Chiral sedimentation of extended objects in viscous media

We study theoretically the chirality of a generic rigid object’s sedimentation in a fluid under gravity in the low Reynolds number regime. We represent the object as a collection of small Stokes spheres or stokeslets and the gravitational force as a constant point force applied at an arbitrary point of the object. For a generic configuration of stokeslets and forcing point, the motion takes a simple form in the nearly free draining limit where the stokeslet radius is arbitrarily small. In this case, the internal hydrodynamic interactions between stokeslets are weak, and the object follows a helical path while rotating at a constant angular velocity $\omega$ about a fixed axis. This $\omega$ is independent of initial orientation and thus constitutes a chiral response for the object. Even though there can be no such chiral response in the absence of hydrodynamic interactions between the stokeslets, the angular velocity obtains a fixed nonzero limit as the stokeslet radius approaches zero. We characterize empirically how $\omega$ depends on the placement of the stokeslets, concentrating on three-stokeslet objects with the external force applied far from the stokeslets. Objects with the largest $\omega$ are aligned along the forcing direction. In this case, the limiting $\omega$ varies as the inverse square of the minimum distance between stokeslets. We illustrate the prevalence of this robust chiral motion with experiments on small macroscopic objects of arbitrary shape.

Figure 1

The stokeslet configuration used to check the scaling of $\lambda$ with the size of the object. To the left is a perspective view from an arbitrary direction. The pulling direction is toward the bottom of the cube in the $-z$ direction. The center and right views show projections of the stokslets onto the $xy$ and $xz$ planes, respectively.

Figure 2

Numerical results showing the scaling of $\lambda$ with (a) $R$, (b) $R_{\perp }$, and (c) $R_{\parallel }$ for configurations like that of Fig. 1. In (a) there is a single scaling exponent of $-2$. In (b), for $R_{\perp }⪢R_{\parallel }=1$, we have a scaling exponent of $-5$ and for $R_{\perp }⪡R_{\parallel }=1$, it is constant. In (c), we have an exponent of 3 for $R_{\parallel }⪡R_{\perp }=1$ and $-2$ for $R_{\parallel }⪢R_{\perp }=1$.

Figure 3

The chiral coefficient $\lambda$ for a three-stokeslet object whose transverse projection is an equilateral triangle of side length $a$. Two of the longitudinal coordinates are fixed at $\pm 5$, and the third is varied over $z$ values between them.

Figure 4

The $g$ function plotted versus $\Delta$ and ${\left(Z/R_{\parallel }\right)}^2$. We can see that it is bounded and prefers high $\Delta$ and $Z/R_{\parallel }$.

Figure 5

Projections of the four stokeslet configurations used in Sec. 7A. In each image, the gray circles represent stokeslets, a small square marks the average stokeslet position, and arrows point to the forcing points used. Objects A and B are identical except for the positions of their forcing points and are shown on the left. Objects C and D are shown in the middle and on the right. Each object has been rotated so that the coordinate axes are aligned with the principal axes of the inertia tensor, with $\widehat{z}$ and $\widehat{x}$ corresponding to the largest and smallest of these, respectively. The size of the gray circles corresponds to the largest the stokeslets can be without causing the object to enter the tumble zone. In the case of the leftmost images, this is done with respect to object B.

Figure 6

A comparison between the predicted values of the axis of rotation and angular velocity in the nearly free draining limit with the full results valid for all values of the drag coefficient. On the left vertical axis, the solid line represents the full $\omega$ when the object is undergoing the globally stable chiral motion, and the dotted line represents the value computed from the perturbative expressions in Sec. 5. On the right vertical axis is the cosine of the angle $\theta$ between the axis of rotation and the axis of rotation computed in the nearly free draining limit. (a)–(d) correspond to objects A–D, respectively. In the case of (a), the object does not enter the tumble zone at all; before this happens, $\rho$ has increased to the unphysical point where the stokeslets overlap. However, the rest of the objects had their forcing points chosen in a manner which required them to be in the tumble zone for larger values of $\rho$, and their plots break off before ${\rho }_{\text{max}}$ is reached.

Figure 7

The skew propeller shape used by [3]. The two orthogonal disks have radius $a$ and are fixed so their centers are a distance $2\ell$ apart. The center of twisting for this object coincides with the origin, so its twist matrix has three real eigenvalues.

Figure 8

Plots of normalized $w\left(t\right)$ and $h\left(t\right)$ as functions of time. The width increases linearly with time for the ellipsoidal particle but remains bounded for the particles with a nonzero twist matrix. The spread of the particles increases linearly with time for both the skew propellers and the ellipsoids, but after an initial transient remains constant for the nearly free draining particles.

Figure 9

(a) A multiple-exposure image of an irregular piece of nylon sedimenting in vegetable oil. The nylon piece is a few millimeters in length, and the pictures were taken about 1 s apart. It is clearly rotating around the vertical axis. (b) A multiple-exposure image of a fine piece of copper wire sedimenting in vegetable oil. The pictures were taken about 0.3 s apart, and the distance scale is the same as in (a). The object is rotating about the vertical axis as it follows a helical path down. (c) A multiple-exposure image of the same piece of nylon in (a) though not at the same time. From above, the helical path is more apparent. More images and movies are available in Ref. [18].

Figure 10

(a) Three multiple-exposure pictures of the 9 mm long object pictured in (b) as it sediments in salt water. The pictures were taken about 0.15 s apart. In each case the twist about the vertical axis as it moves in a helix is clearly visible, indicating that the chiral effects on sedimentation are similar to those in the viscous solvent of Fig. 9. Each picture corresponds to a different initial orientation of the object. Though the transient motion was different in each case, it always ended up turning to the same preferred orientation and twisting in the same direction. (c) Another object cut from a plastic spoon. This object has two stable orientations, which lead to twists about the vertical axis in opposite directions.