Lord Kelvin's isotropic helicoid

Nearly 150 years ago, Lord Kelvin proposed the isotropic helicoid, a particle with isotropic yet chiral interactions with a fluid so that translation couples to rotation. An implementation of his design fabricated with a three-dimensional printer is found experimentally to have no detectable translation-rotation coupling, although the particle point-group symmetry allows this coupling. We explain these results by demonstrating that in Stokes flow, the chiral coupling of such isotropic helicoids made out of nonchiral vanes is due only to hydrodynamic interactions between these vanes. Therefore it is small. In summary, Kelvin's predicted isotropic helicoid exists, but only as a weak breaking of a symmetry of noninteracting vanes in Stokes flow.

Figure 1

(a) Model used to 3D-print the left-handed isotropic helicoid. (b) Anisotropic helicoid oriented with the equatorial vanes reflected. (c) Experimental measurements of the particle orientation around its sedimentation axis as a function of time. Isotropic helicoid (red $\square$ with best fit $\omega =d\alpha /dt=-0.003$ rad/s), anisotropic helicoid with initial orientation as in (b) (blue $\diamond$, $\omega =-0.258$ rad/s), and with reflected vanes on a meridian (black $★$, $\omega =0.135$ rad/s).

Figure 2

Illustration of the point-group symmetries of different isotropic helicoids. (a), (b) Chiral octahedral symmetry O, four ${\mathrm{C}}_3$ rotation axes, three ${\mathrm{C}}_4$ axes, and six ${\mathrm{C}}_2$ axes. (c), (d) Chiral tetrahedral symmetry T, four ${\mathrm{C}}_3$ rotation axes, and three ${\mathrm{C}}_2$ axes (the cubes were drawn after Table 7.2 in Ref. [17]).

Figure 3

Theory. (a) Schematic of an isotropic helicoid made out of 24 spheres (radius $a$) arranged into 12 dumbbells. Each dumbbell (distance $b=5a$ between centres of spheres) represents a vane of the particle shown in Fig. 1. The dumbbells are assumed to be rigidly connected to each other. Three dashed great circles (radius $c$) are guides to the eye. (b) Trace $\mathrm{Tr}\mathbb{B}$ of the translation-rotation coupling matrix for the isotropic helicoid from (a), as a function of particle size $c/a$. Numerical result (solid line), Eqs. (5) and (6) (dashed line).

Figure 4

Numerical result for trace of ${\mathbb{B}}^{\left(\mathrm{p}\right)}$ of a two-armed propeller made out of two dumbbells, as a function of the half-distance between the dumbbells, $c$, normalized by $b$ ($a$ equals unity), solid line. Dashed-line shows the theoretical large-$c$ limit, Eq. (C5).

Figure 5

Higher-Re experiments. (a) Single four-bladed propeller. (b) The cubic isotropic helicoid constructed from six of the propellers shown in (a).