Propulsion and Instability of a Flexible Helical Rod Rotating in a Viscous Fluid

We combine experiments with simulations to investigate the fluid-structure interaction of a flexible helical rod rotating in a viscous fluid, under low Reynolds number conditions. Our analysis takes into account the coupling between the geometrically nonlinear behavior of the elastic rod with a nonlocal hydrodynamic model for the fluid loading. We quantify the resulting propulsive force, as well as the buckling instability of the originally helical filament that occurs above a critical rotation velocity. A scaling analysis is performed to rationalize the onset of this instability. A universal phase diagram is constructed to map out the region of successful propulsion and the corresponding boundary of stability is established. Comparing our results with data for flagellated bacteria suggests that this instability may be exploited in nature for physiological purposes.

Figure 1

(a) Experimental apparatus: a helical rod (1), is rotated by a motor (2), inside a glycerin bath (3), that is enclosed by an external water tank (4) for temperature control. Two orthogonal video cameras (5) and (6) record the rod. (b) Schematic diagram of the rod. (c),(d) Examples of the deformed rod at $\omega =0.6\mathrm{rad}/\mathrm{s}$, from both experiments and simulations: (c1),(c2) helical and (d1),(d2) buckled configurations (movie in Ref. [23]). See text for the properties of rod and fluid.

Figure 2

(a) Experimental images of (a1) straight rod, (a2) $R/L=0.56$, and (a3) $R/L=0.29$ in their undeformed (top row) and deformed state (bottom row) at $\omega =3.14\mathrm{rad}/\mathrm{s}$. (b) Normalized suspended height $\overline{h}$ versus angular velocity $\omega$ for experiments and simulations. (c) Simulation data of normalized propulsive force ${\overline{F}}_p$ versus $\omega$.

Figure 3

(a) Sequence of experimental images at $\omega =0.6\mathrm{rad}/\mathrm{s}$. Material properties are provided in the text (movie in Ref. [23]). Time series of (b) normalized height $\overline{h}(t)$ and (c) normalized propulsive force ${\overline{F}}_p(t)$. (d) Normalized height $\overline{h}$ in steady state ($t>360\mathrm{s}$) versus $\omega$. (e) Normalized propulsive force ${\overline{F}}_p$ in steady state versus $\omega$, with the shaded region representing the standard deviation.

Figure 4

Buckling velocity ${\omega }_b$ versus (a) characteristic angular velocity $E{r}_0^4/\mu l^4$, (b) normalized pitch $\lambda /l$, and (c) normalized helix radius $R/\lambda$. In (a), one of the four parameters $\{E,r_0,\mu ,l\}$ was varied keeping the others fixed. In (b) and (c), the helix pitch and radius, respectively, were varied while fixing all the other parameters. (d) Dependence of propulsive force (color bar) on both the normalized pitch $\lambda /l$ and $\overline{\omega }(R/\lambda {)}^m$, with $m=1.96\pm 0.05$, obtained at $R/\lambda =0.2$ (see Ref. [23]). The symbols correspond to (1) Caulobacter crescentus (wild) (circle), (2) Rhizobium lupini (curly) (upward triangle), (3) Salmonella (wild) (right pointing triangle), (4) Rhizobium lupini (semicoiled) (downward triangle), (5) Escherichia coli (diamond), (6) Vibrio alginolyticus (left pointing triangle) [12, 17]. Filled (and open) symbols correspond to the lower (and upper) bound estimates $EI=10^{-23}\mathrm{N}·{\mathrm{m}}^2$ (and $EI=10^{-24}\mathrm{N}·{\mathrm{m}}^2$). The error bars represent the range in angular velocity, $\omega /(2\pi )=500\pm 200\mathrm{Hz}$.