tail of the swimmer.

The experimental investigation obtained two main results. First, despite such a large change in the fluid rheology, the cells behave in exactly the same way in a complex fluid and in a Newtonian one. The authors found no measurable change in the worm waving amplitude, the speed of the waves, the waving frequency and the swimming speed. These results are consistent with the asymptotic study of Vélez-Cordero & Lauga (Reference Vélez-Cordero and Lauga2013) on the small-amplitude waving locomotion of an inextensible sheet, but not with the numerical results of Montenegro-Johnson et al. (Reference Montenegro-Johnson, Smith and Loghin2013). However, that study focused on swimming spermatozoa-like waveforms with a head and increasing head-to-tail amplitude, whereas the waving motion of the head-less C. elegans has actually a decreasing amplitude.

The second main result is that, despite undergoing no change in their swimming kinematics, the organisms generate flows that display a number of important differences. To characterize the flows, the authors employ velocimetry with small tracer particles seeding the fluids. In the Newtonian case, the instantaneous flow streamlines around the organism are shown in figure 1(b) at the moment in the beating cycle where the instantaneous shear rates are the largest. In contrast, the flow induced in the shear-thinning fluid is displayed in figure 1(c) at the same instant in the waving motion. The main difference between the two is the displacement of the head and tail vortices, which have moved towards the head and display stronger flow gradients. Using measurements of the vorticity field, the authors further confirm that the head vortex increases in both size and magnitude in the non-Newtonian fluid.

In order to further contrast the two flows, the authors compute the difference between the velocity magnitudes (normalized by their maximum values) in the shear-thinning fluid and in the Newtonian one. These results are shown in figure 1(d), and as is quantified in more detail in the paper, show that non-Newtonian stresses impact the swimmer in an asymmetric way. The fluid velocities undergo a decrease near the head of the swimmer and an increase near the tail. Furthermore, this asymmetry in the velocity profile is seen to increase with the amount of shear thinning in the fluid until the typical rate of deformation is of the same order as the critical shear rate at which the shear viscosity displays a transition from Newtonian to non-Newtonian.

3. Future

Like many thought-provoking experiments, the study by Gagnon et al. (Reference Gagnon, Keim and Arratia2014) leads to more questions than answers. The worms are self-propelled, and it is the balance between waving propulsion and drag that leads to their swimming speed (Lauga & Powers Reference Lauga and Powers2009). For the speed to remain unchanged, the stresses leading to both drag and thrust would need to change in exactly the same manner with changes in the rheology. How this is possible while at the same time undergoing significant changes in the flow structure is not clear. Using headless synthetic swimmers with constant-amplitude waveforms, Dasgupta et al. (Reference Dasgupta, Liu, Fu, Berhanu, Breuer, Powers and Kudrolli2013) found a systematic decrease of the swimming speed in shear-thinning fluids. Why such a different answer?

Biologically, how do the results of this study extend to much smaller organisms, such as flagellated algae or bacteria? One key difference lies perhaps in the flexibility of the organisms. The nematode C. elegans is large and uses muscular contractions to generate waving, and has bending rigidities $B$ in the range $B\approx 10^{-16}{-}10^{-13}\ \text{N}\ \text{m}^{2}$ (Backholm, Ryu & Dalnoki-Veress Reference Backholm, Ryu and Dalnoki-Veress2013). In contrast, eukaryotic flagella are orders of magnitude softer, with $B\approx 10^{-23}{-}10^{-21}\ \text{N}\ \text{m}^{2}$ (Hines & Blum Reference Hines and Blum1983). The waving motion of small cells is therefore likely to be impacted by changes in the fluid rheology, leading to an additional feedback mechanism for the fluid rheology to affect locomotion. These small cells might find so much gooeyness to be unbearable.