Sedimentation of a rigid helix in viscous media

We consider sedimentation of a rigid helical filament in a viscous fluid under gravity. In the Stokes limit, the drag forces and torques on the filament are approximated within the resistive-force theory. We develop an analytic approximation to the exact equations of motion that works well in the limit of a sufficiently large number of turns in the helix (larger than two, typically). For a wide range of initial conditions, our approximation predicts that the center of the helix itself follows a helical path with the symmetry axis of the trajectory being parallel to the direction of gravity. The radius and the pitch of the trajectory scale as nontrivial powers of the number of turns in the original helix. For the initial conditions corresponding to an almost horizontal orientation of the helix, we predict trajectories that are either attracted towards the horizontal orientation, in which case the helix sediments in a straight line along the direction of gravity, or trajectories that form a helical-like path with many temporal frequencies involved. Our results provide insight into the sedimentation of chiral objects and might be used to develop new techniques for their spatial separation.

Figure 1

(a) Definitions of the coordinate systems $\{X,Y,Z\},\{x^{\prime },y^{\prime },z^{\prime }\}$ and $\{1,2,3\}$. (b)–(d) Definition of the Euler angles used in this work.

Figure 2

Geometry of the helix: $R$ and $\lambda$ are the radius and the pitch, correspondingly, while $L$ is the total length of the helix along its symmetry axis; ${\alpha }_0$ is defined as the angle between the shortest distance between the end point of the helix and its axis of symmetry, and the line obtained by rotation of the former in the 12-plane until it is parallel to the 1-direction. It serves as a measure of how much the number of helical turns, defined formally as $L/\lambda$, deviates from an integer (see the main text). (a) Side view. (b) Top view.

Figure 3

Comparison between the predictions of Eqs. (36, 37, 38) (dashed lines) and the exact numerical solutions (solid lines) of the full equations of motion for a helix with $N=4,{\alpha }_0=0$, and $\chi =0.733\left(\epsilon =0.25\right)$: (a) $\theta \left(\tilde{t}\right)$, (b) $\varphi \left(\tilde{t}\right)$, and (c) $\psi \left(\tilde{t}\right)$. The initial conditions are ${\theta }_0=1.0,{\varphi }_0=0.2$, and ${\psi }_0=0.5$.

Figure 4

Same as Fig. 3 for a helix with $N=2,{\alpha }_0=-1.34$, and $\chi =0.733\left(\epsilon =0.41\right)$. The initial conditions are ${\theta }_0=1.0,{\varphi }_0=0.0,{\psi }_0=0.0$.

Figure 5

Example of a superhelical trajectory traced by the origin of the body frame for a helix with $N=4,{\alpha }_0=0$, and $\chi =0.733$. The initial values of the Euler angles are ${\theta }_0=1.0,{\varphi }_0=0.2$, and ${\psi }_0=0.5$, while center of the helix is at (0,0,0) at time $\tilde{t}=0$. The dashed line represents the trajectory of the origin of the body frame, given by Eqs. (45, 46, 47), while colors correspond to the instantaneous orientation of the helix at various times: $\tilde{t}=0$ (orange), $\tilde{t}=\tilde{T}/4$ (green), $\tilde{t}=\tilde{T}/2$ (red), $\tilde{t}=3\tilde{T}/4$ (violet), and $\tilde{t}=\tilde{T}$ (brown). The radius of the helix is increased by a factor of 5 compared to its actual value for visualization purposes. (a) Side view. (b) Top view.

Figure 6

Comparison between the superhelical trajectories predicted by Eqs. (45, 46, 47) (dashed lines) and the numerical solutions of the full equations (circles and solid lines). (a) Long helix: $N=4,{\alpha }_0=0$, and $\chi =0.733$ with ${\theta }_0=1.0,{\varphi }_0=0.2$, and ${\psi }_0=0.5$. (b) Short helix: $N=2,{\alpha }_0=-1.34$, and $\chi =0.733$, with ${\theta }_0=1.0,{\varphi }_0=0.0$, and ${\psi }_0=0.0$.

Figure 7

Examples of stable horizontal configurations for a helix with $N=4$: (a) ${\alpha }_0=0.24$, the stable configuration is given by $\psi =0$. (b) ${\alpha }_0=-0.24$, the stable configuration is given by $\psi =\pm \pi$ (both configurations look the same). The relative orientation in the $\tilde{X}\tilde{Y}$ plane is chosen arbitrarily as the helix is rotating around the $\tilde{Z}$ axis.

Figure 8

Numerical solution of the full equations of motion for (a) $\theta \left(\tilde{t}\right)$ and (b) $\psi \left(\tilde{t}\right)$. The parameters of the helix are $N=4$ and $\chi =0.733$, and the initial conditions are given by ${\theta }_0=\pi /2-0.07,{\varphi }_0=0.2$, and ${\psi }_0=0.2$. The solid and dashed lines correspond to ${\alpha }_0=-0.5$ and ${\alpha }_0=0.5$, respectively. In (a) the dotted line is at $\theta =\pi /2$. In (b) the dotted lines are at $\psi =0$ and $\psi =-\pi$.

Figure 9

(a) Spatial trajectory obtained by numerical integration of the exact equations of motion (solid line) for a helix with $N=4,{\alpha }_0=-1.3$ and $\chi =0.733\left(\epsilon =0.226562\right)$ for ${\theta }_0=\pi /2-0.05,{\varphi }_0=0.2$, and ${\psi }_0=0.2$. The analytical superhelical trajectory predicted by Eqs. (45, 46, 47), dashed line, is given for reference. (b) Power spectrum of $\tilde{X}\left(\tilde{t}\right)$ corresponding to the trajectory in panel (a).

Figure 10

State diagram for a helix with $N=4$ and $\chi =0.733$. For a given combination of ${\alpha }_0$ and ${\theta }_0$, the exact equations of motion are numerically integrated with ${\varphi }_0={\psi }_0=0$. The color indicates the amplitude of oscillations of $\theta \left(\tilde{t}\right)$ in a steady state or at long times, if the steady state is not reached; see the main text. (a) The dashed line gives the analytical prediction for the boundary between the superhelical and horizontal orientations ${\theta }_0^{\left(\mathrm{tr}\right)}$, Eq. (50). The solid lines are the analytical predictions, ${\alpha }_0=\pm \pi /4$, for the threshold between the linearly stable horizontal orientations and quasisuperhelical trajectories with large oscillations of $\theta \left(\tilde{t}\right)$. (b) Zoom-in for ${\theta }_0\lesssim {\theta }_0^{\left(\mathrm{tr}\right)}$.