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.