Mixing and pumping by pairs of helices in a viscous fluid

Here, we study the fluid dynamics of a pair of rigid helices rotating at a constant velocity, tethered at their bases, in a viscous fluid. Our computations use a regularized Stokeslet framework, both with and without a bounding plane, so we are able to discern precisely what flow features are unaccounted for in studies that ignore the surface from which the helices emanate. We examine how the spacing and phase difference between identical rotating helices affects their pumping ability, axial thrust, and power requirements. We also find that optimal mixing of the fluid around two helices is achieved when they rotate in opposite phase, and that the mixing is enhanced as the distance between the helices decreases.

Figure 1

Sketch of two upright helices. Each helix has length $L$ and radius $\alpha$ and they are spaced a distance $d$ apart. Both helices rotate counterclockwise from the top view, but with a phase difference $\varphi$.

Figure 2

(a) Velocity field around a portion of a helix projected onto a bisecting plane, (b) transverse velocity contours, (c) axial velocity contours, and (d) vorticity contours. These computational results should be compared with the experimental results shown in Fig. 2 of Ref. [17].

Figure 3

Normalized thrust as a function of pitch angle. Experimental data adapted from Fig. 8 of Ref. [17] (squares), theoretical calculations using approximations by Gray and Hancock [5] (gray solid line), Cox [27] (gray dashed line), and Lighthill [6] (gray dotted line), and our regularized Stokeslet calculations in the presence (black solid line) and absence (black dashed-dotted line) of a no-slip wall. The RFT calculations do not account for a bounding plane. Our simulations in the presence of a wall show very good agreement with the experimental data.

Figure 4

Snapshots of fluid particles in the flow generated by two rotating helices in free space. (a)–(d) Side views and (e)–(h) top views at $t=0$, 20, 40, and 70. Geometries are the same as in the single helix case from Zhong et al. [17], with pitch angle $\beta ={45}^{\circ }$. The bases of the helices are located at $(-2\alpha ,0,0)$ and $(2\alpha ,0,0)$, the phase difference is $\pi$, and colors indicated height. (See Supplemental Material [28] for movies.)

Figure 5

Snapshots of fluid particles in the flow generated by two rotating helices tethered to a wall. (a)–(d) Side views and (e)–(h) top views at $t=0$, 20, 40, and 70. Geometries are the same as in the single helix case from Zhong et al. [17], with pitch angle $\beta ={45}^{\circ }$. The bases of the helices are located at $(-2\alpha ,0,0)$ and $(2\alpha ,0,0)$, the phase difference is $\pi$, and colors indicated height. The blue plane is the location of the no-slip wall at $z=0$. (See Supplemental Material [28] for movies.)

Figure 6

(a)–(c) Instantaneous velocity field, (d)–(f) transverse and (g)–(i) vertical velocities, and (j)–(l) vorticity for two helices tethered to a wall. The separation distance is $3\alpha$, they are rotating at 1 Hz, and they are either in phase (left column), out of phase by $\pi /2$ (middle column), or out of phase by $\pi$ (right column).

Figure 7

Normalized thrust generated by two helices (a), (b) in the presence of a wall and (c), (d) in free space. Thrust is shown as a function of (a), (c) phase difference and (b), (d) separation distance between the two helices.

Figure 8

Normalized power from two helices (a), (b) in the presence of a wall and (c), (d) in free space. Power is shown as a function of (a), (c) phase difference and (b), (d) separation distance between the two helices.

Figure 9

Normalized flux of fluid passing through a flow meter $[-5,5]\times [-5,5]$ at height $z=2.5$ for two pairs of helices at distances $d=0.25$ and $d=0.5$ when either emanating from a wall or rotating in free space as a function of phase difference. The normalized flux is one half of the mass of fluid pumped across the flow meter by the two helices during one rotation divided by the mass of fluid pumped across the flow meter by an identical isolated helix.

Figure 10

(a) and (b) show snapshots of particles seeded in a cylindrical region enclosing two identical helices at time $t=0$ when viewed from (a) the top and (b) the side. (c) and (d) show these same particles and the helices after 20 rotations. These helices are rotated in phase ($\varphi =0$) and are separated by a base distance of $d=3\alpha =0.255$. (e) and (f) show the particles colored by their initial mixing measure $\widehat{M}(p,0)$. (g) and (h) show these particles colored by their mixing measure $\widehat{M}(p,20)$ after 20 rotations. The mixing measure ranges from dark blue (no mixing) to yellow (optimal mixing).

Figure 11

Viewed from above, the first column shows the initial seeding of 5000 green and 5000 magenta particles, where each row depicts simulations with the same distance between helices $[d=3\alpha$ (top) and $6\alpha$ (bottom)]. The phase difference between the helices is varied across columns $[\varphi =0$ (second column), $\varphi =\pi /2$ (third column), and $\varphi =\pi$ (fourth column)]. The last three columns show the positions of the green and magenta particles after 20 rotations.

Figure 12

Here, the particles and helix configurations are the same as in Fig. 11, but the particles are colored by their mixing measures. Viewed from above, the first column shows the initial seeding of the 10 000 particles colored by their mixing measure $\widehat{M}(p,0)$, where each row depicts simulations with the same distance between helices $[d=3\alpha$ (top) and $6\alpha$ (bottom)]. The phase difference between the helices is varied across columns $[\varphi =0$ (second column), $\varphi =\pi /2$ (third column), and $\varphi =\pi$ (fourth column)]. The last three columns show the positions of the particles colored by their mixing measures $\widehat{M}(p,20)$ after 20 rotations.

Figure 13

Mixing as a function of number of helical rotations. Line style denotes helical spacing $d=3\alpha$ (–), $4\alpha$ ($--$), $5\alpha$ ($-·$), $6\alpha$ ($\cdots$), and color denotes phase difference ranging from in phase ($\varphi =0$ dark purple) to completely out of phase ($\varphi =\pi$ light purple).