Nonlocal shear-thinning effects substantially enhance helical propulsion

Helical propulsion is ubiquitously adopted by swimming bacteria and artificial microswimmers to move in biological fluids, which are typically viscoelastic and shear thinning. Here we present a set of theoretical and computational analyses to show that shear-thinning viscosity alone can cause the substantial enhancement reported for helical propulsion in recent experiments. Our analyses provide direct evidence to elucidate the nonlocal nature of the enhancement mechanism. Since the enhancement predicted here can be more substantial than that due to viscoelasticity, our results also suggest that shear-thinning rheology may play a more dominant role in the observed enhancement of bacterial motility in polymeric fluids.

Figure 1

(a) Schematic of a helical body of infinite length rotating about its axis at a prescribed rotation rate, $\mathrm{\Omega }$, and propelling at a speed, $U$, along the axis. (b) The scaled propulsion speed $U/R\mathrm{\Omega }$ as a function of the pitch angle $\theta$ in shear-thinning fluids with different values of power index $n$, at a fixed Carreau number $\text{Cu}=10$, viscosity ratio ${\mu }_{\infty }/{\mu }_0=0.1$, and filament aspect ratio (or slenderness) $a/\mathrm{\Lambda }=0.026$. The value of slenderness considered here is obtained from helical filaments used in previous experiments [11]. In the Newtonian limit ($n=1$), the computational model shows excellent agreement with results based on Lighthill's slender-body theory [39] and experimental results in Ref. [11]. (c) The non-Newtonian propulsion speed relative to the Newtonian speed $U/U_N$ as a function of Cu for different values of filament slenderness $a/\mathrm{\Lambda }$. There exist optimal values of Cu maximizing the speed, with more pronounced enhancement (relative to the corresponding Newtonian value) for a thicker filament ($a/\mathrm{\Lambda }=0.106$; more than 80%) than thinner filaments. Here ${\mu }_{\infty }/{\mu }_0=0.1$ and $n=0.4$. Inset: the scaled propulsion speed as a function of Cu.

Figure 2

The distribution of viscosity and flow speed on a cross section of the rotating helix at varying Cu. (a) The scaled viscosity distribution, $\mu /{\mu }_0$, around the rotating helix. (b) The corresponding scaled flow speed distribution, $\left|\mathbf{u}\right|/R\mathrm{\Omega }$, around the rotating helix. The cross section is taken on the midplane with a normal in the direction of propulsion (see the schematic on the right). In each distribution, only a half helical turn of the helix (out of the plane) is depicted. Here $\theta =\pi /4, a/\mathrm{\Lambda }=0.026, {\mu }_{\infty }/{\mu }_0=0.1$, and $n=0.4$.

Figure 3

The energetic cost of helical propulsion in a shear-thinning fluid. (a) The scaled power expended by the rotating helix during propulsion $\mathcal{P}/{\mu }_0{\mathrm{\Omega }}^2{R}^3$ as a function of Cu for different values of filament slenderness $a/\mathrm{\Lambda }$. Inset: the power consumption relative to the corresponding Newtonian value $\mathcal{P}/{\mathcal{P}}_N$ as a function of Cu; results based on different values of filament slenderness all collapse reasonably well onto the curve describing the viscosity variation given by the Carreau model [Eq. (1) with $|\dot{\mathit{\gamma }}|=\mathrm{\Omega }$]: $\mu /{\mu }_0={\mu }_{\infty }/{\mu }_0+(1-{\mu }_{\infty }/{\mu }_0){(1+{\mathrm{Cu}}^2)}^{(n-1)/2}$ (black solid line). (b) The propulsion efficiency $\mathcal{E}$ of the rotating helix in a shear-thinning fluid for different values of filament slenderness $a/\mathrm{\Lambda }$. The efficiency is consistently enhanced for a wide range of Cu shown, with maximum efficiency occurring at $\text{Cu}=O\left(10\right)$; the enhancement can be more than tenfold compared with the Newtonian efficiency ($\text{Cu}=0$).