Shape Transition and Propulsive Force of an Elastic Rod Rotating in a Viscous Fluid

The deformation of an elastic rod rotating in a viscous fluid is considered, with applications related to flagellar motility. The rod is tilted relative to the rotation axis, and experiments and theory are used to study the shape transition when driven either at constant torque or at constant speed. At low applied torque, the rod bends gently and generates small propulsive force. At a critical torque, the rotation speed increases abruptly, and the rod forms a helical shape with increased propulsive force. We find good agreement between theory and experiment. A simple physical model is presented to capture and explain the essential behavior.

Figure 1

Orthogonal images of steady-state shapes of rotating rod with torque just below (a) and just above (b) the critical torque. The motor (not shown) is at the top, with rotation axis along $z$. Gravity points down. In (a) and (b), the left panel is the side view, and the right panel is the front view. The rod is marked with white dots for contrast. The axes in (b) are the same as in (a). The curved arrow in (b) denotes the sense of rotation of the rod.

Figure 2

Lumped parameter model consisting of two rigid links connected by a torsional spring (open circle). The top link is clamped. All drag is concentrated at the two filled circles.

Figure 3

Dimensionless motor torque $M_{m}L/A$ was measured as a function of dimensionless speed $\chi$ for $L=250\mathrm{mm}$ (◯) and $L=290\mathrm{mm}$ (△) with angle $\theta =26°$. For $L=250\mathrm{mm}$, speed was measured as a function of increasing torque (■) and decreasing torque (◆). Note the hysteresis. The linear (dashed line) and nonlinear (solid line) predictions are shown. The insets show examples of the steady-state filament shapes in the low (left) and high (right) speed regimes.

Figure 4

Steady-state shapes of a rotating rod from experimental measurements for $\chi =1.38$ (◯), 4.25 (□), 5.91 (◊) (before transition), and 164.63 (▿) (after transition), along with the shapes calculated from the nonlinear (solid line) theory. The rod has $L=210\mathrm{mm}$ and $\theta =20°$.

Figure 5

Theoretical calculation of the dimensionless propulsion force $F_p$ as a function of dimensionless $M_m$ for $\theta =20°$, using the nonlinear equations for the ideal case of zero gravity. The arrows denote the transition for ascending (↑) and descending (↓) torque.