Deformation of a helical filament by flow and electric or magnetic fields

Motivated by recent advances in the real-time imaging of fluorescent flagellar filaments in living bacteria [Turner, Ryu, and Berg, J. Bacteriol. 82, 2793 (2000)], we compute the deformation of a helical elastic filament due to flow and external magnetic or high-frequency electric fields. Two cases of deformation due to hydrodynamic drag are considered: the compression of a filament rotated by a stationary motor and the extension of a stationary filament due to flow along the helical axis. We use Kirchhoff rod theory for the filament, and work to linear order in the deflection. Hydrodynamic forces are described first by resistive-force theory, and then for comparison by the more accurate slender-body theory. For helices with a short pitch, the deflection in axial flow predicted by slender-body theory is significantly smaller than that computed with resistive-force theory. Therefore, our estimate of the bending stiffness of a flagellar filament is smaller than that of previous workers. In our calculation of the deformation of a polarizable helix in an external field, we show that the problem is equivalent to the classical case of a helix deformed by forces applied only at the ends.

Figure 1

(a) A helix with pitch $P$ and radius $R$. (b) One turn of a helix “unrolled,” showing the pitch angle $\alpha$, and the orientation of the Serret-Frenet frame $\{\widehat{\mathbf{t}},\widehat{\mathbf{n}},\widehat{\mathbf{b}}\}$.

Figure 2

(a) The material frame for a straight and untwisted rod. (b) The material frame for a straight rod with right-handed (positive) twist; ${\widehat{\mathbf{e}}}_1$ and ${\widehat{\mathbf{e}}}_2$ rotate about ${\widehat{\mathbf{e}}}_3$ as $s$ increases.

Figure 3

Dimensionless extension $f_{\mathrm{t}}$ vs pitch angle $\alpha$ for a helix with four turns, $R=0.22\mu \mathrm{m}$, and $P=2.4\mu \mathrm{m}$, subject to axial flow. The hydrodynamic forces are calculated using resistive-force theory. The solid line is calculated using the exact expression for $f_{\mathrm{t}}$, and the dotted line is $f_{\mathrm{t}}=1$, the approximate dimensionless extension to leading order in $R∕L$.

Figure 4

Dimensionless extension $f_{\mathrm{r}}$ vs pitch angle $\alpha$ for a helix with four turns, $R=0.22\mu \mathrm{m}$, and $P=2.4\mu \mathrm{m}$, subject to rotational flow. The solid line is calculated using the exact expression for $f_{\mathrm{r}}$, and the dotted line is $f_{\mathrm{r}}=\cot \alpha$, the approximate dimensionless extension to leading order in $R∕L$.

Figure 5

The viscous drag force per unit length $\mathbf{K}$, in units of $R\eta \omega$, for an open-coiled helix rotating with speed $\omega$ vs arc length $s$ in units of $L$. The filament radius is $a=0.01\mu \mathrm{m}$, the helix radius is $R=0.22\mu \mathrm{m}$, the pitch is $P=2.4\mu \mathrm{m}$, and the total arc length is $L\approx 11\mu \mathrm{m}$.

Figure 6

The viscous drag force per unit length $\mathbf{K}$, in units of $\eta v$, for a close-coiled helix subject to axial flow with speed $v$ vs arc length $s$ in units of $L$. The helix parameters are taken from [24]: the filament radius is $a=0.01\mu \mathrm{m}$, the helix radius is $0.6\mu \mathrm{m}$, the pitch is $0.5\mu \mathrm{m}$, and the total number of turns is 11.5.

Figure 7

Dimensionless extension $f_{\mathrm{f}}$ vs pitch angle $\alpha$ for a helix with four turns, $R=0.22\mu \mathrm{m}$, and $P=2.4\mu \mathrm{m}$, subject to an external field. The solid line is calculated using the exact expression for $f_{\mathrm{f}}$, and the dotted line is $f_{\mathrm{f}}=2\cos \alpha$, the approximate dimensionless extension to leading order in $R∕L$.

Figure 8

Dimensionless extension $f_{\mathrm{w}}$ vs pitch angle $\alpha$ for a helix with four turns, $R=0.22\mu \mathrm{m}$, and $P=2.4\mu \mathrm{m}$, subject to external force per unit length $\mathbf{K}=w\widehat{\mathbf{z}}$. The solid line is calculated using the exact expression for $f_{\mathrm{w}}$, and the dotted line is $f_{\mathrm{w}}=1∕2$, the approximate dimensionless extension to leading order in $R∕L$.