Instabilities of a rotating helical rod in a viscous fluid

Bacteria such as Vibrio alginolyticus swim through a fluid by utilizing the rotational motion of their helical flagellum driven by a rotary motor. The flagellar motor is embedded in the cell body and turns either clockwise (CW) or counterclockwise (CCW), which may lead to straight forward or backward swimming, or reorientation of the cell. In this paper we investigate the dynamics of the helical flagellum by adopting the Kirchhoff rod theory in which the flagellum is described as a space curve associated with orthonormal triads that measure the degree of bending and twisting of the rod. The hydrodynamic interaction with the flagellum is described by the regularized Stokes formulation. We focus on two different types of instabilities: (1) whirling instability of a rotating helical filament in the absence of a hook and (2) buckling instability of a flagellum in the presence of a compliant hook that links the flagellar filament to the rotary motor. Our simulation results show that the helical filament without a hook displays three regimes of dynamical motions: stable twirling, unstable whirling, and stable overwhirling motions depending on the physical parameters, such as rotational frequency and elastic properties of the flagellum. The helical filament with a hook experiences buckling instability when the motor switches the direction of rotation and the elastic properties of the hook change. Variations of physical parameter values of the hook such as the bending modulus and the length make an impact on the buckling angle, which may subsequently affect the reorientation of the cell.

Figure 1

A schematic view of a computational model for a left-handed helical flagellum. The helical flagellar filament is linked to a compliant hook, which is linked to a rotary motor that is tethered in space and turns either counterclockwise (CCW) or clockwise (CW). The motor is represented by a single material point.

Figure 2

Time evolution of two stable dynamical motions of a rotating helical filament with the path (blue) traced out by the free end. The filament is intrinsically helical and the initial configuration is tilted away from the axis of rotation. The filament reaches either a stable twirling (top) or a stable overwhirling (bottom) when the rotational frequency is given as 1400 and 1500 Hz, respectively. The motor turns CCW in both cases.

Figure 3

The dynamical states of the rotating filament for various (a) bending modulus of the filament, (b) twist modulus of the filament, (c) fluid viscosity, and (d) the filament length. Red circles represent stable overwhirling motions, and blue crosses represent stable twirling motions. The dashed lines indicate the best fit of the critical frequency ${\omega }_c$ to a function of the form $y=c{x}^m$.

Figure 4

The dynamical states of the rotating filament as functions of the helical radius. We choose two different bending moduli with a fixed filament length in (a) and three different filament lengths with a fixed bending modulus in (b). The helical pitch and the twist modulus are fixed at $2\pi p$ with $p=1.5\mu \mathrm{m}$ and $a_3=0.05\mathrm{g}\mu {\mathrm{m}}^3/{\mathrm{s}}^2$, respectively. Red circles represent overwhirling motions, and blue crosses represent twirling motions. The dashed lines are best fitted to a function, ${\omega }_c\sim r_0^{-0.4}$, where $r_0$ is the helical radius.

Figure 5

Subcritical bifurcation diagram. The gray area refers to stable twirling state and the white area refers to stable overwhirling state. The solid lines represent the stable states and the dashed lines represent the unstable states. In the region between the two vertical dotted lines, which is determined by HB (Hopf bifurcation point) and CF (cyclic fold), the helical rod is in the bistable states separated by the dashed curve, which corresponds to the unstable whirling state. The empty circles in the bistable region that are obtained from simulations indicate the critical values of motor frequency and the initial mean curvature of the axis of the helical filament. The filled circles are representative simulations that induce stable overwhirling state.

Figure 6

Dynamical motion of a rotating helical flagellum of V. alginolyticus (top) and its corresponding buckling angle (bottom) when the motor is tethered in space. The angular frequency of the motor is set to be 570 Hz. The motor initially turns CW and then switches to CCW at $t=0.06$ s. The bending moduli of the hook in relaxed and loaded states are given as $a_{\mathrm{relax}}=2\times {10}^{-4}\mathrm{g}\mu {\mathrm{m}}^3/{\mathrm{s}}^2$ and $a_{\mathrm{load}}=11\times {10}^{-4}\mathrm{g}\mu {\mathrm{m}}^3/{\mathrm{s}}^2$, respectively.

Figure 7

Buckling angles with various bending moduli of the hook. The first two panels show the time evolution of the buckling angle of the hook for various bend moduli when the bending modulus of the relaxed hook is fixed as $a_{\mathrm{relax}}=2\times {10}^{-4}\mathrm{g}\mu {\mathrm{m}}^3/{\mathrm{s}}^2$ (left) and when the bending modulus of the loaded hook is fixed as $a_{\mathrm{load}}=11\times {10}^{-4}\mathrm{g}\mu {\mathrm{m}}^3/{\mathrm{s}}^2$ (middle). Right panel illustrates the maximum angle of the buckled hook with various pairs of relaxed and loaded bending moduli. The filled circles indicate the threshold whether or not the bucked hook comes back toward the straight state within $1^{\circ }$.

Figure 8

Time evolution of buckling angles for various twist moduli of the hook (a) and hook lengths (b). The bend moduli of the relaxed hook and loaded hook are fixed at $a_{\mathrm{relax}}=2\times {10}^{-4}\mathrm{g}\mu {\mathrm{m}}^3/{\mathrm{s}}^2$ and $a_{\mathrm{load}}=11\times {10}^{-4}\mathrm{g}\mu {\mathrm{m}}^3/{\mathrm{s}}^2$, respectively. The twist modulus of the hook in the right panel is $a_3^{\mathrm{hook}}=0.05\mathrm{g}\mu {\mathrm{m}}^3/{\mathrm{s}}^2$.