Elastohydrodynamics of swimming helices: Effects of flexibility and confinement

Motivated by bacterial transport through porous media, here we study the swimming of an actuated, flexible helical filament in both three-dimensional free space and within a cylindrical tube whose diameter is much smaller than the length of the helix. The filament, at rest, has a native helical shape modeled after the geometry of a typical bacterial flagellar bundle. The finite length filament is a free swimmer and is driven by an applied torque as well as a countertorque (of equal strength and opposite direction) that represents a virtual cell body. We use a regularized Stokeslet framework to examine the shape changes of the flexible filament in response to the actuation as well as the swimming performance as a function of the nondimensional Sperm number that characterizes the elastohydrodynamic system. We also show that a modified Sperm number may be defined to characterize the swimming progression within a tube. Finally, we demonstrate that a helical filament whose axis is not aligned with the tube axis can exhibit centering behavior in the narrowest tubes.

Figure 1

Computational helical filament consisting of a network of springs. The figure shows the equilibrium configuration. The motion is generated by the activation of a torque at the center of the front cross section and by the countertorque, which is placed a small distance in front of the first cross section where the cell body might be.

Figure 2

Achieved shapes of the helical filament for a fixed stiffness constant $k=300$ and varying torque magnitudes (from left to right, first eight panels). Each of these panels shows the filament at two different phases for visualization purposes. The leftmost image corresponds to the equilibrium configuration without the dynamics ($\sigma =0$). Subsequence images correspond to increasing the regularized rotlet magnitude by one dimensionless unit. The rightmost two panels show the envelope formed by the centerline of the filament as it rotates through one revolution for both the equilibrium (rigid) configuration and for the flexible case with $\sigma =7$.

Figure 3

Achieved shapes of the helical flagellum for a fixed rotlet magnitude of 5 dimensionless units and varying spring stiffness constants. All images are shown at the same time of simulation.

Figure 4

(a)–(c) Swimming speed, rotational frequency, and average swimming distance per revolution as a function of regularized rotlet magnitude. The different curves correspond to different spring stiffness constants. (d) The same data as in (c) plotted as a function of sperm number. Different colors correspond to different spring stiffness constants and the data points of a given color correspond to different rotlet magnitudes.

Figure 5

[(a)–(d)] Snapshots from a simulation with applied rotlet strength of 7 dimensionless units and relatively low stiffness constant $k=75$. (e) Buckling observed experimentally (Reprinted figure with permission from M. K. Jawed et al., Phys. Rev. Lett. 115 168101 (2015). Copyright (2015) by the American Physical Society).

Figure 6

Swimming speed, rotational frequency, and swimming distance per revolution computed as a function of tube radius for the case of a rigid helical flagellum whose axis is aligned with the tube axis. The tube radius is scaled by the maximum flagellum amplitude. The computed quantities are scaled by their corresponding value in the absence of tube.

Figure 7

(a) Swimming speed plotted as a function of frequency for $k=300$ (other values of $k$ are similar). Each curve corresponds to a different tube radius and the points on each curve were computed by applying different torque magnitudes. (b) Swimming distance per revolution plotted as a function of the Sperm number. The colors indicate the varying spring stiffnesses, $k$, and each curve corresponds to a different tube radius. The points of a given color along a curve were computed by varying the rotlet strength. (c) Same data as in panel (b) but plotted as a function of the modified Sperm number in a tube given in Eq. (16)

Figure 8

Initial orientation of a flagellum angled 0.1 rad away from the central axis of a tube. The centerline of the tube is indicated by the blue line and the black line is the centerline of the targeted configuration of the filament.

Figure 9

Side view of the flagellum in three tubes of different radii. Each column shows five snapshots corresponding to a particular tube radius. Each filament was initially angled 0.1 rad away from the central axis of a tube.

Figure 10

Axial view of the snapshots in Fig. 9. Each flagellum was initially angled 0.1 rad away from the central axis of a tube. (a) Flagellum and trajectory of the centerpoint of the front cross section for a tube radius of $R=0.675$. (b) Flagellum and trajectory of the centerpoint of the front cross section for a tube radius of $R=0.725$. (c) Flagellum and trajectory of the centerpoint of the front cross section for a tube radius of $R=0.775$. [(d)–(f)] Perspective views of the trajectory of the front point as the flagellum swims down the tube.